Aspects of Classical and Quantum Dynamics of

Aspects of Classical and Quantum Dynamics of Canonical General Relativity

Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) in der Wissenschaftsdisziplin Physik.

¨ (Uberarbeitete Fassung)

Vorgelegt von Bianca Dittrich aus Berlin, angefertigt am Max–Planck–Institut f¨ ur Gravitationsphysik (Golm) und am Perimeter Institute for Theoretical Physics (Waterloo, Canada), eingereicht an der Mathematisch–Naturwissenschaftlichen Fakult¨ at der Universit¨ at Potsdam.

Potsdam im Juni 2005

Gutachter: Prof. Dr. Martin Wilkens (Universit¨ at Potsdam) Prof. Dr. Thomas Thiemann (Albert–Einstein Institut, Potsdam, und Perimeter Institute for Theoretical Physics, Waterloo) Dr. habil. Renate Loll (Institute for Theoretical Physics, Universiteit Utrecht)

Abstract: This thesis is concerned with the implications of diffeomorphism invariance for the canonical formulation of general relativity. This diffeomorphism invariance leads to gauge symmetries, which encode also the dynamics of the theory. This means that for the canonical quantization of general relativity one needs to construct gauge invariant observables and a Hilbert space of gauge invariant states. In the first part of this thesis we consider the method of partial and complete observables to construct gauge invariant observables, in the second part we test the Master Constraint Programme, which is a proposal for the construction of a Hilbert space of gauge invariant states for quantum general relativity.

Zusammenfassung: Diese Arbeit besch¨ aftigt sich mit den Implikationen der Diffeomorphismen–Invarianz f¨ ur die kanonische Formulierung der Allgemeinen Relativit¨ atstheorie. Diese Diffeomorphismen– Invarianz spiegelt sich in Eichsymmetrien wieder, die auch die dynamische Information des Systems enthalten. F¨ ur die kanonische Quantisierung der Allgemeinen Relativit¨ atstheorie bedeutet dies, dass man eichinvariante Observablen konstruieren muss sowie einen Hilbertraum eichinvarianter Zust¨ ande. Im ersten Teil dieser Arbeit betrachten wir die Methode der partiellen und vollst¨ andigen Observablen um eichinvariante Observablen zu konstruieren, im zweiten Teil testen wir das Master Constraint Programm. Dieses Programm ist ein Vorschlag zur Konstruktion eines Hilbertraums eichinvarianter Zust¨ ande f¨ ur die Quantentheorie der Allgemeinen Relativit¨ atstheorie.

Contents Introduction

I

Introduction to Gauge Systems

I.1 I.2

II

III

3

Parametrized Systems . . I.1.1 Dirac Observables I.1.2 Dirac Quantization Previous Works . . . . . .

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Complete Observables

II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2 Preliminaries and Notation . . . . . . . . . . . . . . . . . . . . . . . . . II.3 Complete and Partial Observables for Systems with One Constraint . . II.4 Complete and Partial Observables for Systems with Several Constraints II.5 A System of Partial Differential Equations for Complete Observables . . II.6 Weak Abelianization of the Constraint Algebra . . . . . . . . . . . . . . II.7 Bubble Time Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . II.8 Gauge Fixing and Dirac Brackets . . . . . . . . . . . . . . . . . . . . . . II.9 τ –Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.10 Partially Invariant Partial Observables . . . . . . . . . . . . . . . . . . . II.11 Complete Observables for General Relativity . . . . . . . . . . . . . . . II.11.1 Complete Observables for Field Theories . . . . . . . . . . . . . . II.11.2 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . II.11.3 Complete Observables Associated to Space–Time Scalars . . . . II.11.4 The Metric as a Space–Time Scalar . . . . . . . . . . . . . . . . II.11.5 Solving the Diffeomorphism Constraints . . . . . . . . . . . . . . II.12 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .

Testing the Master Constraint Programme

III.1 Introduction . . . . . . . . . . . . . . . . . . III.2 The Master Constraint Programme . . . . . III.2.1 Direct Integral Decomposition . . . III.2.2 Representation of Dirac Observables III.3 Finite Dimensional Examples . . . . . . . . 1

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6 9 13 17

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22 22 23 28 32 38 41 42 45 50 54 54 55 59 61 65 69

73

74 77 78 81 86

III.3.1 III.3.2 III.3.3 III.3.4

III.4

III.5

III.6 III.7

Finite Number of Abelean First Class Constraints Linear in the Momenta A Second Class System . . . . . . . . . . . . . . . . . . . . . . . . . . . . SU (2) Model with Compact Gauge Orbits . . . . . . . . . . . . . . . . . . A Model with Structure Functions rather than Structure Constants Linear in the Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.3.5 Another Model with Pure Point and Absolutely Continuous Spectra . . . SL(2, R) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.4.1 A Model with Gauge Group SO(2, 1) . . . . . . . . . . . . . . . . . . . . III.4.2 A Model with Two Hamiltonian Constraints and Non – Compact Gauge Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.5.1 Maxwell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.5.2 Linearized Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.7.1 Review of the Representation Theory of SL(2, R) and its various Covering Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.7.2 Completeness of the basis . . . . . . . . . . . . . . . . . . . . . . . . . . .

Discussion and Outlook

86 91 92 95 102 105 107 112 121 121 132 141 145 145 159 162

Bibliography

166

Danksagung

173

2

Introduction In modern physics our understanding of nature is based on two very different theories, namely quantum theory and general relativity. Whereas in general relativity space–time is dynamical and no reference frame is preferred the standard formulation of quantum theory requires a fixed background and a preferred splitting of space–time into space and time. One approach to reconcile these two theories is to apply the canonical quantization procedure to general relativity. In doing that one has to solve problems which appear because in general relativity there are no preferred reference frames, which implies that it is a system with gauge symmetries. For the description of a system with gauge symmetries one has to choose some kind of auxilary structures, e.g. in order to describe a particular space–time in general relativity we have to choose some reference frame. The main problem in the theory of gauge systems is to extract the gauge invariant information, i.e. information which does not depend on the specific choice of the auxilary structures. This is especially important if one wants to quantize a gauge system. A quantization of a gauge system should model the quantum fluctuations of the physical, i.e. gauge invariant degrees of freedom. But it would be unphysical to include fluctuations in the gauge degrees of freedom, since these do not decribe an inherent property of the physical system but rather our choice of gauge. General relativity is a particularly complicated gauge system. This is the reason why the construction of a quantum theory of general relativity is so difficult: Because of the particular type of gauge symmetries in general relativity gauge invariant phase space functions have to be constants of motions. This means that one has to solve at least partially the dynamics of general relativity in order to obtain gauge invariant quantities. This dynamics is described by a complicated system of highly non–linear partial differential equations and hence it is not surprising that there are almost no1 gauge invariant phase space functions known. However one has to look for a Hilbert space where one can represent the gauge invariant phase space functions by (self–adjoint) operators and such that the wave functions in this Hilbert space model fluctuations only in the gauge invariant directions. Hence a first task would be to gain more information on the set of gauge invariant functions and to develop methods how one could calculate or at least approximate these functions. But as we will see, it is also possible to construct a Hilbert space with gauge invariant wave functions without having available a complete set of gauge invariant classical functions. Gauge invariant quantum observables would then be given by the self–adjoint operators on this Hilbert space. Nevertheless for an interpretation of the quantum theory one needs operators that correspond to classical gauge invariant phase space functions. Also, the construction of such a Hilbert space is facilitated very much if one already knows at least some of the gauge invariant classical 1

For the case of gravity in four space–time dimensions and for asymptotically flat boundary conditions there are 10 gauge invariant phase space functions known. These are the ADM charges [1] given by the generators of the Poincare transformations at spatial infinty. Additionally an observable is known, which takes only a few discrete values and is trivial on almost all points in phase space [2]. For gravity coupled to matter, in some cases gauge invariant functions describing matter are known but in general no phase space functions which describe the gravitational degrees of freedom (with the exception of the ADM charges). Yet there are infinitely many gauge invariant degrees of freedom.

3

functions. This thesis is therefore divided into several parts. In Part II we will develop methods how to extract the gauge invariant information on the classical level. First we will consider general gauge systems but at the end we will concentrate on general relativity. In Part III we will switch to the quantum theory and discuss a certain proposal, namely the so called Master Constraint Programme [3], for the construction of a Hilbert space with gauge invariant wave functions. This Master Constraint Programme is motivated very much by the current situation of Loop Quantum Gravity, which is a very promising framework for the canonical quantization of general relativity, see [4, 5, 6] for recent reviews. Until recently the Master Constraint Programme has just been a proposal and had not even been tested on simpler systems. We will do exactely that in Part III of the thesis. The thesis ends with a discussion of the results obtained and an outlook. But to begin with we will give an introduction to the theory of gauge systems in Part I. We will do that by means of an example, namely a so called parametrized system. Thereby we will explain the problems we will be concerned with in Part II and Part III in more detail.

4

Part I

Introduction to Gauge Systems

5

I.1

Parametrized Systems

This section serves as an introduction to the subject of gauge systems. Such systems are also called first class constrained systems for reasons that will be explained lateron. To begin with we will consider just a very simple subclass of constrained systems, namely those which arise from parametrizing a system of classical mechanics. 2 Such systems are called parametrized, because their time evolution is not described as an evolution in a physical time parameter but in an evolution parameter, which does not have a priori a physical interpretation and moreover can be chosen arbitrarily, as will be clarified later. The reason why we are considering these particular examples of constrained systems is that also general relativity is a parametrized system. This means that one describes the evolution of the system with respect to an a priori unphysical evolution parameter, namely coordinate time. Therefore one can understand many important features of general relativity by considering examples that arise from parametrizing a classical mechanics system. However we will also mention problems which arise in more complicated systems. Usually one describes the evolution of a classical mechanics system with respect to a time parameter t, which is assumed to be given from the outset. This time parameter t – the Newtonian absolute time – is non–dynamical, that is, it is not influenced by the state of the physical system. In particular the time t is not included into the configuration space variables q = (q 1 , . . . , q n ) of the system. The evolution q(t) from a configuration q1 at time t1 to a configuration q2 at time t2 is that path which extremizes the action ! t2 S= L(t; q 1 , . . . , q n , q˙1 , . . . , q˙n )dt . (I.1.1) t1

Here L is the Lagrange function of the system and we will allow for an explicit time dependence of this Lagrange function. A dot atop a variable denotes differentiation with respect to time t. One has to vary over all paths in configuration space with fixed endpoints q1 (t1 ) and q2 (t2 ). Now, following [8], we want to treat also the time t as a configuration variable. Hence t cannot be taken as the evolution parameter anymore and we have to choose another evolution parameter s such that dynamics is now described through parametrized orbits (t(s), q(s)) in the extended configuration space (t, q). These orbits should be such that they coincide with the graph of q(t), i.e. the solutions of (I.1.1), over t. In other words, if one inverts t(s) for s, und uses this to express q as a function of t, then q(s(t)) should coincide with the function q(t). Here one can already see that the dynamics displays a certain gauge symmetry, namely reparametrization invariance: If (t(s), q(s)) satisfies the above requirement, then the same holds ˜ (˜ for the reparametrized path (t˜(˜ s), q s)) := (t(f −1 (˜ s)), q(f −1 (˜ s))) where s˜ is a new parametriza˜ (˜ tion defined by s˜ = f (s). But two paths (t(s), q(s)) and (t˜(˜ s), q s)) which just differ by a reparametrization s˜ = f (s) are physically equivalent, since the parameter s was introduced as an unphysical parameter, which cannot be measured in any way. The physical information is represented by the orbit in the extended configuration space (t, q), not by its particular parametrization. Analogously in general relativity two solutions to the Einstein equations are equivalent, if they just differ by a diffeomorphism. Also here, the time and spatial coordinates, in which the solutions are expressed are auxilary structures, which do not have an a priori physical meaning. 2

For a general introduction to constrained systems see [7].

6

We now seek for an action Sp , defined in the extended configuration space (t, q), the extrema of which should give paths (t(s), q(s)) which should satisfy the requirement mentioned above. Such an action is ! s2 " q1 q n" Sp = L(t; q 1 , . . . , q n , " , . . . , " ) t" ds . (I.1.2) t t s1

Here a prime denotes differentiation with respect to s and one varies over all paths, with fixed endpoints (t1 (s1 ), q1 (s1 )) and (t2 (s2 ), q2 (s2 )). That the action (I.1.2) gives the same physical solutions as (I.1.1) can be seen by checking that Sp gives the same values if evaluated on paths (t(s), q(s)) as the action S on the path q(s(t)) Moreover the action Sp is reparametrization invariant, i.e. it gives the same values on paths that just differ by a reparametrization. Analogously the Einstein–Hilbert action for general relativity is invariant under diffeomorphisms. In order to perform a canonical quantization of the system, we have to change to the Hamiltonian picture. Therefore we perform a Legendre transformation for the Lagrange function Lp := L t" of the action Sp . That is we go over to the (extended) phase space (t, q, pt , p) of the system, whereby the momenta are related to the velocities (the s-derivatives of the configuration variables) by "# n

pt =

∂Lp =L− ∂t"

pi =

∂Lp ∂L = i i " ∂(q ) ∂ q˙

i=1

$ n # ∂L (q i )" " t = −( pi q˙i − L) = −h ∂ q˙i (t" )2 i=1

(I.1.3)

,

where h is the Hamiltonian function for the Lagrange function L. In order to find the Hamiltonian Hp conjugated to the Lagrange function Lp one has to invert the equations (I.1.3), that is to find the velocities (t" , q" ) as functions of the phase space variables. However this is not possible, since on the right hand sides of the equations (I.1.3) the velocities appear only in the scale invariant combination (q " )i /t" = q˙i . The reason for this is that in this case the Legendre transformation is neither a surjective nor an injective map from the tangent space of the (extended) configuration space to the (extended) phase space. Rather the Legendre transformation maps the whole tangent space onto a certain subspace of the phase space. This subspace is described by the first line in (I.1.3) which can be seen as a relation which has to hold for the momenta pt and p: pt = −h(t, q, p)

(I.1.4)

.

In other words we have the constraint C = pt + h(t, q, p)

,

(I.1.5)

the vanishing of which defines the so called constraint hypersurface. Only for phase space points on this constraint hypersurface one can relate the momenta to the velocities (t" , q" ). However this relation is not one–to–one, since if one rescales all velocities by an arbitrary non–vanishing factor, they will correspond to the same momenta as before the rescaling. Since there is no relation between phase space points off the constraint hypersurface to the points in the tangent space to the configuration space, points off the constraint hypersurface do not have any physical meaning. 7

Let us try to find the Hamiltonian Hp despite of the fact, that we cannot invert the Legendre transformation. If we apply the usual definition for the Hamiltonian we will get H p = p t t" +

n #

i=1 n #

% = t" p t + "

pi (q i )" − Lt"

i=1

= t (pt + h)

pi

& (q i )" −L " t

(I.1.6)

i !

where in the last line we used that (qt!) = q˙i . So the Hamiltonian Hp actually vanishes on the constraint hypersurface! However this does not mean that one cannot write down meaningful Hamiltonian equations, in the form f " = {f, Hp } . (I.1.7)

Here {·, ·} are the Poisson brackets for the extended phase space. But in order to have such equations one needs to express the Hamiltonian entirely as a function of the canonical coordinates and this is prevented by the factor t" which cannot be expressed by canonical coordinates. We will now argue that one can replace the factor t" by an arbitrary non-vanishing factor, which may even depend on s and on the extended phase space variables. The first point is, that as was mentioned above, one can rescale the velocities (t" , q" ) by an arbitrary factor without changing the canonical momenta (pt , p). According to the last line in (I.1.6) under such a rescaling Hp is also changed by multiplication with the arbitrary factor. Hence one can indeed multiply the Hamiltonian with an arbitrary factor, which we will call the lapse function N . Despite this, the value of the Hamiltonian is uniquely determined on the constraint hypersurface, namely there it is zero. Only off the constraint hypersurface the Hamiltonian is not uniquely defined. The second point is that it does not matter, if this lapse function is phase space dependent since in the equations of motion (I.1.7) the term with the derivative of the lapse function vanishes on the constraint hypersurface: f " = {f, Hp } = N {f, C} + {f, N }C " N {f, C}

(I.1.8)

where we introduced the symbol " to denote an equation which does only hold on the constraint hypersurface, that is modulo the constraint. Such equations are called weak equations. We can conclude that dynamics is described by the Hamiltonian Hp = N C where N is an arbitrary non-vanishing phase space function. The equations of motion given by f " = {f, Hp }

C=0

(I.1.9)

are consistent in the sense that a phase space point on the constraint hypersurface remains under the flow of the Hamiltonian on the constraint hypersurface. This follows from C " = {C, Hp } " 0 .

(I.1.10)

Now, because the lapse N is arbitrary, we are in the situation that one set of initial data in in (tin , qin , pin t , p ) for some initial evolution parameter s , which satisfies the constraint C = 0, out out out out can lead to different outcomes (t , q , pt , p ) at a ‘later’ evolution parameter sout . Hence the system seems to be non–deterministic. 8

Of course we know that different choices for the lapse function should correspond to different choices of parametrizations for the evolution orbit in the extended phase space, that is different gauges. Therefore one adopts the point of view that the different outcomes obtained by choosing differing lapses are gauge connected and should all be identified to one physical state. That is, physical states are not represented by points (on the constraint hypersurface) but by gauge orbits. The difference between two infinitesimal evolutions of a quantity f with two different lapse functions N(2) and N(1) is given by f(2) (sin + ε) − f(1) (sin + ε) " ε(N(2) − N(1) ){f, C} .

(I.1.11)

Since f(2) and f(1) should be gauge connected, this means that the constraint C generates not only the evolution but also gauge transformations! Hence the gauge orbits coincide with the dynamical orbits obtained by evolving phase space points. Since one gauge orbit represents one physical state, this seems to lead to the conclusion that physical states do not evolve at all. However, one must not forget that the extended phase space includes the physical time t and its conjugated momentum pt . For instance, phase space points which coincide in all coordinates except for the time coordinate, will in general represent different physical states and this can be used to regain a notion of evolution in a certain sense, as will be shown later. Moreover we arrived now at the same number of physical degrees of freedom as for the unparametrized phase space, which has dimension 2n. The extended phase space has two dimensions more, since it also includes the time t and the conjugated momentum pt . One degree of freedom is lost by implementing the constraint C = 0, another by identifying the one–dimensional gauge orbits to physical states. The phenomena which we described here, will occur for all systems with gauge symmetries. A gauge transformation can map an extremum of the action, describing the system, to another extremum with the same boundary conditions, that is the same endpoints for the paths in configuration space, over which one varies the action. One can show [9] that this leads always to a singular Legendre transformation and hence to a certain number N of constraints. The number of constraints reflects the number of gauge degrees of freedom, hence there may be also infinitely many constraints as is the case in field theories. In this work we will only consider systems, the constraints of which arise from gauge symmetries. Such systems are called first class constrained systems. For such systems, the Poisson bracket between two constraints Ci , Cj can always be written as a linear combination of constraints: {Ci , Cj } = fijk Ck

(I.1.12)

where the fijk are called structure constants if they are constant on phase space and structure functions otherwise. Because of relation (I.1.12) the orbits generated by the constraints lie inside the constraint hypersurface. These orbits will be interpreted as gauge orbits.

I.1.1

Dirac Observables

We have seen that in the extended phase space for a parametrized system physical states are not represented by phase space points but by gauge orbits generated by the constraint C. This means that not all phase space functions can be taken as physical observables: The value of an 9

arbitrary phase space function could differ on two phase space points on the same gauge orbit, that is on two phase space points that represent the same physical state. Such functions are hence gauge dependent. Physical observables should be gauge independent and therefore be represented by phase space functions which are constant along the gauge orbits. Since the constraint C generates the gauge orbit, a physical observable F has to fulfill {F, C} " 0

,

(I.1.13)

i.e., its Poisson bracket with the constraint has to vanish at least on the constraint surface. Phase space functions which satisfy this conditions are also called Dirac observables. In more general first class constrained systems a Dirac observable is a phase space function which has vanishing Poisson brackets with all constraints. As we will see, Dirac observables are especially important for the quantization of constrained systems. The intuitive reason for this is that Dirac observables do not carry any gauge information, hence do not include fluctuations in the unphysical gauge direction. Since in a parametrized system the Hamiltonian is a multiple of the constraint, we are in the situation that physical observables do not evolve (in parameter time s), i.e. are constants of motion. One might therefore think, that there can be almost never enough Dirac observables: As already established there are 2n physical degrees of freedom, hence there should be 2n independent Dirac observables. However, even if the unparametrized system would be completely integrable, there would be in general just n independent constants of motion (not counting the constraint, which is also a constant of motion), namely the n action variables. Moreover the question arises how one should describe the dynamics of the system if one can only use constants of motion. The first objection arises actually only, if one asks for Dirac observables, which are globally defined. Locally, a gauge orbit generated by one constraint is one-dimensional and hence it should3 be possible to describe it as the intersection of the level sets of 2n+1 linearly independent functions. (Remember that the extended phase space has 2n + 2 dimensions.) One of these functions is the constraint, therefore we are left with 2n functions which are constant along the gauge orbits. However in general these functions may be well–defined only locally. From a global point of view a one-dimensional orbit may densely fill out a higher dimensional surface. Whether this will happen as well as the dimension of this surface depends on the global properties of the dynamics of the system. But how can one find Dirac observables at least locally? We will here adopt a method suggested in [11] which serves simultaneously as a reply to the second objection. To this end one has to realize, what kind of quantities are “constants of motions” in principle. Remember that the gauge orbitsare one-dimensional. Hence we can parametrize a gauge orbit with the values of a phase space function T , if this phase space function changes strictly monotonously along the orbit. I.e. there should be only one point on each gauge orbit where the function T assumes a given value τ . For a parametrized system, it is natural but not always necessary to choose as the phase space function T the time variable t. Because of the pt term in the constraint (I.1.5) it satisfies the above condition. Since the function T should satisfy the conditions for a clock (namely be strictly monotonous along the evolution orbits), we will call it a clock variable. 3

A proof that locally there are always enough Dirac observables can be found in [10].

10

Now given another phase space function f and a phase space point x one can calculate the value the function f will assume at that point of the gauge orbit through x at which T assumes the value τ . In other words, we start at a phase space point x, for which the value of T is in general not equal to τ , evolve this phase space point (that is calculate the orbit through x) and predict the value of f for that moment for which the clock T shows the value τ . We will call this predicted value F[f ;T ] (τ, x). Since it depends on the phase space point x, we have actually constructed a phase space function F[f ;T ] (τ, ·). Following [11] we will call it the complete observable associated to f and T . The functions f and T are called partial observables. The complete observable is constant along the gauge orbits, hence a Dirac observable: If we would have started at another phase space point y on the same gauge orbit, we would still evaluate f at the same point on that gauge orbit as before, namely at that point, at which T assumes the value τ . Hence the phase space function F[f ;T ] (τ, ·) is a constant of motion. That is, to every evolution orbit it assigns one value, namely the prediction of f for that moment at which T is equal to τ . But this value will change if we change τ , i.e. predict f for another value of the clock variable T . In this sense we can ‘evolve’ F[f ;T ] (τ, ·) through all values of the clock variable T . This is the reason why complete observables were also called evolving constants of motion [11]. Let us consider a simple example, namely the parametrized free (non-relativistic) particle in one dimension. The constraint for this system is C = pt + h = pt +

p2 2m

(I.1.14)

where p is the conjugated momentum to the position variable q and m the mass of the particle. The orbit generated by C through a phase space point x = (t, q, pt , p) can be described in parametric form by t(s) := q(s) :=

∞ # 1 k s {t, C}k = t + s k!

k=0 ∞ # k=0

1 k p s {q, C}k = q + s k! m

(I.1.15)

where s is the (unphysical) parameter and {·, C}k are the iterated Poisson brackets defined by {·, C}k+1 = {{·, C}k , C} and {·, C}0 = id. The momenta pt and p are constant along the orbits. We will choose as clock variable T the time t. In order to find the value of the complete observable F[q;t] (τ, x) associated to the position variable q at the phase space point x, we have to find first the parameter s for which t assumes the value τ . This parameter is given by s = τ − t and if we insert it into the second equation of (I.1.15) we will get for the complete observable F[q;t] (τ, x) = q +

p (τ − t) . m

(I.1.16)

It has indeed vanishing Poisson brackets with the constraint (I.1.14). The complete observables associated to the canonical coordinates t, q, pt , p are given by F[t;t] (τ, x) = τ F[q;t] (τ, x) = q +

F[pt ;t] (τ, x) = pt " −

p (τ − t) m

F[p;t] (τ, x) = p . 11

p2 2m (I.1.17)

Hence these four complete observables give just two independent Dirac observables (apart from the constraint), for instance F[q;t] (τ, ·) and F[p;t] (τ, ·). These correspond to the canonical coordinates (q, p) of the non-extended phase space, they are even canonically conjugated for all values of τ : {F[q;t] (τ, ·), F[p;t] (τ, ·)} = 1 . (I.1.18)

Because the Poisson brackets are independent of τ (i.e. changing τ is a symplectic transformation) one may ask for a function, which generates the change in τ . Indeed with Htτ := F[−pt ;t] (τ, ·) " h =

p2 2m

(I.1.19)

we have {F[q;t] (τ, ·), Htτ } =

d F (τ, ·) dτ [q;t]

{F[p;t] (τ, ·), Htτ } =

d F (τ, x) dτ [p;t]

.

(I.1.20)

2

p Here one can take either −pt or h = 2m as Htτ . They differ by a term proportional to the constraint, but this does not matter, since the constraint Poisson commutes with the complete observables. We will call it the physical Hamiltonian associated to the clock variable T = t. It is a Dirac observable and in contrast to the Hamiltonian (constraint) H = N C it does not vanish on the constraint hypersurface. Indeed it gives the energy associated to the choice of t as the physical time. Of course it cannot generate the change in Ftτ , since the latter is just given by the constant (on phase space) τ . But this is similar to the status of time in ordinary classical mechanics: There t is treated as a constant on phase space and the Hamiltonian does not generate the ‘time change’ for the variable t. In summary we have found two complete observables F[q;t] (τ, ·), F[p;t] (τ, ·), which represent the two physical degrees of freedom of the parametrized system. The evolution in physical time t can now be described by the change in the parameter τ : Given a phase space point x on the constraint hypersurface the values F[q;t] (τ, x), F[p;t] (τ, x) give the values which the phase space functions q, p will assume if the phase space function t assumes the value τ . Changing τ we see that a phase space point can indeed represent the whole evolution orbit. Moreover the change in τ is generated by the physical Hamiltonian Htτ , which coincides with the Hamiltonian h for the non-parametrized system.4 As already mentioned the canonical formulation of general relativity has infinitely many constraints. In Part II of this work we will therefore generalize the concept of complete observables to systems with an arbitrary number of constraints. For general relativity complete observables can serve for several purposes: Firstly, this concept can help to find methods in order to compute Dirac observables. There are almost no Dirac observables known for general relativity and hence it is important to know, how one could find Dirac observables at least in principle. From this, one might be able to develop approximation methods for Dirac observables. Moreover, complete observables can serve as a tool for the interpretation of the (quantized) theory. As for the parametrized system, for general relativity a physical time is not given from the outset, but has to be identified from the phase space variables. This means that general 4 However we have to keep in mind that the physical Hamiltonian Htτ depends on the choice of the phase space function t as the clock variable T and as we will see in section II.9 on the choice of the momentum conjugated to the clock variable, i.e. one has to fix a phase space function ΠT with {T, ΠT } = 1. Here we used ΠT = pt . A physical Hamiltonian associated to another choice may have completely different properties.

12

relativity is also a totally constrained system, i.e. the Hamiltonian is a linear combination of the constraints. Since Dirac observables have to commute with all the constraints, they have also to be constants of motion. This has led to the titling “frozen formalism” for the canonical formulation of general relativity [12]. However in a similar fashion as for the parametrized particle one can identify clock variables and describe the evolution of the system with respect to these clock variables. This would also allow to make statements which are localized in space and time: As we will see in section II.11.3 clock variables will serve as a kind of physical coordinate system, and one can now make predictions for space time regions specified by these physical coordinates.

I.1.2

Dirac Quantization

In this section we want to consider the quantization of a parametrized system and compare it to the standard quantum theory of the unparametrized theory. There are two main methods for the canonical quantization of a gauge system: Reduced Phase Space Quantization and Dirac Quantization. In the Reduced Phase Space Quantization one only attempts to quantize the gauge invariant functions, that is the Dirac observables. Hence one has to find a complete set of gauge invariant functions and furthermore an irreducible representation for them on a Hilbert space. But in practice it is often extremely difficult to find a complete set of Dirac observables.5 An alternative to Reduced Phase Space Quantization is Dirac Quantization. Here one tries to quantize all variables, even the gauge ones. That is, one looks for a representation of a complete set of phase space functions6 on some linear space. In most approaches this linear space is also a Hilbert space and is called the kinematical Hilbert space. The constraints are now linear operators on this Hilbert space. A general state in the kinematical Hilbert space carries also unphysical gauge information, or in other words is not being left invariant under a gauge transformation generated by the quantized constraints. Also, observables which differ by a term at least linear in the constraints, give different results if applied to a general state. Yet such observables are equivalent and should therefore give the same result. Hence one requires from a physical state, that it should be annihilated by the constraints. Let us consider this condition for a parametrized particle with the constraint (I.1.14). As the kinematical Hilbert space for this system we choose the space of square integrable functions over the configuration variables: Hkin = L2 (R2 , dtdq). Kinematical states are therefore square integrable functions of t and q, the configuration operators tˆ, qˆ act by multiplication and the momentum operators pˆt , pˆ by differentiation: tˆψ(t, q) = tψ(t, q) ∂ pˆt ψ(t, q) = −i! ψ(t, q) ∂t

qˆ ψ = qψ(t, q) ∂ pˆ ψ = −i! ψ(t, q) ∂q

5

.

(I.1.21)

If there exist a perfect gauge fixing, i.e. the gauge fixing conditions should specify on each gauge orbit one and only one point, it is possible to represent the gauge invariant functions by their gauge fixed counterparts and then to quantize the gauge fixed theory. But such a perfect gauge fixing is usually very difficult to find and may even not exist at all. 6 A complete set of phase space functions is a set such that for an arbitrary pair of phase space points m1 != m2 there exists a function f such that f (m1 ) != f (m2 ).

13

The quantization of the constraint (I.1.14) is straightforward. The condition for a state to be physical is 2 2 ˆ = −i! ∂ ψ − ! ∂ ψ = 0 , Cψ (I.1.22) ∂t 2m ∂q 2 i.e. it is given by the Schr¨ odinger equation for the unparametrized particle. If one quantizes the unparametrized particle in the standard Schr¨ odinger representation, states are given as wave functions of just q whereas t is an external parameter. In particular the inner product between two wave functions is given by the integration over q. In our kinematical Hilbert space we integrate over q and t. Indeed we have the problem, that there are no solutions to the constraint equation (I.1.22), which are normalizable with respect to the inner product given in the kinematical Hilbert space Hkin . This is a general problem: one cannot expect to have normalizable gauge invariant wave functions (unless the gauge directions are compact) if the inner product includes integration over the gauge degrees of freedom. Hence one has to choose a physical inner product for the space of solutions to the constraint equations. The solution space with this physical inner product can then be completed to a physical Hilbert space Hphys . Still, there may be many solutions to the constraint equations which are also not normalizable with respect to the physical inner product, hence are not proper physical states. On the other hand there may be solutions which will have zero norm with respect to the physical inner product, these will also not appear in the physical Hilbert space. Indeed the choice of a physical inner product is a crucial step in the Dirac quantization process, which influences very much the “size” of the physical Hilbert space. In order to construct a physical inner product for the parametrized particle consider the resolution of a kinematical state ψ ∈ Hkin into the generalized eigenfunctions of the constraint operator (I.1.22): ! ! ˜ E) e−iEt/! eikq ψ = dk dE ψ(k, R R ! ! ! !2 2 ˜ E) e−iEt/! eikq = dλ dk dE δ(λ − ( k − E)) ψ(k, 2m R R R ! ! ˜ E(k) − λ) e−i(E(k)−λ)t/!eikq = dλ dk ψ(k, R R ! =: dλ Ψ(λ) (I.1.23) R

! where E(k) = 2m k2 . The Ψ(λ) are generalized eigenfunctions of Cˆ with generalized eigenvalue λ. ' A solution ψphys to the constraint equation (I.1.22) is of the form ψphys = dλδ(λ)Ψ(λ) = Ψ(0) and the delta–funtion in the integral representation leads to a divergence if one computes the norm in the kinematical Hilbert space. To get rid of this divergence notice that we can write the inner product between two kinematical states ψ, ψ " as ! < ψ, ψ " >kin = dλ < Ψ(λ), Ψ" (λ) >λ (I.1.24) 2

R

where < ·, · >λ defines an inner product on the space of generalized eigenfunctions to the 14

eigenvalue λ: "

< Ψ(λ), Ψ (λ) >λ = 2π! =

√

!

˜ E(k) − λ)ψ˜" (k, E(k) − λ) dk ψ(k,

R!

2π!

dq Ψ(λ)Ψ(λ)

(I.1.25)

.

R

Since solutions to (I.1.22) are generalized eigenfunctions to the eigenvalue λ = 0, i.e. ψphys = Ψ(0) we can define an inner product on the space of solutions by using (I.1.25) with λ = 0: ! " " < ψphys , ψphys >phys = dq ψphys (tf ix , q)ψphys (tf ix , q) . (I.1.26) R

Here tf ix is an arbitrary fixed value for t. The inner product (I.1.26) between two solutions of the constraint equation (I.1.22) is independent from the choice of tf ix . The physical Hilbert space Hphys is given by the completion of the space of solutions with respect to the physical inner product. Note that the physical inner product (I.1.26) coincides with the inner product in the standard Schr¨ odinger quantization. That (I.1.26) is independent from the choice of tf ix is just the statement that time evolution is unitary. Indeed we can construct a unitary map U between the Hilbert space L2 (R, dq) of the Schr¨ odinger quantization and the physical Hilbert space Hphys . This unitary map is given by: U : L2 (R, dq) → Hphys

iˆ ψ(q) &→ ψphys (t, q) := exp(− ht)ψ(q) !

,

(I.1.27)

ˆ = pˆ is the Hamiltonian for the unparametrized particle. where h 2m Let us turn to the observables in the Dirac quantization and compare them to the ones in the quantization for the unparametrized particle. On the kinematical Hilbert space we defined the action of all the elementary phase space functions t, q, pt , p by (I.1.21). But not all of these operators preserve the solution space to the constraint equation (I.1.22). The condition that an operator Fˆ preserves the solution space is that it commutes with the constraint Cˆ at least on the solution space. That is exactly the “quantized” condition for a Dirac observable! So formally 7 Dirac observables can be promoted into operators on the physical Hilbert space and hence into physical quantum observables. In fact only the Dirac observables are physical observables, since they do not include any gauge information and hence do not lead to fluctuations in gauge directions. Since the gauge directions are conjugated to the constraints (remember that gauge transformations are generated by the constraints) and physical states have a sharply vanishing expectation value for the constraints, these fluctuations would otherwise tend to infinity due to the Heisenberg uncertainty relation. For our parametrized particle we therefore have to choose a complete set of Dirac observables, for instance the complete observables Fqτ , Fpτ associated to the clock variable t for some fixed 2

7 It could happen, that the commutator between constraints and Dirac observables obtains quantum corrections and hence does not vanish. So one has the technical condition to quantize the Dirac observables in such a way that they preserve the physical Hilbert space. This might not always be possible, one might try to alter the physical inner product or the quantization of the constraints then.

15

value of τ . Their quantizations pˆ Fˆ[q;t] (τ ) = qˆ + (τ − tˆ) , m

Fˆ[p;t] (τ ) = pˆ

(I.1.28)

commute with the constraint (I.1.22) and by choosing appropriate domains they can be defined as self–adjoint operators in Hphys . What is the relation between these observables and the observables qˆ(S) and pˆ(S) in the standard Schr¨ odinger quantization? Here the subindex (S) distinguishes the operators qˆ(S) , pˆ(S) in the standard Schr¨ odinger quantization, that is on L2 (R, dq) from the operators qˆ, pˆ on Hkin . If we apply the map U from (I.1.27) to the observables qˆ(S) , pˆ(S) , we will get U q(S) U −1 = Fˆ[q;t] (τ = 0)

U pˆ(S) U −1 = Fˆ[q;t] (τ = 0) .

,

(I.1.29)

Since U identifies the wave function ψ(q) as the one at time t = 0 (see (I.1.27)) the observables q(S) and Fqτ =0 carry indeed he same meaning: namely to give the position at that moment at which t = 0. The same holds for the (time-independent) momentum operator. As the states in L2 (R, dq) represent wave functions at a fixed time t = 0, they are Heisenberg picture states. To describe dynamics we have to make the observables qˆ(S) and pˆ(S) time– dependent. The Heisenberg equation of motions are given by d 1 ˆ , qˆ (τ ) = [ˆ q , h] dτ (S) i! (S)

d 1 ˆ pˆ (τ ) = [ˆ p , h] dτ (S) i! (S)

(I.1.30)

where we called – in anticipation of the result below – the time evolution parameter τ . The solutions to these equations can be calculated to qˆ(S) (τ ) = qˆ(S) +

pˆ(S) τ m

pˆ(S) (τ ) = pˆ(S)

,

(I.1.31)

.

Now, if we map these Heisenberg picture observables onto the physical Hilbert space we will get U q(S) (τ ) U −1 = Fˆ[q;t] (τ ) ,

U pˆ(S) (τ ) U −1 = Fˆ[p;t] (τ )

.

(I.1.32)

Indeed both q(S) (τ ) and Fˆ[q;t] (τ ) should give the position of the particle at the moment at which t = τ. Hence we have found that the standard quantization of the unparametrized free particle and the Dirac quantization of the parametrized particle give completely equivalent type of predictions and results. Firstly the dynamics in the Dirac quantization seemed to be “frozen”, but this just appeared so with respect to the unphysical evolution parameter. By choosing an internal time t we reproduced all the results of the standard quantization, and in particular the dynamics with respect to the physical time t: In the standard Schr¨ odinger quantization the time t is treated as a parameter, which is not quantized. In the Dirac quantization for the parametrized particle we quantized the phase space variable t (on the kinematical Hilbert space) but now the parameter τ remains classical. Both quantizations have a classical time parameter. This ends our discussion of the Dirac quantization of the parametrized particle. For the Dirac quantization of general relativity one of the main open problems is the construction of the physical Hilbert space. Our construction of the physical inner product (I.1.26) was based on a spectral analysis of the constraint operator. For general relativity we will have 16

infinitely many constraint operators with complicated commutations relations. Hence it is not possible to find (generalized) eigenfunctions which simultaneously diagonalize all the constraint operators. We will discuss in Part III a method, proposed in [3], where one replaces all constraint operators by just one constraint operator. This constraint operator is called the Master Constraint Operator. The infinitely many constraint equations which have to be satisfied by physical states are replaced by one Master Constraint Equation. With the introduction of the Master Constraint Operator it is now possible to base the construction of the physical Hilbert space on the spectral analysis of this operator. Indeed we will see that a spectral analysis will ensure that the physical wave functions are not only solutions to the Master Constraint equation but also to the original constraint equations. The availability of spectral analysis for the construction of the physical Hilbert space was not the only motivation for the introduction of the Master Constraint. In fact as we will explain in section III.1, it is tailored to the current situation in Loop Quantum Gravity. However, applying the Master Constraint to Loop Quantum Gravity will be very complicated and take a huge amount of work. But so far it has not even been tested on simpler examples. The purpose of Part III of this thesis is to test it on a broad selection of examples. These include a system with an Abelian constaint algebra, a constraint algebra with structure functions, constraint algebras which generate compact and non–compact Lie groups and free field theories with constraints. Each example will highlight a different aspect of the Master Constraint Programme. As we will see the application of the Master Constraint Programme to some of the examples is technically involved, since it requires a complete spectral analysis of the Master Constraint Operator. We will give an introduction to the necessary mathematical details at the beginning of Part III. There we will also discuss the representation of Dirac observables on the physical Hilbert space.

I.2

Previous Works

Both problems we will consider in this thesis, i.e. the construction of Dirac observables and the construction of a physical Hilbert space, are rather old ones. We can therefore mention only a few works out of the huge amount of literature. One of the first who developed a systematic approach to quantize gauge systems was Dirac [13]. He suggested that physical wave functions have to be solutions to the constraints and that a physical Hilbert space has somehow to be formed out of these solutions. On this physical Hilbert space one has to quantize the Dirac observables, which access the gauge–independent information of the system. In the following it turned out that this Dirac Quantization Programme needs many refinements. E.g. the following questions arise: Do the constraints have to be quantized on a (kinematical) Hilbert space or rather implemented as operators on some vector space? Should one then require some properties of this implementation of the constraints? What is the significance of this kinematical Hilbert space or vector space for the resulting physical Hilbert space? Which kind of solutions should one consider? If one has chosen to work with a kinematical Hilbert space, then in general there are no solutions to the constraints which are elements 17

of this kinematical Hilbert space. So one has to look for solutions in some bigger space. A priori there are many choices for this bigger space. But one has to keep in mind that one has to implement an inner product on the solution space and that one then has to ensure a self–adjoint representation of the Dirac observables. How should one construct a physical inner product? This construction should be such, that is ensures the self–adjoint representation of the Dirac observables. Additionally it is by no means trivial to find solutions to all the constraints or even to gain control over the ‘space of solutions’. These considerations presuppose that an (anomaly–free) quantization of the constraint algebra is possible, which usually is an already very difficult problem. A more specific suggestion how one could perform a Dirac quantization of a constrained system was made in [14]. This approach is called Algebraic Quantization. There one implements the constraints as operators on some auxilary vector space. This vector space has to carry a representation of the kinematical observables (i.e. phase space functions, which are not Dirac observables). From the physical Hilbert space it is required, that it should carry a self–adjoint representation of a complete set of Dirac observables. Hence one needs to know a complete set of Dirac observables and one needs also to have some control on the representation theory of the algebra of these Dirac observables. However in such complicated theories as general relativity this is difficult to obtain. In Refined Algebraic Quantization [15, 16, 17] one works with a kinematical Hilbert space Hkin , on which one has to quantize the constraints. Then one has to choose a dense subspace D, which has to serve as a common dense domain for the constraint operators. One has to find all solutions to the constraints, which are elements of the algebraic dual D ∗ of D. This space will be ‘bigger’ than the kinematical Hilbert space Hkin . These solutions can be found, if there is a Rigging map, which maps an arbitrary element in D to a solution of the constraints in D ∗ (and satisfy some technical requirements, see the references). With the help of such a Rigging map one can define a physical inner product and hence a physical Hilbert space. Also one has then given a representation of the Dirac observables on this physical Hilbert space. The main problem then is the construction of the Rigging map. A heuristic idea how to obtain a Rigging map is via group averaging. For this the constraints have to generate a Lie group, which ideally should be compact. One can then average a wavefunction over this gauge group and in this manner obtain a gauge–independent wave function. For a non–compact Lie group it is non–trivial to find a measure for the averaging over the gauge group such that the group averaging procedure converges on the subspace D, see [18]. It is not clear how the resulting physical Hilbert space depends on the choice of D, as is remarked in [18]. For constraint algebras with structure functions group averaging does not work since we cannot exponentiate the constraint algebra any longer. Also if one has just one constraint with a mixed spectrum (e.g. with an absolutely continuous part and a pure point part), it is difficult to define an averaging procedure which leads to meaningful results. The construction of the physical inner product in the Master Constraint Programme is based on the spectral analysis of the Master Constraint Operator. For systems with one constraint a similar suggestion was made in [19]. However there one assumes that the spectrum of the 18

constraint operator is homogeneous near zero, i.e. has constant multiplicity near zero. With this assumption it is not necessary to apply the apparatus of the so–called Direct Integral Decomposition, as we will do in Part III of the thesis. To assume constant multiplicity for the Master Constraint Operator does not make sense – typically the multiplicity will be discontinuous at zero. Hence we have to use the Direct Integral Decomposition, which is a well developed matematical method but to our knowledge was not applied to physical examples before. Other approaches for constraint quantizations rely on BRST methods (see the book [7] and references therein). There one usually has to include ghosts and anti–ghosts as variables into the phase space and all constraints are summarized in a BRST operator. The physical inner product can be defined via a (formal) BFV–path integral. One has then to check whether this physical inner product is positive definite. Also we want to mention the Affine Quantization Programme [20]. There one tries to define a projector onto the physical states via path integral methods. The Master Constraint Programme [3] was proposed in 2003. As will be explained in section III.1, it is motivated very much by the current situation in Loop Quantum Gravity. The advantage which the Master Constraint Programme has to offer for Loop Quantum Gravity is that it replaces the infinitely many constraints, which furthermore display a complicated constraint algebra with structure functions, by just one constaint, which has a trivial constraint algebra. The physical inner product is then defined via a Direct Integral Decomposition and is by construction always positive definite. In the case of general relativity it was hoped [21] that one can construct a physical Hilbert space in a way similar to the definition in (I.1.26) for the parametrized particle. There in contrast to the kinematical inner product where we integrated over all configurational variables (q, t) we just integrated over the configuration variable q and held the ‘time variable’ t fixed in order to get the physical inner product. We were also able to establish a unitary transformation between the physical Hilbert space and the standard Schr¨ odinger quantization, i.e. a Hilbert space which represents the wave functions at a certain fixed time. In particular one could map the operators on the standard Schr¨ odinger quantization Hilbert space to Dirac observables on the physical Hilbert space. From another viewpoint, since one has established this relationship between the Schr¨ odinger quantization and the Dirac quantization, it is not necessary anymore to have explicit expressions for the Dirac observables. One can just work with the operators in the standard Schr¨ odinger quantization since one knows that these will be mapped to Dirac observables by the unitary transformation mentioned above.8 In order to obtain such a Schr¨ odinger quantization for quantum general relativity one has to find a canonical transformation such that the new canonical coordinates are devided into two sets: one set should include variables which play a similar role as the time t for the parametrized particle and the conjugated momenta. The other set should include the rest of the canonical variables. In particular the new variables should be such that one ends up with constraints which are linear in the momenta conjugated to the ‘time variables’. (We will explain this idea in more detail in section II.7.) Such transformations were found for some symmetry reduced models of gravity, for instance [23], and for gravity coupled to some phenomenological matter, e.g. dust [24, 25]. The quantization of these models was only achieved on a very formal level. For more realistic models such transformations were not found and it is very questionable whether these exist at all [22, 26, 27]. The problem there is to find transformations which are globally well 8

This seems to be the reason, why Kuchaˇr claims [22], that a physical observable does not need to commute with the Hamiltonian constraints in the case of general relativity.

19

defined and not only locally. In particular fixing the ‘time variables’ to arbitrary fixed values should give a good gauge fixing as will be defined in section II.8. However it is known that such a gauge fixing may not exist.9 The approach pursued in the framework of Loop Quantum Gravity is therefore to look for a method in order to apply Dirac Quantization, i.e. to construct a physical Hilbert space, without having a complete knowledge of the set of Dirac observables or have found ‘time variables’ beforehand. Such a method is now available with the Master Constraint Programme [3]. If one has managed to construct the physical Hilbert space quantum Dirac observables are given by the (self–adjoint) operators on this space. However for an interpretation in classical terms one needs Dirac observables which allow for a classical limit. These one will usually get by quantizing classical Dirac observables, i.e. by finding a representation of these on the physical Hilbert space. This brings us to the subject of Dirac observables and the problem to express a dynamical evolution with ‘constants of motions’. In the case that one has canonical coordinates as described above, it was known that one can implement dynamics in the same way as was done for the parametrized particle in sections I.1.1, I.1.2 see [29, 26]. Also for the example of cylindrically symmetric waves with one polarization, where Kuchaˇr has found such canonical coordinates [23], Torre [30] calculated Dirac observables by determining the complete observables for a fixed value of the parameters τ . Rovelli introduced the terminology of partial and complete observables [11]. For systems with one constraint the idea is that the relation between the two partial observables f and T is expressed by the complete observable associated to the two partial observables F[f ;T ] (τ, ·). For systems with one constraint this concept is more general than the situation described above, since T does not need to be part of the canonical coordinates. Rovelli called the complete observables also evolving constants of motion, indicating that with the change of the parameter τ in the complete observable F[f ;T ] (τ, ·) one can implement an evolution for the complete observable. That one should use some kind of relational observables for general relativistic systems was emphasised by a number of authors [31, 32, 11]. However a canonical formulation of this concept in the case for general relativity, i.e. a system with more than one constraint, was missing. It was even suggested in [22] that it is not possible to generalize the concept of partial and complete observable to field systems. But Kuchaˇr remarks there also that for the program described above, i.e. finding ‘time variables’ as part of a canonical coordinate system, it is important to choose these as phase space functionals which can be interpreted as space–time scalar fields. He found a clear formulation in the canonical framework (where one rather works with fields on a spatial manifold and not on a space–time manifold) of this condition in [33]. We will make use of this idea in section II.11. In section II.8 we will see that the concept of partial and complete observables can also be expressed as gauge invariant extensions of gauge fixed functions as for instance defined in [7]. This of course presupposes a good gauge fixing as will be explained in section II.8. We hope that we can clarify the relations between the different approaches, indicated here. Our approach is, to develop systematically the concept of partial and complete observables for systems with an arbitrary number of constraints and to apply this concept and additionaly some new ideas to the case of general relativity.

9

This may be caused by the appearence of Gribov ambiguities, see [28].

20

Part II

Complete Observables

21

II.1

Introduction

In this part of the thesis we will generalize the concept of partial and complete observables introduced by Rovelli in [11] for systems with one constraint, to systems with an arbitrary number of first class constraints. This is necessary in order to apply these concepts to general relativity which is a theory with infinitely many constraints. In section II.3 we will review the definition for systems with one constraint and by means of examples discuss the existence of complete observables. Section II.4 generalizes complete observables to systems with an arbitrary number of first class constraints. In section II.5 we will give a system of partial differential equations for complete observables and a formal solution to this system in form of a power series. Section II.6 explains that an essential technique is to introduce constraints, such that the associated flows commute on the constraint hypersurface. Such constraints will be called weakly Abelian. In section II.7 and II.8 we will see that complete observables generalize also other ideas developed for first class systems. In II.7 we will review the Bubble Time Formalism [21]. There one works with strongly Abelian constraints, i.e. the flows associated to these constraints commute everywhere on phase space. In section II.8 we will show that one can understand complete observables also as gauge invariant extensions of gauge fixed functions [7]. This viewpoint allows us to write down the Poisson algebra of complete observables and to compare this algebra with the algebra of gauge fixed functions. We will need the results of this section for the sections II.9,II.10 and II.11. Section II.9 generalizes the definition of physical Hamiltonians introduced in section I for the parametrized particle, to arbitrary first class systems. These physical Hamiltonians generate the change of complete observables with respect to the clock variables. In section II.11 we will explain that in the case of general relativity clock variables define a physical coordinate system. Physical Hamiltonians will then generate translations in these coordinates. Section II.10 discusses how one can calculate complete observables in stages, that is firstly with respect to a subset of the full constraint set and afterwards with respect to the remaining constraints. In II.11 we will apply this technique to general relativistic systems and consider methods with which one can reduce the number of constraints one has to deal with from infinitely many constraints to one constraint. We will end this part of the thesis in section II.12 with a discussion.

II.2

Preliminaries and Notation

To begin with we will consider finite dimensional phase spaces. Here a phase space is a 2pdimensional smooth manifold M with a non-degenrate Poisson bracket {·, ·}, such that there are canonical coordinates (q a , pa , a = 1, . . . , p) with {q a , pb } = δba

.

(II.2.1)

C ∞ (M) is the space of smooth phase space functions and throughout this work we will work in the category of smooth function, also if this is not explicitely mentioned. (However this is not always the case in the examples.) We will denote phase space points by x = (q a , pa ). We will shortly introduce some notation and facts concerning first class constrained systems. First class constrained systems are characterized by the fact that the admissible initial data are restricted to a submanifold of the phase space M which is called the constraint hypersurface 22

C. This hypersurface is described by the vanishing of n < p constraints Cj , j = 1, . . . , n, which are functions on M. We will always assume that the constraints are algebraically independent, i.e. that the maximal submanifold on which they vanish simultanously is (2p − n)-dimensional, where 2p is the dimension of the phase space M. A gauge transformation is a transformation which is generated by the constraints Cj . To explain this in more detail we will introduce the notion of a phase space flow αtC generated by a smooth phase space function C. Firstly we note that to every phase space function C one can associate a so called Hamiltonian vector field χC defined by the condition {f, C} = χC · f

(II.2.2)

which has to be satisfied for arbitrary smooth functions f . Here χC · f denotes the action of the vector field χC on the function f . Then we can define the flow αtC (x) of a phase space point x by demanding that for the tangent of the curve c : R ' t &→ αtC (x) ∈ M the following holds: d t α (x) = χC (αtC (x)) dt C

.

(II.2.3)

(s+t)

The flow satisfies the group property αsC ◦ αtC = αC . It also acts on phase space functions, that is we have a map atC : C ∞ ' f → & αtC (f ) ∈ C ∞ . The value of the function αtC (f ) at the point x is given by αtC (f )(x) = f (αtC (x))

.

(II.2.4)

It can be calculated with the help of the series αtC (f )(x)

=

∞ r # t r=0

r!

{f, C}r (x)

(II.2.5)

where {f, C}0 := f and {f, C}r+1 = {{f, C}r , C}. Now a gauge transformation g : M ' x &→ g(x) ∈ M is a transformation which can be written as a composition of flows generated by the constraints Cj , j = 1, . . . , n. The first class property (I.1.12) guarantees that for x ∈ C the set {g(x) | g is a gauge transformation} is an n-dimensional submanifold of the constraint hypersurface C – the gauge orbit through x, which we will denote by Gx . For constraint algebras with structure constants a gauge orbit can also be defined for points not contained in the constraint hypersurface C. Finally we would like to mention that one can replace the set of constraints {C1 , . . . , Cn } by another set of constraints {C˜1 , . . . , C˜n } as long as the constraint ( hypersurfaces defined by these two sets coincide. This is guaranteed if one can write C˜j = k Ajk Ck where (Ajk )nj,k=1 is a matrix of phase space functions with non-vanishing determinant on M. The two sets of constraints lead also to the same gauge orbits Gx if x ∈ C.

II.3

Complete and Partial Observables for Systems with One Constraint

Here we will define complete observables for systems with one constraint. The concepts of partial observables and complete observables for such systems were introduced by Rovelli in 23

[11]. A partial observable is understood as “a physical quantity to which we can associate a (measuring) procedure leading to a number” ,[11]. We will assume here, that one can associate to an arbitrary phase space function such a measuring procedure. A partial observable is then a phase space function, which does not need to be a Dirac observable, i.e. it does not have to commute with the constraints. A complete observable is “a quantity whose value can be predicted by the theory (in classical theory)”. We will understand here complete observables as phase space functions which commute (weakly) with the constraints, i.e. phase space functions, that are invariant under gauge transformations generated by the constraints. As outlined in [11] if one has a system with one constraint C(x), one can associate to two partial observables (that is phase space functions) f (x), T (x) a family of complete observables F[f ;T ] (τ, x) labelled by a parameter τ . This complete observable is defined in the following way: Consider the flow αtC (x) generated by the constraint C starting from the phase space point x. The function F[f ;T ] (τ, x) gives the value that the function fx (t) := f (αtC (x)) assumes if the function Tx (t) := T (αtC (x)) assumes the value τ . Hence the definition is F[f ;T ] (τ, x) = αtC (f )(x)|αt

(II.3.1)

C (T )(x)=τ

One can interpret T as a kind of clock, whose values parametrize the gauge orbit of C. The complete observable F[f ;T ] (τ, x) predicts the value of f for the ‘time’ τ . Here we can already see the conditions under which F is well defined: The function T has to provide a good parametrization of the gauge orbit through x, that is the function Tx (t) := αtC (T )(x) has to be invertible. Now, locally this will be the case as long as ) * ) * d t αC (T ) (x) = αtC ({T, C}) (x) )= 0 . (II.3.2) dt

However, Tx (t) does not have to be globally invertible, it suffices that the function fx (t) := αtC (f )(x) fulfils fx (t) = fx (s) for all s, t for which Tx (t) = Tx (s). One can get a still weaker condition by considering a fixed value of τ . Then fx (ti ) has to be the same for all values ti in the preimage (Tx )−1 (τ ). We will now prove that F[f ;T ] (τ, x) is indeed a Dirac observable. To this end we have to show that αεC (F[f ;T ] (τ, ·)) = F[f ;T ] (τ, ·): αεC (F[f ;T ] (τ, ·))(x) = F[f ;T ] (τ, αεC (x)) = αtC (f )(αεC (x))|αtC (T )(αεC (x))=τ = αεC ◦ αtC (f )(x)|αεC ◦αtC (T )(x)=τ t+ε = αC (f )(x)|αt+ε (T )(x)=τ C

(II.3.3)

In the last expression, s = t + ε is just a dummy variable – one has to solve αsC (T )(x) = τ for s and then to replace s in αsC (x). This term is therefore equal to F[f ;T ] (τ, x) and we have proved the theorem: Theorem II.3.1. Let f, T be two phase space functions and x ∈ M a phase space point, fulfilling the condition: αtC (f )(x) = αsC (f )(x) for all values s, t ∈ R for which αtC (T )(x) = αsC (T )(x). Then F[f ;T ] (τ, ·) is invariant under the flow generated by C. Example 1: We will consider the constraint C = q 1 p2 − q 2 p1 24

(II.3.4)

on phase space M = R2q × R2p . We will choose f = q2 and T = q1 as partial observables. To calculate the associated complete observable, we have to evolve both partial observables under the flow of C: + αtC (T )(q1 , q2 , p1 , p2 ) = q1 cos(t) − q2 sin(t) = − q12 + q22 sin (t − arctan(q1 /q2 )) + αtC (f )(q1 , q2 , p1 , p2 ) = q2 cos(t) + q1 sin(t) = q12 + q22 cos (t − arctan(q1 /q2 )) (. II.3.5)

Now we have to invert Tx (t) := αtC (T )(x) and to find all values in the preimage of τ . Since Tx (t) is a periodic function the inverse ,will be a multi–valued function. The - equation Tx (t) = τ q1 π q1 π is uniquely solvable on the interval t ∈ − 2 − arctan( q2 ), 2 − arctan( q2 ) , where the solution is given by τ q1 t10 = − arcsin( . 2 ) + arctan( ) . (II.3.6) 2 q2 q1 + q2

(If not otherwise specified, we will always take the positive branch of the square root.) All other solutions are given by t1k = t10 + 2πk t2k = π − t10 + 2 arctan(

q1 ) + 2πk q2

,

k∈Z

(II.3.7)

.

Evaluating fx (t) := αtC (f )(x) at these points gives + + 2 2 2 fx (t1k ) = q1 + q2 − τ and fx (t2k ) = − q12 + q22 − τ 2

,

(II.3.8)

. hence F[f =q2 ;T =q1 ] (τ, x) = ± q12 + q22 − τ 2 is a double valued function. Even thought the condition in Theorem II.3.1 is not exactly fulfilled both branches of F[q2 ;q1 ] (τ, x) are Dirac observables, i.e. commute with the constraint C.

This example suggests to introduce a two-dimensional configuration space of partial observables N , which is coordinatized by values of the partial observables T and f .10 That is, we have a map P : M →N given by x &→ (T (x), f (x)). Fix a point x ∈ M. Then the flow αtC (x) of the point x in M induces a flow of the point P (x) = (T (x), f (x)) by αtC (T (x), f (x)) := P (αtC (x)) = (T (αtC (x)), f (αtC (x))). One has to keep in mind that this flow in N is not necessarily uniquely determined by the initial point (T (x), f (x)) ∈ N but it is determined by the initial point in x ∈ M (see next example). In this way each point in M defines a gauge orbit in N , namely the set {P (αtC (x)) |t ∈ R}. (Of course, gauge equivalent points in M define the same gauge orbit in N . It may also happen, that gauge inequivalent points in M define the same gauge orbit in N as is the case in example 1, where the gauge orbits do not depend on the momenta.) The functions (Tx (t) = T (αtC (x)), fx (t) = f (αtC (X)) provide a parameter description of this gauge orbit. One can interpret the complete observable F[f ;T ] (x, ·) as a function of the first coordinate of N , whose graph (·, F[f ;T ] (x, ·)) coincides with the gauge orbit. Of course, it is in general not possible to describe a curve (namely the gauge orbit) by a graph of a single valued function – exactly this is the case in example 1, where one needs a double valued function. Another way to describe 10

The topology of N should be determined by the properties of the partial observables.

25

a surface or curve is as a level set of a function from N to R. In example 1 this description is given by T (x)2 + f (x)2 = q12 + q22 = const.. Example 2: Here we will consider the constraint C = 12 (p21 + ω1 q12 ) − 12 (p22 + ω2 q22 )

(II.3.9)

where ω1 , ω2 are (constant) frequencies. The phase space is again M = R2q × R2p . The clock variable T will be T (x) = q1 , and the other partial observable is f (x) = q2 . The evolution of these partial observables under the flow of C is again periodic: + p1 sin(ω1 t) = − q12 + (p1 /ω1 )2 sin (ω1 t + arctan(ω1 q1 /p1 )) αtC (T )(q1 , q2 , p1 , p2 ) = q1 cos(ω1 t) + ω1 + p2 t αC (f )(q1 , q2 , p1 , p2 ) = q2 cos(ω2 t) − sin(ω2 t) = − q22 + (p2 /ω2 )2 sin (ω2 t − arctan(ω2 q2 /p2 )) . ω2 (II.3.10) The function Tx (t) = αtC (T )(x) is uniquely invertible on the intervall [− 2ωπ 1 −arctan(ω1 q1 /p1 ), 2ωπ 1 − arctan(ω1 q1 /p1 )] (and all intervals which one can get by translating the first interval by an amount kπ, k/ω1 ∈ Z). The solutions of Tx (t) = τ are given by / / 0 0 " $ 1 ω1 τ ω1 q1 t1k = − arcsin . 2 2 + arctan + 2πk ω1 p1 ω1 q1 + p21 / / 0 0 " $ 1 ω1 τ ω1 q1 t2k = − π − arcsin . 2 2 + arctan + 2πk k ∈ Z (II.3.11) . ω1 p1 ω1 q1 + p21

Applying the function fx (·) = αtC (f )(x) to these values we will get the complete observable F[f ;T ] (τ, x), which in general will be multi–valued. We will label these multiple values by {i, k}, i = 1, 2; k ∈ Z: / / / 0 0 " $ 1 p ω ω τ ω q 2 2 1 1 1 F[f ;T ] (τ, x)1k = q22 + ( )2 sin arcsin . 2 2 + arctan + 2πk ω2 ω1 p1 ω1 q1 + p21 " $0 ω2 q2 + arctan p2 / / / 0 0 " $ 1 p2 2 ω2 ω1 τ ω1 q1 2 q2 + ( ) sin π − arcsin . 2 2 + arctan + 2πk F[f ;T ] (τ, x)2k = ω2 ω1 p1 ω1 q1 + p21 " $0 ω2 q2 + arctan . (II.3.12) p2 In spite of the multi–valuedness of F[f ;T ] (τ, x) all the phase space functions F[f ;T ] (τ, x)i,k commute with the constraint C. (All the functions in (II.3.12) are combinations of hj (x) := qj2 + (pj /ωj )2 , j = 1, 2 and ω2 g(x) := arctan ω1

"

ω1 q1 p1 26

$

+ arctan

"

ω2 q2 p2

$

(II.3.13)

and C commutes with hj and g.) In the projected phase space N = R2 the gauge orbits of C are Lissajous figures. If the frequencies ω1 and . ω2 are non–commensurable these curves . . . fill densely the rectangle 2 2 2 2 2 2 [− q1 + (p1 /ω1 ) , q1 + (p1 /ω1 ) ] × [− q2 + (p2 /ωj ) , q22 + (p2 /ω2 )2 ]. Therefore it is not astonishing that the prediction of fx (t) given the value of Tx (t) is highly ambiguous. Nevertheless it is possible to obtain a Dirac observable in this way. In practice it will not be possible to obtain such global information on the behavior of the partial observables T, f as is needed for the proof of the theorem II.3.1. In the above example we could have done the following: To invert the flow of the clock variable only on some suitable small interval and to calculate the complete observable just for parameter values in this interval. We would have obtained not all values of the multi–valued complete observable but just one of them. Nevertheless in the above example this value turns out to be a Dirac observable. The questions arises whether this argumentation works also in more complicated or chaotic systems. Of course one could take the viewpoint that one can at least try to calculate complete observables and then check afterwards whether the complete observables obtained are Dirac observables, i.e. commute with the constraints. We will proceed in this way in some examples for practical reasons. The next example gives a glimpse of what might happen in chaotic systems. Example 3: The Partial Observable Method promises to give a Dirac observable (or constant of motion if one identifies the constraint with the Hamiltonian) for arbitrary pairs of phase space functions satisfying the conditions mentioned in theorem II.3.1. But we know that in ergodic systems there do not exist analytic constants of motion (see [34]). It would be therefore interesting to know how the method applies to such systems. Since it is very difficult to perform explicit calculations for such systems, we will leave this question open for further research and consider here a very simple ergodic systems, which evolves in discrete steps. We will investigate the so called Baker transformation, see for instance [35]. This system is defined on the phase space M = [0, 1]x1 × [0, 1]x2 (identified to a torus) and evolves in discrete steps in the following way 2 (2x1 , 12 x2 ) if 0 ≤ x1 < 12 α1 ((x1 , x2 )) = (II.3.14) (2x1 − 1, 12 x2 + 12 ) if 12 ≤ x1 < 1 where αK (·) denotes the evolution after K steps. As partial observables we choose T = x1 and f = x2 , i.e the phase space coordinates. Now one has to consider the flow of these functions. Assume that x1 < 2−N for some number N ∈ N. Then αK (T )(x) = 2K x1 for K ≤ N and we can invert the function Tx (K) := αK (T )(x) on the interval K ∈ [0, N ]: The solution of the equation Tx (K) = τ in this interval is K0 = log2 (τ /x1 )

(II.3.15)

.

Inserting this solution into the function fx (K) := αK (f )(x) = 2−K x2 gives F[f ;T ] (τ, x) = fx (K0 ) =

27

x1 · x2 τ

.

(II.3.16)

Now, assuming x1 < 2−N , F[f ;T ] (τ, x) is not invariant under the flow αJ (·) for arbitrary J ∈ N, but it is invariant for J ≤ N . Hence these J’s lie exactly in the interval, on which we inverted the function Tx (K). Beginning with the assumption 1 > x1 ≥ 1 − 2−N one can invert the function Tx (K) on the interval K ∈ [0, N ] and one finds the complete observable F[f" ;T ] (τ, x) = 1 −

(1 − x)(1 − y) 1−τ

.

(II.3.17)

Again, assuming x1 ≥ 1 − 2−N , this phase space function is invariant under the flow αJ (·) for J ∈ [0, N ]. The complete observable F[f ;T ] (τ, x) for x1 < 2−N corresponds to the fact that the phase space function g(x) = x1 · x2 is invariant under the evolution law which applies in the region x1 < 12 , whereas F " [f ; T ](τ, x) corresponds to the phase space function g" (x) = (1 − x1 )(1 − x2 ) which is invariant under the evolution law which applies in the region x1 ≥ 12 . Because the above model evolves in discrete steps it is difficult to draw conclusions for continuous models. It is possible to show that for certain models there do not exist analytic single–valued constants of motion, see [34]. So if one would calculate complete observables for those models the result may be still a Dirac observable, but a non–analytic or multi–valued one.

II.4

Complete and Partial Observables for Systems with Several Constraints

The aim of this chapter is to define complete observables for systems with several constraints. For systems with one constraint we had the following geometrical picture: The constraint C generates a one–dimensional gauge orbit. We used the clock variable T to parametrize this gauge orbit. In the ideal case each point of the gauge orbit was uniquely specified by a value τ of the phase space function T . We then evaluated the function f (·) at the phase space point labelled by τ . Now for a system with n independent (first class) constraints the gauge orbits generated by these constraints will be n–dimensional.11 Hence we need n clock variables (i.e. phase space functions) Tj , j = 1, . . . , n to parametrize a gauge orbit. Ideally each point of a (fixed) gauge orbit should be uniquely specified by the values (τ1 , . . . , τn ) of the clock variables (T1 , . . . , Tn ). It is then non-ambiguous to evaluate a phase space function f at that point of a (fixed) gauge orbit, on which the functions (T1 , . . . , Tn ) give the values (τ1 , . . . , τn ). Locally this amounts to the condition det({Ti , Cj }ni,j=1 ) )= 0 (II.4.1) which has to be satisfied on the gauge orbit through x.

11 The first class property of the constraints guarantees, that the flow of the constraints is integrable to an n dimensional surface – the gauge orbit. For constraint algebras with structure functions the integrability conditions are only fulfilled on the constraint hypersurface, hence one can only expect on the constraint hypersurface well defined gauge orbits. In this case the method following below will work only for points x on the constraint hypersurface. Moreover if one replaces the constraints by another equivalent set of constraints, as was mentioned in section II.2, the new constraints will only generate the same gauge orbits as the old constraints on the constraint hypersurface.

28

Again, the above procedure is still well defined, if several points of the gauge orbit correspond to the values (τ1 , . . . , τn ) but if f evaluated at these points always gives the same number. Otherwise the result will be (possibly extremely) multi–valued. Analogously to the previous section we therefore define the complete observable F[f,T1 ,...,Tn ] associated to the partial observables f, T1 , . . . , Tn as F[fτ ;T1 ,...,Tn ] (x) = αβj Cj (f )(x)|αβ

j Cj

(Ti )(x)=τi

(II.4.2)

.

P (·), that is the flow generated by the phase space function Here αβj Cj (·) is defined to be αt=1 j βj Cj ( j βj Cj evaluated at the parameter value t = 1. Here and in the following we will assume that all points of the gauge orbit Gx can be reached by transformations of the form {αβj Cj (x); βj ∈ R}. (We defined Gx to be the set which can be reached by all forms of compositions of flows of the form αβj Cj (x).) The complete observable F[f ; T1 ,...,Tn ] (τ1 , . . . , τn , x) is not defined for those parameter values τi for which the intersection of the level set {x |Ti (x) = τi for i = 1, . . . , n} and the gauge orbit through x is empty. To calculate the complete observable F[f ; T1 ,...,Tn ] one has at first to solve the system of equations αβj Cj (Ti )(x) = τi , i = 1, . . . , n (II.4.3)

for β1 , . . . , βn . Next one has to plug in these solutions βk = Bk (τi , x) into the function fx (β1 , . . . , βn ) := αβj Cj (f )(x). This gives the complete observable F[f ; T1 ,...,Tn ] (τ1 , . . . , τn , x) = fx (B1 (τi , x), . . . , Bn (τi , x)). Different solutions βk = Bk (τi , x) of (II.4.3) may correspond to the same phase space point on the gauge orbit in question or to different points of this gauge orbit. In the latter case F[f ; T1 ,...,Tn ] (τi , x) is still well defined, if f (·) evaluated at these points gives always the same number. Otherwise F[f ; T1 ,...,Tn ] (τi , x) will be multi–valued. We will now verify that F[f ; T1 ,...,Tn ] (τi , x) is indeed an n–parameter family of Dirac Observables. To this end we have to show that it remains constant under the flow of αεj Cj (·) = P αt=1 εj Cj (·), where εj , j = 1, . . . , n are arbitrary fixed numbers: j

αεj Cj (F[f ;Tk ] (τm , ·))(x) = F[f ;Tk ] (τm , αεj Cj (x)) = αβj Cj (f )(αεl Cl (x))|αβ

j Cj

= αεl Cl ◦ αβj Cj (f )(x)|αε C

l l

(Ti )(αεl Cl (x))=τi ◦αβj Cj (Ti )(x)=τi

.

(II.4.4)

The key point in the last expression is that αεl Cl ◦ αβj Cj (x) is still a point on the gauge orbit Gx through x. (For constraint algebras with structure functions this holds only on the constraint hypersurface.) Hence solutions of αεl Cl ◦ αβj Cj (Ti )(x) = τi correspond to the same points on the gauge orbit as solutions of αβj Cj (Ti )(x) = τi , namely to the intersection of the level sets {Ti−1 (τi )}ni=1 and Gx . Both expressions in (II.4.4) and (II.4.3) prescribe to evaluate the phase space function f (·) at exactly these points. If f (·) gives always the same value at these points the expressions in (II.4.4) and (II.4.3) are well defined and coincide. Thus we have the theorem: Theorem II.4.1. Let Cj , j = 1, . . . , n be n independent first class constraints and x ∈ M a phase space point on the constraint hypersurface. Given n phase space functions Ti , i = 1, . . . , n and a phase space function f , assume that f evaluated at the points in the intersection of the 29

level set {x |Ti (x) = τi for i = 1, . . . , n} and the gauge orbit through x gives always the same result. Then the complete observable F[f,Ti ] (τi , x) is well defined and is invariant under the flow generated by the constraints. Analogously to the last section we can introduce a configuration space of partial observables, which is an (n + 1)-dimensional manifold coordinatized by values of the partial observables (T1 , . . . , Tn , f ). The ‘projection’ P : M → N is defined by P : x &→ (T1 (x), . . . , Tn (x), f (x)). Fix a point x. Then the flow αβj Cj (x) induces a flow of the point (T1 (x), . . . , Tn (x), f (x)) in N by αβj Cj ((T1 (x), . . . , Tn (x), f (x))) = P (αβj Cj (x)). Again one has to keep in mind that this flow needs not to be uniquely determined by the initial point (T1 (x), . . . , Tn (x), f (x)) ∈ N , but it is uniquely determined by the point x ∈ M. Since in M the flow generated by the constraints integrates to a hypersurface, namely the gauge orbit (at least on the constraint hypersurface), this is also the case in the space N . This gauge orbit can be described in a parametric way, i.e. as the set {P (αβj Cj (x)) |βj ∈ R}. The complete observable F[f,Ti ] (τi , x) provides a description of the gauge orbit as the graph of a function depending on n variables, i.e. the gauge orbit is the set {(τ1 , . . . , τn , F[f,Ti ] (τ1 , . . . , τn , x)) |τ1 , . . . , τn ∈ R}. Example 4: Here we will consider the phase space M = R3q × R3p and take the coordinates of the angular momentum as constraints: Ci =

3 #

,ijk qj pk

, i = 1, 2, 3

(II.4.5)

j,k=1

where ,ijk is the totally antisymmetric tensor with ,123 = 1. ( ( But the three constraints (II.4.5) are not algebraically independent since xi Ci = pi Ci = 0. Hence it is sufficient to consider C1 and C2 , which form a constraint algebra with structure functions: 1 1 {C1 , C2 } = C3 = − (q1 C1 + q2 C2 ) = − (p1 C1 + p2 C2 ) . (II.4.6) q3 p3 Therefore we need two clock variables which we will choose as T1 = q1 and T2 = q2 . The flow of these variables generated by the constraints is " $ + + β2 (β2 q1 − β1 q2 ) β2 q3 2 2 . α(β1 C1 +β2 C2 ) (T1 )(x) = q1 + cos( β1 + β2 ) − 1 + sin( β12 + β22 ) β12 + β22 β12 + β22 " $ + + β1 (β1 q2 − β2 q1 ) β1 q3 2 2 α(β1 C1 +β2 C2 ) (T2 )(x) = q2 + cos( β1 + β2 ) − 1 − . 2 sin( β12 + β22 ) β12 + β22 β1 + β22 (II.4.7) Next we have to solve the system of equations α(β1 C1 +β2 C2 ) (T1 )(x) = τ1 α(β1 C1 +β2 C2 ) (T2 )(x) = τ2

(II.4.8)

for β1 and β2 . But already in this relatively simple example, it is very difficult to find the solutions of (II.4.8), since on the left hand side the βi ’s appear in the argument and as coefficients of transcedental functions. In the next section we will explain a different method to calculate complete observables, but here we will use that one can replace the constraints C1 and C2 by an 30

equivalent set of constraints, obtained from the first ones by multiplying them with (nowhere vanishing) phase space functions. In this example we will use the modified constraints C1" =

1 q 3 p2 C1 = p3 − q2 q2

C2" = −

1 q 3 p1 C1 = p3 − q1 q1

(II.4.9)

.

The flow generated by these constraints is especially simple for the phase space function q3 , so this will be one of our clock variables T1 := q3 : αβ1 C1! +β2 C2! (T1 )(x) = q3 + β1 + β2

(II.4.10)

.

To find a suitable second clock observable, we will consider the flow of the phase space function g = q1 . Its Poisson bracket with the constraints is {β1 C1" + β2 C2" , q1 } = −β2

q3 q1

(II.4.11)

.

The function q1 in the denominator of the right hand side vanishes if we choose the function T2 := g2 = q12 as a second clock variable. One obtains the following differential equation for the flow T2x (t) := αtβ1 C ! +β2 C2 (T2 )(x): 1

d T2x (t) = −2β2 T1x (t) dt

(II.4.12)

where T1x (t) := αtβ1 C ! +β2 C2 (T1 )(x) = q3 −(β1 +β2 )t. This differential equation is easily integrable 1 and gives for the flow of the second clock variable αβ1 C1! +β2 C2! (T2 )(x) = T2x (t = 1) = q12 − β2 (β1 + β2 ) − 2β2 q3

.

(II.4.13)

Analogously, choosing as the third partial observable f = q22 , its evolution under the constraints is αβ1 C1! +β2 C2! (f )(x) = q22 − β1 (β1 + β2 ) − 2β1 q3 . (II.4.14) Now we have to solve the system of equations

αβ1 C1! +β2 C2! (T1 )(x) = q3 + β1 + β2 αβ1 C1! +β2 C2! (T2 )(x) =

q12

= τ1

− β2 (β1 + β2 ) − 2β2 q3 = τ2

(II.4.15)

for the parameters β1 and β2 . The solutions are given by β1 = B1 (τ1 , τ2 , x) = β2 = B2 (τ1 , τ2 , x) =

−q12 − q32 + τ12 + τ2 q 3 + τ1 2 q 1 − τ2 . q 3 + τ1

(II.4.16)

Replacing the βi ’s in the evolution of f in (II.4.14) by the solutions Bi (τ1 , τ2 , x) we obtain finally the complete observable F[f ;T1 ,T2 ] (τ1 , τ2 , x) = q12 + q22 + q32 − τ12 − τ2 This phase space function is indeed a Dirac observable. 31

.

(II.4.17)

II.5

A System of Partial Differential Equations for Complete Observables

As we have seen in example 4 it may be very difficult to invert the flow of the clock variables Ti in order to find the complete observable F[f ;Ti ] . In this section we will derive a system of first order partial differential equations for F[f ; Ti ] (τj , x) as a function of the τj ’s. To this end we will make the following assumptions: Consider a phase space point x ∈ M and the gauge orbit Gx through x in M. We define the map Tx : Gx → Tx (Gx ) ⊂ Rn

y &→ (T1 (y), . . . , Tn (x))

(II.5.1)

.

The index x in Tx represents the gauge orbit Gx through x. Our assumption is, that Tx is uniquely invertible as a function from Gx to Tx (Gx ). In other words, to each point (τ1 , . . . τn ) ∈ Tx (Gx ) there exists a unique point y ∈ Gx which solves Tk (y) = τk for k = 1, . . . , n. We will denote this point by y = T−1 x (τi ). We defined the complete observable F[f ; Ti ] as F[f ; Ti ] (τi , x) = αβk Ck (f )(x)|αβ

k Ck

(Ti )(x)=τi

(II.5.2)

.

The value of F[f ; Ti ] at the slightly displaced point (τ1 + ε1 , . . . , τn + εn ) is F[f ; Ti ] (τi + εi , x) = αβk Ck (f )(x)|αβ

k Ck

(Ti )(x)=τi +εi

.

(II.5.3)

Now we know that F[f ; Ti ] is gauge invariant, therefore we can replace x by the point T−1 x (τi ) ∈ Gx on the right hand side of (II.5.3): F[f ; Ti ] (τi + εi , x) = αβk Ck (f )(T−1 x (τi ))|αβ

k Ck

(Tj )(T−1 x (τi ))=τj +εj

As by definition Tj (T−1 x (τi )) = τj , we can solve the equations # 2 αβk Ck (Tj )(T−1 βk {Tj , Ck }(T−1 x (τi )) = τj + x (τi )) + O(ε ) = τj + εj

(II.5.4)

(II.5.5)

k

for the βk ’s to the first order in the εi ’s. (Here O(ε2 ) denotes terms of higher than first order in the εi ’s.) To this end we define the matrix of phase space functions Ajk := {Tj , Ck }

(II.5.6)

−1 −1 and its inverse A−1 jm by Akj Ajm = δkm = Akj Ajm . (The inverse exists at least on the gauge orbit Gx because of the assumptions we made for the map Tx .) The solution to the equations (II.5.5) can then be written as # −1 2 βk = εj A−1 (II.5.7) kj (Tx (τi )) + O(ε ) . j

Inserting these values into −1 αβk Ck (f )(T−1 x (τi )) = f (Tx (τi )) +

# k

32

2 βk {f, Ck }(T−1 x (τi )) + O(β )

(II.5.8)

we arrive at F[f ; Ti ] (τi + εi , x) = f (T−1 x (τi )) +

# j,k

= F[f ; Ti ] (τi , x) +

3 4 2 εj A−1 {f, C } (T−1 k x (τi )) + O(ε ) kj

# j,k

3 4 2 εj A−1 {f, C } (T−1 k x (τi )) + O(ε ) kj

This gives for the partial derivative of F[f ; Ti ] with respect to τm / 0 # ∂ −1 −1 F (τi , x) = A−1 km {f, Ck } (Tx (τi )) =: gm (Tx (τi )) . ∂τm [f ; Ti ]

. (II.5.9)

(II.5.10)

k

The function gm (T−1 x (τi )) on the right hand side of (II.5.10) (can again be written as a complete observable associated to the partial observable gm (x) = k A−1 km {f, Ck }: gm (T−1 x (τi )) = gm (αβk Ck (x))|Ti (αβ

k Ck

(x))=τi

= F[g; Ti ] (τi , x) .

(II.5.11)

In summary we have the partial diffential equations ∂ F (τi , x) " F[P A−1 {f,Ck }; Ti ] (τi , x) k km ∂τm [f ; Ti ]

(II.5.12)

F[f ; Ti ] (τi = Ti (x), x) " f (x) .

(II.5.13)

with initial conditions following from the definition of complete observables (II.4.2):

Partial derivatives with respect to the parameters τm of complete observables are again complete observables (which is not surprising, since a complete observable is a Dirac observable for all values of the parameters τm ). Now the problem is, that in general the complete observables F[gm ,Ti ] (τi , x) are unknown functions. In this case one has to add the partial differential equations (PDE’s) for the F[gm ,Ti ] . Again, it may happen, that these involve unknown functions. In this case one has to iterate the procedure until one obtains a closed system of PDE’s for a set of functions F[f ; Ti ] , F[gm ,Ti ] , F[gmm! ,Ti ] , . . .. Using this system it may be possible to derive a higher order PDE for the primary function F[f ; Ti ] . However one can limit the number of necessary iterations if one realizes that if a function g is composed from m phase space functions fh the associated complete observable is F[g(f1 ,...,fm ); Ti ] (τi , x) = αβj Cj (g(f1 , . . . , fm ))(x)|αβ

j Cj

(Ti )(x)=τi

= g(αβj Cj (f1 ), . . . , αβj Cj (fm ))(x)|αβ

j Cj

= g(F[f1 ; Ti ] (τi , x), . . . , F[fm ; Ti ] (τi , x))

(Ti )(x)=τi

.

(II.5.14)

That is F[Ti ] (τi ) : f &→ F[f ; Ti ] (τ, x) is an algebra homomorphism with respect to multiplication and addition. Hence we just need to choose dim(M) = 2p algebraically independent phase space functions fh (for instance the canonical coordinates) and to consider the PDE’s (II.5.10) for the associated complete observables. The right hand side of these PDE’s will then be expressible through the complete observables associated to the fh , h = 1, . . . , 2p. Therefore putting the PDE’s for the complete observables associated to the fh ’s together one obtains a closed system of partial 33

differential equations for 2p unknown functions. Moreover if n of the functions fh coincide with the constraints and another set of n functions coincides with the clock variables Ti , the associated complete observables vanish weakly or are constants respectively: F[Ck ; T1 ,...,Tn ] (τ, x) " 0

F[Tk ; T1 ,...,Tn ] (τ, x) = τk

(II.5.15)

and one is left with a set of (2p − 2n) unknown functions. The property (II.5.14) also shows that the set {F[qa ; T1 ,...,Tn ] (τ1 , . . . , τn , ·), F[pa ; T1 ,...,Tn ] (τ1 , . . . , τn , ·) | a = 1, . . . , p}

(II.5.16)

provides an over-complete basis of the space of Dirac observables: If d is a Dirac observable then F[d; Ti ] (τi , x) " d(x = (qa , pa )) " d(F[qa ; Ti ] (τi , x), F[pa ; Ti ] (τi , x))

.

(II.5.17)

Hence d can be expressed as a combination of the complete observables associated to the canonical coordinates. The above mentioned over-completeness is described by F[Ck ; Ti ] (τi , x) " 0 " Ck (F[qa ,Ti ] (τi , x), F[pa ,Ti ] (τi , x))

F[Tk ; Ti ] (τi , x) " τk " Tk (F[qa ,Ti ] (τi , x), F[pa ,Ti ] (τi , x))

,

(II.5.18)

that is we have 2n relations between the 2p complete observables. At the end of this section we will give a (formal) solution to the PDE’s (II.5.10) as a power series in the τi ’s. Before we come to an example, we will prove, that the PDE’s (II.5.10) are consistent, i.e. satisfy the integrability conditions ∂2 ∂2 F[f ; Ti ] (τi , x) = F (τi , x) ∂τl ∂τm ∂τm ∂τl [f ; Ti ]

(II.5.19)

at least on the constraint hypersurface. Since the partial derivative of F[f ; Ti ] (τi , x) is again a complete observable, we can apply equation (II.5.10) to obtain the second partial derivatives of F[f ; Ti ] (τi , x): ∂2 ∂ F (τi , x) = F (τi , x) ∂τl ∂τm [f ; Ti ] ∂τl [gm ; Ti ] −1 −1 = {{f, Ck }A−1 km , Cj }Ajl (Tx (τi )) 3 4 −1 −1 −1 −1 = {f, Ck , }, Cj }Akm Ajl + {f, Ck }{Akm , Cj }Ajl (T−1 x (τi ))

(II.5.20)

where here and in the following we sum over repeated indices. Multiplying both sides of this equation with Ali Amh gives Ali Amh

% & ∂2 F[f ; Ti ] (τi , x) = {{f, Ch }, Ci } + {f, Ck }{A−1 , Ci }Amh (T−1 x (τi )) km ∂τl ∂τm % & −1 = {{f, Ch }, Ci } − {f, Ck }{Amh , Ci }A−1 km (Tx (τi ))

(II.5.21)

34

where we used that −1 −1 0 = {δhk , Ci } = {A−1 km Amh , Ci } = Akm {Amh , Ci } + Amh {Akm , Ci } .

(II.5.22)

The anti–symmetrization of equation (II.5.21) in the indices i, h is Al [i A|m| h]

∂2 F (τi , x) = ∂τl ∂τm [f ; Ti ] = =

1 2

%

{{f, Ch }, Ci } − {{f, Ci }, Ch }

& −1 −1 −{f, Ck }{Amh , Ci }A−1 km + {f, Ck }{Ami , Ch }Akm (Tx (τi )) % 1 2 {{f, Ch }, Ci } − {{f, Ci }, Ch } & −1 −{f, Ck }A−1 km ({{Tm , Ch }, Ci } − {{Tm , Ci }, Ch }) (Tx (τi )) % & −1 −1 1 2 {f, {Ci , Ch }} − {f, Ck }Akm {Tm , {Ci , Ch }} (Tx (τi )) (II.5.23)

where in the second line we used Amj = {Tm , Cj } and in the last line we used the Jacobi identity for the Poisson brackets. Now we apply the first class property of the constraints, i.e. that their algebra closes (on the constraint hypersurface): {Ch , Ci } = fhij Cj and use again that Amj = {Tm , Cj } Al [i A|m| h]

% ∂2 F[f ; Ti ] (τi , x) = {f, fihj }Cj + {f, Cj }fihj ∂τl ∂τm & −1 −{f, Ck }A−1 km ({Tm , fihj }Cj + {Tm , Cj }fihj ) (Tx (τi )) % % & & −1 = Cj {f, fihj } − {f, Ck }A−1 (Tx (τi )) km {Tm , fihj } " 0.

(II.5.24)

Therefore (since it follows from our assumption that Amh is an invertible matrix) the integrability condition for the PDE’s (II.5.10) are satisfied everywhere in M for constraint algebras with structure constants and at least on the constraint hypersurface for constraint algebras with structure functions. Example 5: Here we will consider a kind of deformed SO(3) algebra given on the phase space R3q × R3p by n3 m2 3 C1 = q2n2 pm 3 − q 3 p2

n1 m3 1 C2 = q3n3 pm 1 − q 1 p3 n2 m1 2 C3 = q1n1 pm 2 − q 2 p1

(II.5.25)

( n m that is Ci = jk ,ijk qj j pk j . A special case of this example is quantized in section III.3.4. The constraint algebra is given by # {Ci , Cj } = ,ijk nk mk qknk −1 pknk −1 Ck , (II.5.26) k

hence it is a first class algebra with structure functions. The set {C1 , C2 , C3 } is not an independent set of constraints: because of the anti-symmetry of ,ijk we have the relations m2 m3 1 q1n1 C1 + q2n2 C2 + q3n3 C3 = pm 1 C1 + p2 C2 + p3 C3 = 0 .

35

(II.5.27)

We will therefore choose as a set of independent constraints {C1 , C2 }. Now we choose as clock variables the functions T1 = q1 and T2 = q2 . The PDE’s (II.5.10) for q3 (τ1 , τ2 ) = F[q3 ; T1 ,T2 ] (τ1 , τ2 , ·) are 3 −1 n1 ∂ m3 p m q1 3 q3 (τ1 , τ2 ) = − (τ1 , τ2 ) m −1 1 ∂τ1 m1 p 1 q3n3

∂ m3 p3k3 −1 q2n2 q3 (τ1 , τ2 ) = − (τ1 , τ2 ) ∂τ2 m2 p2m2 −1 q3n3

(II.5.28)

where we abbreviated xi (τ1 , τ2 ) = F[xi ; T1 ,T2 ] (τ1 , τ2 , ·) with either xi = qi or xi = pi . Since q1 , q2 are the clock variables we have q1 (τ1 , τ2 ) = τ1 and q2 (τ1 , τ2 ) = τ2 . This leaves us with the unknown functions pi (τ1 , τ2 ) in the PDE’s (II.5.28). Hence we consider also the partial derivatives of these functions: 3 ∂ n1 q1n1 −1 pm 3 p1 (τ1 , τ2 ) = (τ1 , τ2 ) m −1 n ∂τ1 m1 p 1 1 q 3 3

∂ p1 (τ1 , τ2 ) = 0 ∂τ2

∂ p2 (τ1 , τ2 ) = 0 ∂τ1 ∂ n 3 p1 p3 (τ1 , τ2 ) = − (τ1 , τ2 ) ∂τ1 m1 q 3

3 ∂ n2 q2n2 −1 pm 3 p2 (τ1 , τ2 ) = (τ1 , τ2 ) ∂τ2 m2 p2m2 −1 q3n3 ∂ n 3 p2 p3 (τ1 , τ2 ) = − (τ1 , τ2 ) . ∂τ2 m2 q 3

(II.5.29)

On the constraint surface we have 3 2 1 pm pm pm 3 2 1 n3 " n2 " n1 q3 q2 q1

(II.5.30)

,

n3 3 therefore we can replace in (II.5.29) the term pm 3 /q3 according to (II.5.30). q1 (τ1 , τ2 ) = τ1 and q2 (τ1 , τ2 ) = τ2 we obtain

∂ n 1 p1 p1 (τ1 , τ2 ) " (τ1 , τ2 ) ∂τ1 m1 τ 1 ∂ p2 (τ1 , τ2 ) = 0 ∂τ1

∂ p1 (τ1 , τ2 ) = 0 ∂τ2 ∂ n 2 p2 p2 (τ1 , τ2 ) " (τ1 , τ2 ) ∂τ2 m2 τ 2

Using also

(II.5.31)

These equations are easily integrable to 1 1 pm pm pm1 (τ10 , τ20 ) 1 1 (τ1 , τ2 ) " 1 n1 (τ1 , τ2 ) = n1 n1 q1 τ1 τ10

2 2 pm pm pm2 (τ10 , t20 ) 2 2 (τ1 , τ2 ) " 2 . n2 (τ1 , τ2 ) = n2 n2 q2 τ2 τ20 (II.5.32)

n2 m1 n1 m3 n3 2 Hence F1 := pm 2 /q2 " p1 /q1 " p3 /q3 is conserved and indeed it commutes weakly with the constraints. Using the relations (II.5.29) and the solutions (II.5.32) the equations (II.5.28) can be written as n /m

n /m

n /m

n /m

∂ m3 p 1 m3 p1 (τ10 , τ20 ) τ1 1 1 q3 3 3 (τ10 , τ20 ) q3 (τ1 , τ2 ) = − (τ1 , τ2 ) = − ∂τ1 m1 p 3 m1 p3 (τ10 , τ20 ) τ n1 /m1 q n3 /m3 (τ1 , τ2 ) 10 3 ∂ m3 p 2 m3 p2 (τ10 , τ20 ) τ2 2 2 q3 3 3 (τ10 , τ20 ) q3 (τ1 , τ2 ) = − (τ1 , τ2 ) = − ∂τ2 m2 p 3 m2 p3 (τ10 , τ20 ) τ n2 /m2 q n3 /m3 (τ1 , τ2 ) 10 3

36

.

(II.5.33)

The equations can be integrated to (n3 /m3 )+1

q3

(n /m )+1

(τ1 , τ2 )

n /m3

(n3 + m3 )q3 3

(τ10 , τ20 )

q3 (τ10 , τ20 )p3 (τ10 , τ20 ) τ1 1 1 − p (τ , τ ) n /m 1 10 20 n 3 + m3 (n1 + m1 )τ101 1

p2 (τ10 , τ20 ) "

(n2 /m2 )+1

−

τ2

p (τ , τ ) n /m2 2 10 20

(n2 + m2 )τ202

(II.5.34)

From here and (II.5.32) we obtain that F2 :=

q 1 p1 q 2 p2 q 3 p3 + + n 1 + m1 n 2 + m2 n 3 + m3

.

(II.5.35)

is a Dirac observable. Hence we found two (independent) Dirac observables F1 , F2 which describe the two-dimensional reduced phase space. Next we will give a formal solution to the system of PDE’s (II.5.10). To this end consider the (formal) power series of F[f ; Ti ] (τi , x) in the τi ’s around the point τi = Ti (y) where y is a point in the gauge orbit Gx through x: F[f ; Ti ] (τi , x) =

∞ #

k1 ,...,kn =0

1 ∂ k1 ···kn F (Ti (y), x) (τ1 − T1 (y))k1 · · · (τn − Tn (y))kn k1 ! · · · kn ! ∂ k1 τ1 · · · ∂ kn τn [f ; Ti ]

(II.5.36)

We know that all the partial derivatives appearing in (II.5.36) can be written as complete observables associated to some phase space function g(k1 ,...,kn ) ∂ k1 ···kn F =: F[gk ,...,kn ; Ti ] 1 ∂ k1 τ1 · · · ∂ kn τn [f ; Ti ]

.

(II.5.37)

Furthermore by definition (II.4.2) we have F[gk

1 ,...,kn

; Ti ] (Ti (y), x)

= g(k1 ,...,kn) (y)

(II.5.38)

so that we can replace the partial derivatives in (II.5.36) by g(k1 ,...,kn ) (y). Because of equation (II.5.10) these functions are given by g(k1 ,...,kn ) = (S1 )k1 ◦ · · · ◦ (Sn )kn (f )

(II.5.39)

where Sj is the map Sj : C ∞ (M) → C ∞ (M)

h &→ A−1 lj {h, Cl }

.

(II.5.40)

Here we made the assumption that A−1 jl has smooth matrix entries. The order of the operators Sj in (II.5.39) does not matter (at least on the constraint hypersurface) because of the consistency conditions (II.5.19).

37

Therefore the formal solution to the PDE’s (II.5.10) can be written as12 F[f ; Ti ] (τi , x) =

∞ #

k1 ,...,kn =0

1 (y) (τ1 − T1 (y))k1 · · · (τn − Tn (y))kn g k1 ! · · · kn ! (k1 ,...,kn )

(II.5.41)

where the functions g(k1 ,...,kn ) (y) are defined in (II.5.39,II.5.40) and y is a point on the gauge orbit Gx through x. In particular we can choose y = x. Assume that the sum (II.5.41) converges for fixed values of the parameters τ0i in some neighbourhood of a phase space point x0 . Furthermore assume that one can permute differentiation with respect to phase space variables and summation. Then it is straightforward to show directly, that (II.5.41) Poisson commutes (weakly) with the constraints. Act on (II.5.41) with the differential operator Sj and use that they commute up to terms proportinal to the constraints Sj ◦ Sk (h)(x) = Sk ◦ Sj (h)(x) + λj (x)Cj (x)

(II.5.42)

as well as Sj (Ti ) = δij . The result {F[f ; Ti ] (τ0i , x), Ck }A−1 kj "

∞ #

k1 ,...,kn =0

−

1 k +1 S k1 ◦ · · · ◦ Sj j ◦ Snkn (f )(y) k1 ! · · · kn ! 1

(τ1 − T1 (y))k1 · · · (τn − Tn (y))kn ∞ # kj S k1 ◦ · · · ◦ Snkn (f )(y) k1 ! · · · kn ! 1

k1 ,...,kj =1,...,kn =0

(τ1 − T1 (y))k1 · · · (τj − Tj (y))kj −1 · · · (τn − Tn (y))kn

(II.5.43)

vanishes weakly, which can be seen by relabelling the index kj in the last sum to kj" = kj − 1. Hence, if the above mentioned convergence conditions are satisfied, (II.5.41) provides a local definition of a complete observable. This is in contrast to theorem II.4.1 where we made global assumptions on the properties of the partial observables with respect to the constraint flow.

II.6

Weak Abelianization of the Constraint Algebra

In this section we will remark, that one can understand the result (II.5.41) also from another viewpoint: The proof for the consistency conditions (II.5.19) proceeded by showing that −1 −1 −1 {{f, Cj }A−1 jm , Ck }Akl − {{f, Cj }Ajl , Ck }Akm

(II.6.1)

vanishes weakly for an arbitrary phase space function f . Therefore the flows generated by the vector fields χ˜m := A−1 jm χCj commute weakly. Since we have −1 {f, A−1 jm Cj } " Ajm {f, Cj }

(II.6.2)

the flows generated by χ˜m and the flows generated by χC˜m where C˜m := A−1 jm Cj 12

(II.6.3)

A similar formula appeared in [36] where Dirac observables are constructed as gauge invariant extensions of gauge fixed functions, see section II.8.

38

coincide on the constraint hypersurface. Hence also the flows generated by the C˜m commute on the constraint hypersurface and indeed one can calculate directly that {C˜m , C˜j } contains only terms which are at least quadratic in the constraints. We will call this property of the constraint set {C˜m ; m = 1, . . . , n} weakly Abelian. That is, a constraint set is weakly Abelian if the associated structure functions vanish weakly. Moreover the evolution of the clock variables Tj with respect to these new constraints is linear in the evolution parameters (again restricted to the constraint hypersurface): αβj C˜j (Tk )(x) " Tk (x) + δkj βj

(II.6.4)

.

Now we can equally well use the constraints C˜m in the definition of a complete observable: F˜[f ;T1 ,...,Tn ] = αβj C˜j (f )|αβ

˜ (Tk )(x)=τk j Cj

(II.6.5)

.

Then the complete observables F˜[f ;T1 ,...,Tn ] and F[f ;T1 ,...,Tn ] coincide weakly. The advantage in using the constraints C˜m is that now the solution of the equations αβj C˜j (Tk )(x) = τk

(II.6.6)

is very easy, namely given by βj = τj − T (x) for x on the constraint hypersurface. From this we can conclude F˜[f ; T1 ,...,Tn ] (τi , x) = αβj C˜j (f )(x)|βj =τj −T (x)

" αβ1 C˜1 ◦ · · · ◦ αβn C˜n (f )(x)|βj =τj −T (x) =

∞ #

1 S˜k1 ◦ · · · ◦ S˜nkn (f )(x) (τ1 −T1 (x))k1 · · · (τn −Tn (x))kn(II.6.7) k1 ! · · · kn ! 1

k1 ,...,kn =0

where S˜j : C ∞ (M) → C ∞ (M)

˜ h &→ {h, A−1 lj Cl } = {h, Cj }

.

(II.6.8)

Indeed formulas (II.6.7) and (II.5.36) for the power series of F˜[f,Ti ] and F[f ; Ti ] respectively coincide on the constraint hypersurface. More generally, assume that one has found a set of constraints {Cˆj , j = 1, . . . , n} which has the property, that it evolves the clock variables linearly on the constraint hypersurface, i.e. {Tk Cˆj } = δjk + λjkmCˆm

(II.6.9)

where λjkm are smooth phase space functions13 . For such a constraint set one can prove that it is weakly Abelian: One either uses that the matrix Aˆkj := {Tk , Cˆj } coincides weakly with the identity matrix and hence also its inverse coincides weakly with the identity matrix. Therefore ˜ " ˆ ˆ ˆ ˆ Cˆj := Aˆ−1 lj Cl = Cj + λjkl Ck Cl

(II.6.10)

That a function g which vanishes on the constraint hypersurface can always be written as λjkm Cˆm is proven in [7]. 13

39

˜ for certain phase space functions λ"jkl . Since we know that Cˆj are weakly Abelian, we can conclude that the Cˆj ’s are also weakly Abelian. An alternative proof proceeds by calculating {Cˆj , {Cˆi , Tk }} first directly and then by using the Jacobi identity. One then compares the two results and concludes that the structure functions fˆijk defined by {Cˆi , Cˆj } = fˆijk Cˆk vanish weakly. On the other hand we can also argue that if there exists a set of phase space variables {T1 , . . . , Tn } such that (II.6.9) holds with respect to a constraint set {Cˆ1 , . . . , Cˆn }, then the constraints have to be weakly Abelian. This provides a method to show that a certain constraint set is weakly Abelian without calculating the Poisson brackets between the constraints explicitely. Also, if one has two constraint sets {Cˆj }nj=1 and {Cˆj" }nj=1 satisfying {Tk , Cˆj } " δkj

{Tk , Cˆj" } " δkj

and

then the constraint sets are connected by Cˆj" = Djm Cˆm

with

Hence we will have

Djm " δjm

,

.

(II.6.11)

(II.6.12)

{f, Cˆk } " {f, Cˆk" }

(II.6.13)

Πj = Ej (Tk , Ym ) .

(II.6.14)

for an arbitrary phase space function f . This means, that on the constraint hypersurface Cˆk and Cˆk" generate the same gauge transformation. The arguments made above show that in the formal power series of a complete observable (II.6.7) one can replace the constraints C˜j by the constraints Cˆj as long as the latter have the property (II.6.9) with respect to the clock variables. The resulting complete observable will at least weakly coincide with the original (II.6.7) (or(II.5.36)) one. Applying the inverse of the matrix Akj = {Tk , Cj } to the constraints Cj provides one way to obtain a set of constraints satisfying (II.6.11). Another way is to construct (strongly) Abelian constraints in a way which is explained in [7]. We will repeat this construction here for completeness: To obtain Abelian constraints is locally always possible and proceeds in the following way. Assume that the clock variables commute14 . Then it is possible to use these clock variables Tj as a part of a new set of canonical coordinates, such that Πj are the momenta conjugated to Tj . Call the remaining new canonical coordinates Ym , m = 1, . . . 2p − 2n. If det(Akj ) = det({Tk , Cj }) )= 0 it is in principle possible to solve the constraints locally for the momenta Πj , that is the vanishing of the constraints is equivalent to Hence an equivalent set of constraints is given by Cˆj = Πj − Ej (Tk , Ym ) .

(II.6.15)

{Cˆi , Cˆj } = {Ei (Tk , Ym ), Ej (Tk , Ym )} −{ Ei (Tk , Ym ), Πj } −{ Πi , Ej (Tk , Ym )}

(II.6.16)

Obviously we have {Tk , Cˆj } = δjk (strongly) and one can also show that the Cˆj Poisson commute strongly: Since Cˆj are first class constraints their Poisson brackets vanishes on the constraint hypersurface. But

14

In [7] the Abelianization is obtained in a slightly more general setting, in which the Tj ’s are just part of a new set of canonical coordinates, that is the Tj ’s may also contain conjugated variables: {Tj , Tj ! } = ±1 or {Tj , Tj ! } = 0.

40

does not depend on the values of the momenta Πk . Hence the Poisson bracket (II.6.16) has to vanish not only on the constraint hypersurface C but on the whole phase space M. As we will see in the next section, such a constraint set {Cˆj , j = 1, . . . , n} was used before by Kuchaˇr in the construction of the Bubble-Time Canonical Formalism [21]. However to solve the constraints for some of the momenta may be very complicated and globally even not possible. For instance in relativistic systems the constraints are typically quadratic in the momenta. If one attempts to solve the constraints for some of these momenta, the solutions will contain roots. The signs for these roots may not be determined, moreover it may happen that a quantity inside the root assumes negative values. This reflects the fact that the constraint hypersurface may have a complicated structure – different sign choices may correspond to different sheets of the constraint hypersurface, as is the case for the mass shell constraint of the relativistic particle. Whereas different signs reflect the possibility for several solutions, negative values inside a root reflect the possibility that there does not exist a solution for a certain range of values of the remaining phase space variables.

II.7

Bubble Time Formalism

In this section we will explain how the theory of partial observables connects to the Bubble Time Canonical Formalism introduced by Kuchaˇr in [21]. There the Bubble Time Formalism was introduced for General Relativity, i.e. a field theory with a totally constrained Hamiltonian. However it is straightforward to apply this formalism to other first class constrained systems. The Bubble Time Formalism is mainly motivated by the example of cylindrically symmetric gravitational waves with one polarization [23]. For this system one can find a canonical transformation, such that the constraints are linear in a subset of the new momenta. Such a system is then structurally similar to the example of the parametrized particle in Part I. Given a system on a 2p-dimensional phase space M with n constraints the Bubble Time Formalism starts from the following assumption: There exists canonical coordinates (Q1 , . . . , Q(p−n) , P1 , . . . , P(p−n) ; T1 , . . . , Tn , Π1 , . . . , Πn ) such that (i) the pairs (Ti , Πi ) and (Qk , Pk ) are canonically conjugate; (ii) the determinant of ({Tj , Ci })nj,i=1 does not vanish on the constraint hypersurface; (iii) the constraint equations Cj = 0; j = 1, . . . , n can be solved for the momenta Πk ; k = 1, . . . , n. Since according to the last assumptions the constraints can be solved for the momenta Πi = hi (Qk , Pk , Tj ) such that Cj (Qk , Pk , Ti , Πh = Eh (Qk , Pk , Tj )) ≡ 0

(II.7.1)

we can replace the set {Cj , j = 1, . . . , n} by an equivalent set of constraints defined by Cˆj = Πj − Ej (Qk , Pk , Ti ) .

(II.7.2)

As explained in the last section the flow generated by these constraints commutes (strongly) and the clock variables Tj evolve linearly with respect to these flows: αβj Cˆj (Th ) = Th + βh 41

.

(II.7.3)

Consider the flow of the constraint H := βj Cˆj . For either Xl = Ql or Xl = Pl we have the equations d t α (Xl ) = βj αtH ({Xl , Cˆj }) = βj αtH ({Xl , −Ej }) dt H d t α (Tk ) = βj αtH ({Tk , Cˆj }) = βk . dt H

(II.7.4) (II.7.5)

That is we can write d d t α (Xl ) = αtH ({Xl , −Ek }) αtH (Tk ) dt H dt

(II.7.6)

.

In [21] Kuchaˇr introduces a new Poisson bracket [·, ·], which is defined by using (Qk , Pk ), k = 1, . . . , p − n as a complete set of canonical variables. One can then replace {Xl , −Ej } with [Xl , −Ej ]. Varying the βj in H = βj Cˆj one can evolve the phase space functions Xl from the initial data, Qk , Pk , Ti (fulfilling the constraint equations) until one reaches Th = τh , h = 1, . . . , n. By definition the result will be equal to our complete observables F[Xl ; Th ] (τh , x = (Qk , Pk , Ti , Πi = Ei )) = αβj Cˆj (Xk )(x)|βj =Tj (x)−τj

.

(II.7.7)

Hence the Bubble Time Formalism can be reinterpreted using the concepts of complete and partial observables. From equation (II.7.6) one can again conclude that the complete observables have to satisfy the partial differential equations ∂ F = F[{Xl ,−Ej }; Th ] ∂τj [Xl ; Th ]

.

(II.7.8)

As alreday mentioned one can replace the brackets {·, ·} in (II.7.8) by the new Poisson brackets [·, ·] and this is how equation (II.7.8) appears in [21]. (Also there, the integrability conditions for (II.7.8) are proven.) However if one replaces there Xl by a function f , equation (II.7.8) is in general only valid (with the new Poisson brackets) if f does not depend on Tk and Πk . Otherwise one has to use the function {f, Cˆl } instead of [f, −El ]. On the other hand since F[f ; Ti ] (τi , x) gives the value of f at that point in the gauge orbit of x (on the constraint hypersurface) at which the functions Ti give the values τi we have F[f ; Ti ] (τi , ·) " F[f ! ,Ti ] (τi , ·)

(II.7.9)

where f " (Qk , Pk ) = f (Qk , Pk , Tj = τj , Πj = Ej (Qk , Pk , τh )). Hence it seems not to be necessary to consider functions which depend on Ti or Πi . However as we will show in II.9 one has then to keep in mind that there might appear an external τ –dependence of the observables. This is analogous to systems with a time–dependent Hamiltonian. We will clarify this issue in section II.9, where we will consider generalizations to equation (II.7.8).

II.8

Gauge Fixing and Dirac Brackets

The complete observables F[f,Tj ] (τj , ·) can also be understood using Tj (x) − τj " 0, j = 1, . . . , n as gauge fixings. For this the clock variables Tj have to satisfy the following conditions (taken from [7]), describing good gauges: 42

(i) The chosen gauge must be accessible from an arbitrary point on the constraint hypersurface. That is to each point x on the constraint hypersurface there exists a flow of the form αβ 1 Cj ◦ · · · ◦ αβ l Cj with l arbitrary, that maps x to a point y satisfying Ti (y) = τi for j j i = 1, . . . , n. (ii) The conditions Ti (x) " τi must fix the gauge completely, that is there is no gauge transformation other than the identity, that preserves Ti (x) = τi . Locally this means that εj {Tk − τk , Cj } = εj Akj " 0

for

k = 1, . . . , n

(II.8.1)

have the unique solutions εj = 0, j = 1, . . . , n. Hence det(Ajk ) has to be non-vanishing on the constraint hypersurface. Obviously we have that F[f ; Ti ] (τj , x)|Tj (x)=τj " f (x)|Tj (x)=τj

(II.8.2)

,

that is the partial observable f and the associated complete observable F[f,Tj ] (τj , x) coincide on the hypersurface defined by the constraints and the gauge conditions. Now, given a gauge satisfying the above mentioned conditions, for each gauge restricted phase space function f|Tj =τj one can find a gauge invariant extension F away from the gauge conditions, which is uniquely defined at least on the constraint hypersurface (see [7, 37] where this idea is expressed). This extension is (weakly) unique, since through each point x of the constraint hypersurface there is given a gauge orbit Gx and on each gauge orbit Gx there exits exactly one point y with Ti (y) = τi , i.e. which satisfies the gauge conditions. We will call this point y = T−1 x (τj ). Since the extension has to be gauge invariant, it has to be constant along each gauge orbit Gx . The value of F on such a gauge orbit is determined by the gauge restriction f, that is F (x) = f (T−1 x (τj ))

(II.8.3)

.

Again, we recognize our complete observable F (x) " F[f ; Ti ] (τi , x). Hence complete observables are simply gauge invariant extensions of gauge restricted functions. This also shows that F[f ; Ti ] (τj , x) " F[g; Tj ] (τj , x)

(II.8.4)

if the gauge restrictions of f and g coincide because the gauge invariant extension is unique if the above assumptions hold. As is well known, the symplectic structure induced on the hypersurface {Cj = 0, Ti = τi ; j, i = 1, . . . , n} by the symplectic structure of the phase space M is given by the Dirac bracket {·, ·}∗ . Interestingly the Dirac brackets also appear if one calculates the Poisson bracket of two complete observables F[f ; Ti ] (τi , ·) and F[g; Ti ] (τi , ·). (This has been stated in [7] but an explicit proof is not available in the literature to the best knowledge of the author.) To explain this in more detail, we will first define the Dirac bracket (following [7]). Consider the matrix (Bjk )2n j,k=1 defined by Bjk := {χj , χk }

with

2 Cj χj := (Tj−n − τj−n ) 43

for 1 ≤ j ≤ n for n < j ≤ 2n

.

(II.8.5)

On the constraint hypersurface the inverse of Bjk is given by (B −1 )hl " (B

−1

n #

i,m=1

−1 A−1 hi Alm {Ti , Tm }

(B −1 )h(n+l)) " A−1 hl

)(h+n)l " −A−1 lh

(B −1 )(h+n)(n+l) " 0

(II.8.6)

where 1 ≤ h, l ≤ n and Ahl = {Ch , Tl }. Then the Dirac bracket is given by {f, g}∗ = {f, g} − " {f, g} −

2n #

{f, χj }(B −1 )jk {χk , g}

j,k=1 n #

{f, C˜h }{Th , Tl }{C˜l , g} −

h,l=1

n n # # ˜ {f, Ch }{Th , g} + {f, Th }{C˜h , g} . h=1

h=1

(II.8.7)

However there is an alternative way to define the Dirac bracket on the gauge fixed surface (see [7]): For f being an arbitrary phase space function define f ∗ = f − {f, χj }(B −1 )jk χk

(II.8.8)

.

Then we have that {f ∗ , Ci }|Tl =τl " 0 and

{f ∗ , Ti }|Tl =τl " 0 .

(II.8.9)

The first of these equations means that f ∗ Poisson commutes with the constraints to the zeroth order in (Tj −τj ). Moreover f ∗ and F[f ; Ti ] (τi , x) coincide weakly on the gauge fixed surface, hence f ∗ and F[f ; Ti ] (τi , x) coincide up to first order in (Tj − τj ). (The functions have the same zeroth order and commute with the constraints at least to the zeroth order. One can also calculate (II.8.8) explicitly and compare it to the power series of F[f ; Ti ] in (II.5.41).) From this one can conclude that {f ∗ , g}|Tl =τl " {F[f ; Ti ] (τi , ·), g}|Tl =τl (II.8.10)

for an arbitrary phase space function g. Now it is straightforward to compute that

{f ∗ , g}|Tl =τl " {f ∗ , g∗ }|Tl =τl " {f, g}∗|Tl =τl

.

(II.8.11)

Therefore the Poisson and Dirac bracket between two complete observables on the gauge fixed surface is given by {F[f ; Ti ] (τi , ·), F[g; Ti ] (τi , ·)}|Tl =τl " {f, g}∗|Tl =τl " {F[f ; Ti ] (τi , ·), F[g; Ti ] (τi , ·)}∗|Tl =τl

. (II.8.12)

Because we know that the Poisson bracket of two gauge invariant functions is again gauge invariant, we can conclude that the Poisson bracket of two complete observables associated to the functions f and g respectively is the gauge invariant extension of the gauge restricted result, that is the complete observable associated to {f, g}∗ . Moreover, as one can directly verify using formula (II.8.7), the Dirac bracket of two gauge invariant functions is weakly equal to their Poisson bracket. Hence {F[f ; Ti ] (τi , ·), F[g; Ti ] (τi , ·)}(x) " {F[f ; Ti ] (τi , ·), F[g,Ti ] (τi , ·)}∗ (x) " F[{f,g}∗ ; Ti ] (τi , x) 44

. (II.8.13)

That is the map F[Ti ] (τi ) defined by F[Ti ] (τi ) : (C ∞ (M)/I(M), {·, ·}∗ ) → (D(M)/I(M), {·, ·}) &→ F[f ; Ti ] (τi , ·)

f

(II.8.14)

where D((M)) is the space of gauge invariant functions on M and I(M) the ideal of smooth functions vanishing on the constraint hypersurface, is a Poisson algebra homomorphism. ( That F[Ti ] (τi ) is a homomorphism with respect to multiplication and addition was proved in (II.5.14). The space C ∞ (M)/I(M) has a well defined Dirac bracket since {f, Ch }∗ = 0 for arbitrary phase space functions f . ) The appearence of the Dirac bracket in the Poisson bracket of two complete observables is natural since a complete observable F[f ; Ti ] (τi , ·) is determined by the values of f on the submanifold T := {Ti (x) = τi , Ci = 0 for i = 1, . . . , n}. That is, the complete observable is determined by the gauge restriction of f . Hence also the Poisson bracket of two complete observables associated to f and g respectiviely must be determined by the restriction of f and g to T . But the induced Poisson bracket on the submanifold T is given by the Dirac bracket (see [7]). The Dirac bracket appears also in the Reduced Phase Space Quantization where one first performs a gauge fixing and then quantizes. That is one has to represent the algebra of gauge fixed functions equipped with the Dirac bracket as (self–adjoint) operators on a Hilbert space. The aim of the Dirac quantization is to find the physical Hilbert space where the Dirac observables should be represented. The set of complete observables for a fixed choice of clock variables {Ti } and fixed parameters {τi } is, as already mentioned, an over–complete basis for the space of Dirac observables. Moreover we have that if two partial observables coincide on the gauge fixed surface {Ti = τi , i = 1, . . . , n}, then the associated complete observables coincide weakly. On the other hand because of (II.5.13) two complete observables will always differ if the two partial observables differ on the gauge restricted surface. Hence we have a one–to–one correspondence between gauge–fixed functions and complete observables. Also, as (II.8.14) shows, the symplectic structures for both algebras are isomorphic. Hence during Dirac quantization one has to represent the same algebra on the physical Hilbert space as in the Reduced Phase Space Quantization. Therefore both quantizations should give the same result (ignoring quantization ambiguities from other sources). However one should keep in mind that a Reduced Phase Space Quantization is only possible if one has found a good gauge as defined at the beginning of this section. Also, only in this case it is guaranteed that the complete observables are well defined on the whole constraint hypersurface and not multi–valued. But to find a good gauge or to verify that a certain gauge is a good gauge might be very complicated or even impossible.

II.9

τ –Generators

Next we want to define a non–trivial action of gauge transformations on the space of complete observables. This will generalize the idea of ‘evolving constants of motion’ (see [11]) to constraint systems with an arbitrary number of constraints. We will denote this action of αγj Cj by α ˆγj Cj : α ˆ γj Cj : D(M) ' F[f ;Ti ] (τi ; ·) &→ F[αγ

j Cj

(f );Ti ] (τi ; ·)

∈ D(M) .

(II.9.1)

Here it is crucial that the gauge transformation acts only on f and not simultaneously on the clock variables Ti . If that would be the case, then this gauge action would amount to the usual gauge action, i.e. act trivially on the space of complete observables. 45

Alternatively we can also define the gauge action by letting it act on the clock variables (and not on the partial observable f ): F[αγ

j Cj

(f );Ti ] (τi ; x)

= αβj Cj ◦ αγk Ck (f )(x)|αβ

= αβj! Cj (f )(x)|αβ ! C

j

j

= F[f ;(α(γ

j Cj )

j Cj

(Ti )(x)=τi

◦(αγk Ck )−1 (Ti )(x)=τi

)−1 (Ti )] (τi ; x)

(II.9.2)

where the second equation holds because if Bj = βj solves αβj Cj (Ti )(x) = τi then αβj! Cj = αBj Cj ◦ αγk Ck solves αβj! Cj ◦(α(γk Ck ) )−1 (Ti )(x) = τi . Therefore the action α ˆβj Cj of a gauge transformation on a complete observable changes the gauge restriction from Ti (x) = τi to (αβj Cj )−1 (Ti )(x) = τi . Moreover F[αγ

j Cj

(f );Ti ] (τi ; x)

= aγj Cj (f )(T−1 x (τi )) = F[f ;Ti ] (τi" ; x)

with

τi" = αγj Cj (Ti )(T−1 x (τl ))

(II.9.3)

.

If we choose to work with the constraints C˜j (see II.6.3) we obtain F[αγ

˜ (f );Ti ] j Cj

(τi ; x) " F[f ;Ti ] (τi + γi ; x) .

(II.9.4)

Hence we can conclude that for a Poisson bracket between complete observables with differing values of the parameters τi we have {F[f ;Ti ] (τi ; ·), F[g;Ti ] (τi" ; ·)} " F[{f,α(τ

∗ ! ˜ (g)} ;Ti ] k −τk )Ck

(τi ; ·)

.

(II.9.5)

Equation (II.9.3) shows that the gauge transformation act merely on the parameters τ in a complete observable. For instance if we rewrite a Dirac observable d as a complete observable as in (II.5.17) the complete observable does not depend on the parameters τi . Therefore the action of a gauge transformation on a complete observable associated to a Dirac observable is trivial. Hence it is also not surprising that the action of a gauge transformation respects the Poisson brackets between two complete observables: 5 6 5 6 {ˆ αγj Cj F[f ;Ti ] (τi ; ·) , α ˆ γj Cj F[g;Ti ] (τi ; ·) } = {F[αγj Cj (f );Ti ] (τi ; ·), F[αγj Cj (g);Ti ] (τi ; ·)} = {F[f ;Ti ] (τi" ; ·), F[g;Ti ] (τi" ; ·)} = F[{f,g}∗ ;Ti ] (τi" ; ·) = F[αγ

j Cj

5

({f,g}∗ );Ti ] (τi ; ·)

6 = α ˆ γj Cj {F[f ;Ti ] (τi ; ·), F[g;Ti ] (τi ; ·)}

(II.9.6)

where τi" is the same as in the last line in equation (II.9.3). Here it does not matter for which values of τi one calculates the Dirac bracket {f, g}∗ since according to formula (II.8.7) the Dirac bracket is independent from the choice of the τi ’s. Equation (II.9.6) means that the gauge action α ˆγj Cj is a symplectic automorphism on the space of complete observables. Therefore, as suggested in [38] one can ask for phase space 46

functions which generate this gauge action. In particular one would like to find phase space functions Hjτ such that ∂ ∂ F[αγ C˜ (f );Ti ] (τi ; x) " F (τi , x) " {F[f ; Tj ] (τi , ·), Hjτ }(x) j j ∂γj ∂τj [f ; Tj ]

(II.9.7)

where the first equation follows from (II.9.4). We will call such a phase space function Hjτ a τj –generator or as was done in [38] a physical Hamiltonian. Now on the left hand side of (II.9.7) we have a gauge invariant function and in the first slot in the Poisson bracket on the right hand side we have also a gauge invariant function. We will therefore assume that Hjτ is also a gauge invariant function and can hence be written as a complete observable associated to a partial observable hj : Hjτ " F[hj ; Ti ] (τk , ·)

(II.9.8)

.

Then we know that {F[f ; Ti ] (τk , ·), Hjτ } " F[{f,hj }∗ ; Ti ] (τk , ·)

.

(II.9.9)

Comparing with (II.9.7) we have to find functions hj which satisfy ∂ F (τk , ·) " F[{f,C˜j } ; Ti ] (τk , ·) " F[{f,hj }∗ , Ti ] (τk , ·) ∂τj [f ; Ti ]

(II.9.10)

that is {f, C˜j } " {f, hj }∗

(II.9.11)

for every phase space function f . But if we choose f = Tj because of the properties of the Dirac bracket (II.8.9) we will have {Tj , C˜j } " 1

but

{Tj , hj }∗|Ti =τi " 0 .

(II.9.12)

Hence we can never generate the τ -change in F[Tj ; Ti ] (τk , ·) ≡ τj in this manner. Anyway this would be impossible, since F[Tj ; Ti ] (τk , ·) is just a constant. But we can still try to find functions hj such that (II.9.11) holds for all phase space functions, which do not depend on the {Ti }. In the following we will assume that the clock variables {Ti } have vanishing Poisson brackets with each other. If this is the case one can always find a canonical transformation such that the {Ti }ni=1 are part of the new canonical configuration coordinates. The canonical momenta conjugated to the {Ti }ni=1 will be denoted by {Πi }ni=1 and the other canonical pairs by {(Qk , Pk )}p−n k=1 . Let us consider the Dirac bracket (II.8.7) of a new canonical variable, either Xk = Qk or Xk = Pk , with the momentum Πj conjugated to the clock variable Tj : {Xk , Πj }∗ " −{Xk , C˜j }

.

(II.9.13)

I.e. for functions f which depend only on the canonical coordinates {(Qk , Pk )}p−n k=1 the requirement (II.9.11) is satisfied if we choose hj = −Πj . For f = Πi we find {Πi , Πj }∗ " −{Πi , C˜j } − {C˜i , Πj } . 47

(II.9.14)

Hence we also have to exclude a dependence of f in (II.9.11) from the momenta {Πj }nj=1 . In summary we have for a function f which does not depend on the canonical coordinates {Ti , Πi }ni=1 ∂ F (τk , ·) " {F[f ; Ti ] (τk , ·), F[hj ; Ti ] (τk , ·)} " F[{f,hj }∗ ; Ti ] (τk , ·) ∂τj [f ; Ti ]

(II.9.15)

where hj = −Πj . Now we have the problem, that {f, hj }∗ may depend on the {Ti }ni=1 also if the function f does not. This will be the case if {{f, C˜j }, Πk } does not vanish for all k = 1, . . . , n. So we cannot express higher derivatives with respect to the parameters τ by just applying Poisson brackets. But this is similar to the situation one has in classical mechanics with a time–dependent Hamiltonian. There, the Poisson bracket of a time–independent phase space function with the Hamiltonian may be time–dependent. Therefore, to obtain the total time–derivative of such functions one adds the partial time derivative of the function to the Poisson bracket with the Hamiltonian. Also here, if we start with a function f which may be dependent on the {Ti }ni=1 (but not on the momenta {Πi }ni=1 ), we can write ∂ F (τk , ·) " F[{f,C˜j }; Ti ] (τk , ·) " {F[f ; Ti ] (τk , ·), F[hj ,Ti ] (τk , ·)} + F[ ∂ f ; Ti ] (τk , ·) (II.9.16) ∂Tj ∂τj [f ; Ti ] where hj = −Πj . It remains to discuss whether there might arise a dependence on the momenta {Πi }ni=1 under the application of the constraints C˜j to a function f which does not depend on the variables {Πi }ni=1 initially. To this end consider again the constraints {Cˆj }nj=1 of the form Cˆj = Πj − Ej (Qk , Pk , Ti )

(II.9.17)

defined in (II.6.15). At least locally this form of the constraints can always be obtained (if det(Ajk ) )= 0) by solving the constraints {Cj }nj=1 for the momenta Πj . Both sets of constraints {C˜j }nj=1 and {Cˆj }nj=1 satisfy the condition (II.6.11), i.e. they are weakly conjugated to the clock variables {Tk }nk=1 . Hence equation (II.6.13) holds that is we have {f, Cˆj } " {f, C˜j }

(II.9.18)

for an arbitrary phase space function f . Now if f does not depend on the momenta Πi , i.e. {f, Ti } = 0, this holds also for the left hand side of (II.9.18). The right hand side can therefore only depend on the momenta Πj through terms that vanish on the constraint hypersurface and hence are proportional to the constraints. These terms have to be discarded before applying formula (II.9.16) again. In (II.9.16) we concluded that F[hj ,Ti ] (τi , ·) with hj = −Πj generates the transformation in τj . Because of F[C˜j ,Ti ] (τi , ·) " 0 we can add to hj = −Πj any combination of constraints. Using the constraints Cˆj one arrives at hj = −Ej (Qk , Pk , Ti ). This form is also used in the Bubble Time formalism in section II.7. The brackets [·, ·] introduced there coincide with the Dirac bracket if applied to functions which do not depend on the momenta {Πm }nm=1 , as is the case for hj = −Ej (Qk , Pk , Ti ). 48

However the form hj = −Ej (Qk , Pk , Ti ) may be inconvenient because for relativistic systems it will typically involve square roots of functions which may also assume negative values. This comes about because one has to solve constraints which are usually quadratic in the momenta. This can be avoided if one allows for an explicit dependence of the hj on the momenta Πi . Note that the hj do not only depend on the choice of the set of clock variables but also on the momenta conjugated to the clock observables. For instance, if we have a canonical coordinate system with canonical pairs {(Ti , Πi )}ni=1 and {(Qk , Pk )}p−n k=1 we can easily define new canonical coordinates by altering two canonical pairs to T1" = T1

Π"1 = Π1 + P1

Q"1 = Q1 − T1

P1" = P1

(II.9.19)

and leaving the other pairs unchanged. Now the τ1 –generator for the first set of canonical coordinates is H1τ = −F[Π1 ;Ti ] (τ, ·) and for the second H " τ1 = F[Π1 +P1 ;Ti ] (τ, ·). These two functions will in general differ also on the constraint hypersurface. In particular one function may vanish on the constraint hypersurface and the other not. One will obtain vanishing or constant τ -generators if the canonical variables {(Qk , Pk )}p−n k=1 are already invariant with respect to transformations ˜ generated by Ck . Let us consider the question in which cases a dependence on the clock variables does not arise, if one applies C˜j to a function f , which is independent of the canonical coordinates {(Ti , Πi )}ni=1 . To this end we calculate {{f, C˜j }, Πk } = {{f, Πk }, Cj } + {{Πk , C˜j }, f } = {{Πk , C˜j }, f } .

(II.9.20)

Therefore it is guaranteed that a clock variable dependence does not arise, if all the momenta Πk commute with the constraints C˜j , i.e. if all the Πk are Dirac observables. Regarding the last term in (II.9.20), one may think that it is also necessary that the Πk are strong Dirac observables with respect to the constraintset {C˜j }nj=1 . However if the Πk are weak Dirac observables, that is {Πk , C˜j } = µkjmC˜m for some phase space functions µkjm, we can write {{Πk , C˜j }, Th } = {µkjm C˜m , Th } = µkjm (−δmh + O(C)) + {µkjm , Th }C˜m = {{Πk , Th }, C˜j } + {{Th , C˜j }, Πk } = 0 + {O(C), Πk } = O(C)

(II.9.21)

Comparing the right hand side of the first line with the last line we see that µkjm has to be at least of order O(C). So {{f, C˜j }, Πk } vanishes at least weakly. If the momentum Πk is a Dirac observables, then the τk –generator Hkτ := F[−Πk ,Ti ] (τi , ·) does not depend on the parameters τi , i = 1, . . . , n. This is comparable to a time–independent Hamiltonian in classical mechanics. Next we will show that one can always find a set of clock variables {Tk }nk=1 and conjugated momenta such that m = min(n, p − n) of the n τk –generators do not depend on the parameters τi , i = 1, . . . , n. Here 2p is the dimension of the phase space. However this is a local statement, i.e. the clock variables may be defined only locally in phase space. For the proof of this 49

statement we will employ the fact [10], that at least locally one can always find a canonical coordinate chart and a strongly Abelian form of the constraints such that n canonical pairs are of the form {φj , Cˆj }nj=1 , i.e. n momenta coincide with the strongly Abelian constraints Cˆj . The remaining (p − n) pairs {Q"k , Pk" }p−n k=1 are Dirac observables. We then perform a canonical coordinate transformation. For j = 1, . . . , m we define new canonical coordinates by 1 Tj = √ (φj + Q"j ) 2 1 Qj = √ (Q"j − φj ) 2

1 Πj = √ (Cˆj + Pj" ) 2 1 Pj = √ (Pj" − Cˆj ) . 2

(II.9.22)

The remaining canonical pairs are left unchangend. If m = n we already have a complete set of clock variables {Tj }nj=1 . If m = p − n < n we complete the set {Tj }p−n j=1 with the clock variables Tk := φk , k = p − n + 1, . . . , n. Now, as explained above, the τ –generators are given by the complete observables associated to minus the momenta conjugated to the clock variables. Hence we have for the first m τ – generators 1 1 Hjτ = F[−Πj ;Ti ] (τ, ·) " − √ F[Pj! ;Ti ] (τ, ·) " − √ Pj" 2 2

for j ≤ min(n, p − n) . (II.9.23)

The second equation holds because the complete observable associated to a constraint vanishes weakly and the last equation holds because Pj" is a Dirac observable. Equation (II.9.23) gives m τ –generators, that do not depend on the parameters {τi }ni=1 because these are given by complete observables associated to Dirac observables. Moreover these τ –generators Poisson commute. If n − p < n the remaining τ –generators vanish weakly Hkτ = F[−Ck ;Ti ] (τ, ·) " 0

for

n−ph= dν(y) ui (y)vi (y) . (III.2.2) Y

i∈I

The Master Constraint is then defined as M(m) =

1 < C(m), K(m) · C(m) >h 2

(III.2.3)

Here m ∈ M denotes a phase space point and K : m &→ K(m) is a phase space function with values in the set of strictly positive linear operators on h. The vanishing of the Master Constraint is then equivalent to the vanishing of the constraints Ci (y) for all i ∈ I and almost all y ∈ Y. As we will see in the examples in section (III.5) it is necessary to introduce the operators K in order to make M convergent. The operator valued function K and the measure ν should be chosen in such a way, that the Master Constraint M becomes a differentiable function on the phase space M. Now we come to the quantum theory. We will assume that some kinematical Hilbert space HKin has been constructed and that this Hilbert space is seperable. The latter assumption is needed for the Direct Integral Decomposition. Furthermore the quantization of M should result 7 If it is not possible to obtain such an operator with a in a positive, self–adjoint operator M. given choice of ν, K one has to change these ingredients. As already mentioned we want to define the physical Hilbert space as the (generalized) eigenspace to the spectral value zero of the Master Constraint Operator. Hence zero should be 7 of M. 7 If this is not the case we substract from M 7 the finite, positive in the spectrum σ(M) 7 7 7 number λ0 := inf(σ(M)) and redefine M by M −λ0 idHKin . The number λ0 should vanish in 77

the classical limit ! → 0 so that the redefined Master Constraint Operator has still the correct classical limit. 7 on HKin we can completely solve the quantum constraint equations Given such an operator M and explicitly provide the physical Hilbert space with its physical inner product. The main idea is to apply a Direct Integral Decomposition of the kinematical Hilbert space. That is, the kinematical Hilbert space HKin is unitarily equivalent to a direct integral ! ⊕ ⊕ ∼ HKin = dµ(λ) HKin (λ) (III.2.4) R+

⊕ of Hilbert spaces where µ is a spectral measure and HKin (λ) is a separable Hilbert space with 7 inner product induced from HKin . The action of M induces an action on the Hilbert spaces ⊕ ⊕ HKin (λ) and this action is given by multiplication with λ. Hence HKin (λ) can be seen as the generalized eigenspace to the spectral value λ. The physical Hilbert space can then be defined as ⊕ HP hys = HKin (0) . (III.2.5)

In the following we will explain how one can obtain a Direct Integral Decomposition. The first example in section III.3 shows nicely how the method works, so one might consult this example for a first demonstration of the technique.

III.2.1

Direct Integral Decomposition

The Direct Integral Decomposition follows from the spectral theory for self–adjoint operators. Let us therefore recall the spectral theorem for such operators, [35]: Theorem III.2.1. Let a be a self–adjoint operator on a Hilbert space H. Then there exists a projection valued measure E on the measurable space (R, BBorel ) such that ! a= λ dE(λ) (III.2.6) R

where the domain of integration can be restricted to the spectrum σ(a). Here BBorel is the set of Borel sets in R. For B ∈ BBorel one can define E(B) which satisfies the projection property E(B)2 = E(B). Given the projection valued measure E we can define the spectral measures µΩ associated to a normalized vector Ω ∈ H by ! µΩ (B) :=< Ω, E(B)Ω >H =: dµΩ (x) . (III.2.7) B

Also the measure µ in (III.2.4) is a spectral measure associated to a certain normal vector Ωm . This vector has to be maximal in a certain sense, in order that the measure µ has support on the whole spectrum of the operator in question. The sole invariant under unitary equivalence of a seperable Hilbert space is its dimension d(H) ∈ N ∪ ∞. For the construction of (III.2.4) we have therefore to find the dimensions of the 7 at the spectral Hilbert spaces HKin (λ)⊕ . This dimension is defined to be the multiplicity of M 78

7 the multiplicity of this eigenvalue can be easily defined value λ. If λ is a proper eigenvalue of M as the dimension of E({λ})HKin which is the eigenspace for this eigenvalue. However if λ is in the continuous part of the spectrum E({λ})Hkin is zero-dimensional, because for all the spectral measures µΩ the point λ has zero measure. With the following procedure one can determine the multiplicity of a spectral value for all types of spectra. It serves also for the construction of the unitary equivalence in the decomposition (III.2.4). Firstly we have to find a decomposition of the kinematical Hilbert space HKin into cyclic subspaces #⊕ HKin ∼ Hi (III.2.8) = i∈J

where J is some discrete index set. Cyclicity of Hi means that Hi is generated by a vector Ωi in the sense that20 Hi = H(Ωi ) := clos(span{E(B)Ωi , B ∈ BBorel }) .

(III.2.9)

7 on such a cyclic subspace is either one or zero. The multiplicity of M A cyclic decomposition can always be obtained in the following way: Begin with an arbitrary basis {φj }j of HKin , and consider the subspace H(φ1 ). If the latter is a true subspace in HKin then there exist a minimal k, such that the projection P1⊥ (φk ) of φk on 7 leaves H(φ1 ) invariant the orthogonal complement H(φ1 )⊥ of H(φ1 ) in H is non–zero. Since M ⊥ and is selfadjoint it leaves also H(φ1 ) invariant, i.e. it restricts to a selfadjoint operator on 7 on H(φ1 )⊥ , which has a basis H(φ1 )⊥ . One can therefore take as new data the restriction of M {P1⊥ (φj )}j≥k and iterate the above procedure. One ends up with a finite or+ countable (since . ⊥ {φj }j is contable) set of orthonormal vectors {Ω1 := φ1 / ||φ1 ||, Ω2 := P1 (φk )/ ||P1⊥ (φk )||, . . .} which fulfills the above requirements by construction. We now define the spectral measures µi (λ) =< Ωi , E((−∞, λ])Ωi >HKin (III.2.10) ( and choose a sequence {αi }i∈J with i αi = 1 and αi > 0 ∀i. Since E((−∞, ∞)) = 1H is the identity in H we have µi (∞) = 1. The spectral measure µ(λ) which will appear in the Direct Integral Decomposition is then given by # µ(λ) = αi µi (λ) . (III.2.11) i

(

The requirement i αi = 1 ensures that µ(∞) = 1, i.e. µ(λ) = µ((−∞, λ]) is a finite measure on R. Whereas the individual measures µi do not need to have support on the whole spectrum 7 this is the case for the measure µ. of M Furthermore we need the Radon–Nikodym derivatives ρi of µi with respect to µ, which are defined by the equation ! µi (B) = χB (λ)ρi (λ)dµ(λ) (III.2.12) R

b i.e. in the domain of M b n for all n ∈ N, this condition can be replaced by If Ωi is a C ∞ –vector for M, n b Ωi , n ∈ N}). Hi = clos(span{M 20

79

which has to hold for any µ–measurable set B ⊂ R. Here χB is the characteristic function of B. Because µi is absolutely continuous with respect to µ and both are finite measures, the Radon–Nikodym derivative ρi will always exist and be a non–negative L1 (R, dµ) function. Hence it is just determined µ–almost everywhere. We will choose some non–negative representative, also to be called ρi . 7 one defines the multiplicity N (λ) ∈ N ∪ ∞ of λ as the number of indices Now for λ ∈ σ(M) 7 → N ∪ ∞ is also only defined µ–almost i ∈ J for which ρi (λ) > 0. The function N : σ(M) everywhere. ⊕ The dimension of the induced Hilbert spaces HKin (λ) is given by the multiplicity N (λ) of 7 These Hilbert spaces are given by M. ⊕ HKin (λ) = clos(span{ei (λ)|i ∈ J and ρi (λ) > 0})

(III.2.13)

where {ei (λ)|i ∈ J and ρi (λ) > 0} is an ortho–normal basis. This determines an inner product ⊕ in HKin (λ). In order to define the unitary equivalence in (III.2.4) we note that because of (III.2.9) any vector ψ ∈ HKin can be represented as21 # # ψ= aik E(Bik )Ωi . (III.2.14) i∈J Bik ∈BBorel

Here {Bik }k∈N is a family of Borel sets. In this way we can write ! ⊕ V : HKin → HKin (λ)dµ(λ) R # # ˜ ψ= aik E(Bik )Ωi &→ (ψ(λ)) b λ∈σ(M)

(III.2.15)

i∈J Bik ∈BBorel

where

⊕ ˜ HKin (λ) ' ψ(λ) :=

#. # ρi (λ) aik χBik (λ)ei (λ) i∈J

.

(III.2.16)

k

Here χB is the characteristic function of the Borel set B. This ends our construction of the Direct Integral Decomposition. Now the question arises to which extent this construction is unique. For instance the number of cyclic subspaces is not determined. But if one has two Direct Integral Decompositions arrived at by for instance a different choice of cyclic systems these Direct Integral Decompositions are always unitarily equivalent22 . The multiplicity function N (λ) is uniquely defined µ–almost everywhere. Hence it is only uniquely defined at one point λ0 if this point is not a null set of µ, i.e. if λ0 is a proper eigenvalue. If λ0 is also in the continuous spectrum one can show that this does not have an influence on the multiplicity of the point λ0 . That is, the contribution from the continuous spectrum is suppressed and the multiplicity N (λ0 ) is given by the dimension of the proper eigenspace. b a vector ψ is decomposable as ψ = P b If the Ωi are C ∞ –vectors for M, i∈J ψi (M)Ωi where ψi are µ–measurable P b = b functions. In (III.2.14) we have ψi (M) Bik ∈BBorel aik E(Bik ) where E(·) are the spectral projectors of M. 22 i.e. there exist a unitary operator U = (U (λ))λ∈σ(M) between these decompositions. The operators U (λ) are b 21

⊕

! unitary operators between H⊕ Kin (λ) and H Kin (λ) for µ–almost all λ’s.

80

This can be understood if one realizes that the contribution from the continuous spectrum is only determined almost everywhere with respect to the Lebesgue measure. However since λ0 has non–vanishing µ–measure, N (λ0 ) should be uniquely determined and the only way this is possible is to set the contribution from the continuous spectrum to zero. However one may be also interested in the contribution from the continuous spectrum. Therefore, before we perform the Direct Integral Decomposition we divide the kinematical Hilbert 7 has uniformly either pure point, absolutely continuous or space into subspaces on which M continuous singular spectrum: pp ac cs HKin = HKin ⊕ HKin ⊕ HKin

(III.2.17)

.

7 can be restrited to a positive self–adjoint operator on each of This subdivision is unique and M these subspaces. We can therefore perform the Direct Integral Decomposition on each of these subspaces seperately. We alter the definition of the physical Hilbert space from (III.2.5) to ⊕

pp ac ⊕ cs ⊕ Hphys = Hkin (0) ⊕ Hkin (0) ⊕ Hkin (0)

.

(III.2.18)

⊕

pp Now, whereas Hkin (0) is uniquely defined the other parts may depend on the choice of ∗∗ the cyclic system {Ω∗∗ } i i∈J , ∗∗ = ac, cs and on the representatives ρi for the Radon–Nykodym derivatives. To minimize this dependence one can choose for the cyclic basis one with the following property: Choose J such that J = 1, 2, . . . , M with M either finite or equal to infinity. Then we require that µ1 ≥ µ2 ≥ . . . where µ ≥ ν means that µ is absolutely continuous with respect to ν. Such a cyclic system always exists. As we will see in the next section the choice of the representatives ρi influences the representation of the Dirac observables on the resulting physical Hilbert space. The choices should be made in such a way that a complete23 subalgebra of the algebra of Dirac observables is irreducibly and self–adjointly represented on the physical Hilbert space. For practical reasons we will adopt the prescription that for the representatives ρ∗∗ i , ∗∗ = ac, cs of the Radon-Nykodym derivatives one should choose a function that is continuous at λ = 0 from the right if such a representative exists. So far this was always the case in the examples. Next we will discuss the representation of Dirac observables.

III.2.2

Representation of Dirac Observables

Firstly we have to find a criterion for characterizing Dirac observables with the help of the Master Constraint M. The problem is, that because M is quadratic in the constraints, all phase space functions f poisson commute weakly with M. However, a short calculation reveals, that {F, {F, M}}| M=0 = 0

(III.2.19)

is equivalent to the statement that F poisson commutes with (almost) all constraints on the constraint surface and hence is a weak Dirac observable. Equation (III.2.19) is satisfied, if we have {F, M} = 0 23

i.e. this subset should seperate the points of the reduced phase space

81

(III.2.20)

on the whole phase space. We will call phase space functions F which fulfill equation (III.2.20) 7 Phase space functions which satisfy (III.2.19) will strong Dirac observables (with respect to M). be called weak Dirac observables. In the quantum theory condition (III.2.20) translates into 7 =0 [Fˆ , M]

(III.2.21)

,

which has to hold on the kinematical Hilbert space HKin . We will come back to the quantization of the condition for weak Dirac observables (III.2.19) at the end of this section and consider now the representation of strong Dirac observables on the physical Hilbert space. To avoid domain questions for unbounded operators we will say that two self–adjoint opˆ commute, if the associated spectral projection operators E ˆ (B) and E b (B " ) erators Fˆ and M F M commute for all Borel sets B, B " ⊂ R. If this is the case, then one can show that Fˆ preserves the subdivision of the kinematical 7 That is we have Hilbert space according to the spectral types of M. ∗∗ ∗∗ EFˆ (B " )HKin ⊂ HKin

(III.2.22)

for ∗∗ = pp, ac, sc and every Borel set B " ⊂ R. Hence we can consider the restriction Fˆ ∗∗ of ∗∗ . These are unitarily equivalent to a Direct Integral Hilbert Fˆ on each of the subspaces HKin space, that is we have unitary operators V ∗∗ ! ∗∗ ∗∗ ⊕ V ∗∗ : HKin → dµ∗∗ (λ)HKin (λ) (III.2.23) b σ(M)

∗∗ . Now we can consider the operator where we just apply (III.2.15) to each of the subspaces HKin V ∗∗ Fˆ ∗∗ (V ∗∗ )−1 on such a Direct Integral Hilbert space. This operator will preserve the Hilbert ∗∗ ⊕ (λ) in the Direct Integral, that is it decomposes as spaces HKin

V ∗∗ Fˆ ∗∗ (V ∗∗ )−1 = (Fˆ ∗∗ (λ))|λ∈σ(M) b

(III.2.24)

∗∗ ⊕ (λ) for µ∗∗ -almost all λ’s. The decomposition where Fˆ ∗∗ (λ) is a self–adjoint operator on HKin (III.2.24) is unique up to µ∗∗ –null sets. Moreover if Fˆ is bounded then this holds also for Fˆ ∗∗ (λ) (for µ∗∗ -almost all λ’s). This means that strong Dirac observables Fˆ can be represented on the physical Hilbert space pp ⊕ ac ⊕ (0) ⊕ Hcs ⊕ (0) by Hphys = HKin (0) ⊕ HKin Kin

Fˆ (0) := Fˆ pp (0) ⊕ Fˆ ac (0) ⊕ Fˆ cs (0)

.

(III.2.25)

But we can also conclude that if we just consider strong Dirac observables, these will preserve ∗∗ ⊕ (0) and hence we will get superselection sectors. the subspaces HKin We will now derive the matrix elements of the operators F ∗∗ (0). In order to avoid domain questions we will assume that Fˆ is bounded 24 . In the following we will suppress the superindex ∗∗ but it should be understood that all formulas have to be applied to each case ∗∗ = pp, ac, sc separately. 24

If Fˆ is unbouded one can consider functions g such that g(Fˆ ) is bounded.

82

7 Hence we can Assume that {Ωm }m∈J is a cyclic system of C ∞ –vectors with respect to M. find µ–measurable functions Gmn such that # 7 n . Fˆ Ωm = Gmn (M)Ω (III.2.26) n∈J

Remember that a vector ψ is decomposed into ψ=

# i∈J

V

7 i ψi (M)Ω

&→

˜ (ψ(λ)) b = λ∈σ(M)

/ #.

0

ρi (λ)ψi (λ)ei (λ)

i∈J

.

Applying this decomposition to Fˆ ψ " and ψ for arbitrary ψ, ψ " ∈ HKin we will get ! # # " ˆ < ψ, F ψ >= dµ(λ) ρi (λ)ψi (λ) Gji (λ)ψj" (λ) b σ(M)

(III.2.27)

b λ∈σ(M)

(III.2.28)

j∈J

i∈M (λ)

where M (λ) is the subset M (λ) = {j ∈ J |ρj (λ) > 0} of the index set J . On the other hand we can first decompose ψ " and then apply the decomposed operator ∼ ˆ F = (Fˆ (λ))λ∈σ(M) b . Thus we arrive at < ψ, Fˆ ψ >= "

!

b σ(M)

dµ(λ)

#

j,i∈M (λ)

+ ρi (λ)ρj (λ)ψi (λ)Fji (λ)ψj" (λ)

where we defined the matrix elements Fji (λ) for j, i ∈ M (λ) by # Fˆ (λ)ej (λ) = Fji (λ)ei (λ) .

(III.2.29)

(III.2.30)

i∈M (λ)

Hence we have µ–almost everywhere Fji (λ) = χM (λ) (j)χM (λ) (i)

8

ρi (λ) Gji (λ) ρj (λ)

(III.2.31)

where χM (λ) (j) is equal to one if j ∈ M (λ) and zero otherwise. One has to choose representatives ρj , Gji (in their class of µ–equivalent functions) in such a way that Fˆ (λ) becomes self–adjoint and bounded. This is possible µ–almost everywhere. In the examples we will often have the following situation. There is a set of unitary operators 7 on ˆ {Uk }k∈K (where K is some index set) which commute with the Master constraint operator M the kinematical Hilbert space. The cyclic system {ΩIi }I∈I,i∈J (I) , where I is an index set and J (I) an I-dependent index set, can be chosen in such a way that it decomposes into families {ΩIi }i∈J (I) labelled by I ∈ I such that in each family and for each i ∈ J (I) there exists an ˆk(I,i) satisfying unitary operator U ˆk(I,i) ΩIi0 (I) ΩIi = U

(III.2.32)

for some fixed index i0 (I) ∈ J (I). That is the I-family of cyclic vectors is generated by a set of unitary Dirac observables from the vector Ωi0 (I) . 83

The advantage of such a structure is that the spectral measures associated to the cyclic vectors of one family turn out to be equal to each other. Hence also the Radon–Nikodym derivatives will coincide. That is either a whole family of vectors {eIi (0)}i∈J (I) is included into the physical Hilbert space or this is the case for neither of the vectors eIi . Otherwise the unitary ˆk(Ii) are not represented properly on the physical Hilbert space. operators U Because the Radon–Nikodym derivatives ρIi for fixed I coincide, formula (III.2.31) for the matrix elements of strong Dirac observables F simplifies for matrix elements between vectors of one and the same family I. Moreover typically the functions G(Ii)(Ij) in (III.2.31) will be constant in λ. In this case the action of F on the basic vectors of the physical Hilbert space coincides with the action of F on the cyclic vectors. If there are strong Dirac observables F that have non–vanishing matrix elements between cyclic vectors from different families, that is G(Ii)(I ! i! ) )= 0 for I " )= I, this means that there exist strong Dirac observables permuting between the different families of cyclic vectors. Then it may 7 that maps Ωi0 (I) to Ωi0 (I ! ) . be possible to find also a unitary operator UII ! commuting with M That is one can reduce the number of families and consequently the amount of work necessary for the Direct Integral Decomposition. In the cases described above the freedom for the possible choices of the representatives for the Radon–Nikodym derivatives ρIi from their L1 –equivalence classes is reduced considerably, 7 In a more general case because one should choose these such that ρIi (λ) = ρIj (λ) for λ ∈ σ(M). the requirement of a self–adjoint (or unitary) representation of the algebra of Dirac observables will still impose severe constraints on the sets M (λ) = {i ∈ J |ρi > 0} of indices where the Radon–Nikodym derivatives do not vanish. To understand ( this consider three strong Dirac 7 observables Fˆ m , m = 1, 2, 3 with25 Fˆ 1 Fˆ 2 = Fˆ 3 and Fˆ m Ωi = j Gm ij (M)Ωj . It follows for the ( 2 1 3 measurable functions Gm ij that k Gik Gkj = Gij . The corresponding operators in the fibres are ( then given by Fˆ m (λ)ei (λ) = j Fijm (λ)ej (λ) where Fijm (λ) =

+

ρj /ρi (λ)χM (λ) (i)χM (λ) (j)Gm ij (λ)

.

Now a short calculation reveals 8 # ρj (λ) Fˆ 1 (λ)Fˆ 2 (λ) = χM (λ) (i)χM (λ) (j) G2ik (λ)G1kj (λ) ρi (λ)

(III.2.33)

(III.2.34)

k∈M (λ)

which coincides µ−a.e. with 8 8 # ρ (λ) ρn (λ) j χM (λ) (i)χM (λ) (j)G3ij (λ) = χM (λ) (i)χM (λ) (j) G2ik (λ)G1kj (λ) . Fˆ 3 (λ) = ρi (λ) ρm (λ) k (III.2.35) The point is now that (III.2.34) and (III.2.35) differ by the fact that in (III.2.34) the sum over k is restricted to the set M (λ) while in (III.2.35) it is not. Requiring that these two expressions coincide, at least at λ = 0, numerically rather than a.e. should impose restrictions on the choice of the representatives of ρj (0), Fijm (0) and all other Dirac observables. Intuitively, this requirement will amount to choosing representatives ρj (0) which are positive for a maximal 25

We consider as algebra operation multiplication and not the commutator because we have unitary operators in mind.

84

number of j so that we are not missing necessary terms while irreducibility will require to have a maximal number of the ρj (0) vanishing so that at least heuristically these two requirements have the tendency to restrict the freedom. In particular note that there may be operators commuting with the Master Constraint Operator whose classical counterparts vanish on the constraint hypersurface. We will encounter such operators in the first example.26 There we will see that despite the fact that the corresponding classical functions vanish on the constraint hypersurface, such operators might be helpful in the construction of the physical Hilbert space. The representatives for the Radon–Nikodym derivatives ρj should be chosen in such a way that such operators have a trivial representation on the physical Hilbert space. Otherwise the representation of the (non–trivial) Dirac observables will be reducible. This ends our discussion of the representation of strong Dirac observables. The discussion of weak Dirac observables is ( more complicated because they are not necessarily fibre preserving. 7 ∗∗ in HKin where {Ω∗∗ }i∈J ∗∗ are the cyclic systems Consider a vector ψ = ∗∗,i∈J ∗∗ ψi∗∗ (M)Ω i i ∗∗ in the Hilbert spaces HKin , ∗∗ = pp, ac, cs and where ψi∗∗ are µ∗∗ –measurable functions. The Direct Integral representation of ψ is given by #+ ∗∗ ∗∗ ˜ ρ∗∗ (III.2.36) (V ψ)(λ) = ψ(λ) = i (λ)ψi (λ)ei ∗∗,i

where V = V ⊕ V ⊕ V is the unitary operator which realizes the Direct Integral Decomˆ can then be represented as position. A weak bounded self–adjoint Dirac observable D ! ˜ ") ˆ (V Dψ)(λ) = dν D (λ" ) d(λ" , λ)ψ(λ (III.2.37) pp

ac

cs

⊕ ⊕ ⊕ for some measure ν D and some kernel d(λ" , λ) : HKin (λ" ) → & HKin (λ) with HKin (λ) = ac ⊕ (λ) ⊕ Hpp ⊕ (λ) ⊕ Hsc ⊕ (λ). The classical condition for a weak Dirac observable HKin Kin Kin

{D, {D, M}}M=0 = 0

(III.2.38)

−1 should annihilate the fibre H⊕ (0). 7 ˆ [D, ˆ M]]V translates into the condition that Aˆ := V [D, Kin In other words with the integral representation ! ˜ ") ˆ (V Aψ)(λ) = dν A (λ" ) a(λ" , λ)ψ(λ (III.2.39)

for the operator Aˆ we should have a(0, λ) = 0 for µ–almost all λ. In terms of the measure ν D and the kernel d we have explicitly ν A = ν D and ! " a(λ , λ) = dν D (λ"" ) d(λ"" , λ)d(λ" , λ"" )[λ + λ"" − 2λ" ] . (III.2.40)

This condition is implied by d(0, λ) = 0 for ν D −almost all λ and except for λ = 0 which would ⊕ mean that the fibre HKin (0) is preserved but not necessarily the individual sectors H∗∗ ⊕ Kin (0). D D ˆ We conclude that (V Dψ)(0) = ν ({0})d(0, 0)(V ψ)(0) where ν ({0}) )= 0. It would be valuable to learn more on the representation of weak Dirac observables, in particular whether weak Dirac observables may destroy the superselection structure Hphys = ac ⊕ (0) ⊕ Hpp ⊕ (0) ⊕ Hsc ⊕ (0) with respect to strong Dirac observables. HKin Kin Kin 26

Another example is given by the diffeomorphism constraints with respect to the extended Master constraint for LQG defined in [3].

85

III.3

Finite Dimensional Examples

We begin our examination of examples with finite dimensional systems, in particular: a finite number of Abelian constraints linear in the momenta which will also play an important role for section III.5, a system with second class constraints, first class constraints with structure constants at most quadratic in the momenta and first class constraints linear in the momenta with structure functions.

III.3.1

Finite Number of Abelean First Class Constraints Linear in the Momenta

Our first example is a system with configuration manifold Rn , coordinatized by xi , i = 1, . . . , n and m < n commuting constraints Ci = p i

i = 1, . . . , m

(III.3.1)

,

where the pi ’s are the conjugated momenta to the xi ’s. All phase space functions which do not depend on xi are Dirac observables, i.e. functions which commute with the constraints (on the constraint hypersurface). A Dirac observable which depends on pi , i = 1, . . . , m is equivalent to the Dirac observable obtained from the first one by setting pi = 0 (since these two observables will coincide on the constraint hypersurface). Therefore it is sufficient to consider observables which are independent of the first m configuration observables and of the first m conjugated momenta. A canonical choise for an observable algebra is the one generated by xi , pi , i = (m + 1), . . . , n. To quantize the system, we will start with an auxilary Hilbert space L2 (Rn ) on which the operators x ˆi act as multiplication operators and the momenta as derivatives, i.e. we use the standard Schr¨ odinger representation: x ˆi ψ(x) = xi ψ(x)

pˆi ψ(x) = −i! ∂i ψ(x)

.

(III.3.2)

According to the Master Constraint Programme, we have to consider the spectral resolution of the Master Constraint Operator m #

7 = −!2 M

i=1

∂i2 =: −!2 ∆m

.

(III.3.3)

This spectral resolution can be constructed with the help of the spectral resolutions of the operators pˆi = −i!∂i , which are well known to be given by the Fourier transform: ! ψ(x) = < bki , ψ(x1 , . . . , xi−1 , ·, xi+1 , xn ) >i bki (xi )dki (III.3.4) R

with generalized eigenfunctions bki (xi ) =

√1 2π 1

exp(iki xi ) i−1

< bki , ψ(x , . . . , x

and i+1

, ·, x

, x ) >i = n

86

!

R

√1 2π

exp(−iki xi )ψ(x)dxi

(III.3.5)

and ψ ∈ L2 (Rn ). The spectrum of pˆi is therefore spec(ˆ pi ) = R. Moreover the spectral projectors of pˆi and pˆj commute, so we can achieve a simultaneous diagonalization of all the Cˆi ’s (that is the Fourier transform with respect to (x1 , . . . , xm ): "9 $ ! m m 9 i < bki (·), ψ(· · · , xm+1 , . . . , xn ) >(m) bki (x ) dk1 · · · dkm (III.3.6) ψ(x) = Rm

i=1

i=1

with


(m) =

!

Rm

"9 m i=1

$ bki (x ) ψ(x)dx1 · · · dxm i

.

(III.3.7)

This decomposition of a function ψ ∈ L2 (Rn ) corresponds to a decomposition of the Hilbert space L2 (Rn ) into a direct integral of Hilbert spaces ! n L2 (R ) " H(k1 ,...,km ) dk1 · · · dkm . (III.3.8) Rm

where each H(k1 ,...,km ) is isomorphic to L2 (Rn−m ). Since the Master Constraint Operator is a polynomial of the Cˆi ’s, according to spectral calculus we already achieved the resolution of the Master Constraint. The spectrum of (spectral m 2 7 7 7 to M is given by spec(M) = clos{ i=1 ki , ki ∈ R} = R+ . The generalized eigenfunctions of M the (generalized) eigenvalue 0 can be read off from (III.3.6) to be Ψ(x) = =

m # 1 √ exp(i ki xi )|ki =0 ψ(xm+1 , · · · , xn ) ( 2π)m i=1

1 √ ψ(xm+1 , · · · , xn ) ( 2π)m

with

ψ ∈ L2 (Rn−m )

(III.3.9)

i.e. functions which do not depend on (x1 , . . . , xm ). So one could conclude that the physical Hilbert space is H(0,...,0) " L2 (Rn−m ), a space of functions of (xm+1 , . . . , xn ). The action of the elementary Dirac observables xˆi and pˆi for i = (m + 1), . . . , n is well defined on this physical Hilbert space. However we would like to go through the explicit procedure for constructing the Direct Integral Decomposition for a seperable Hilbert space, since we will use it later on for Maxwell theory. We will work in the Fourier–transformed picture, i.e. the Hilbert space is L2 (Rn ) as a function space of the momenta (p1 , . . . , pn ) and the Master Constraint Operator becomes the 7 = (m p 2 . multiplication operator M i=1 i To begin with we have to choose a set of orthonormal vectors {Ωi }i from which H = L2 (Rn ) 7 To this can be generated through repeated applications of the Master Constraint Operator M. end we use the fact (see [51]), that m m # % &% # &k p2j p2j ψs,l |s, k ∈ N, l ∈ ls } {exp − 12 j=1

(III.3.10)

j=1

is a basis for L2 (Rm ). Here ψs,l are harmonic polynomials27 of degree s on Rm , and the index l takes values in some finite index set ls , which depends on m and s. Obviously the set (III.3.10) 27

i.e. homogeneous polynomials annihilated by the Laplace operator on Rm

87

7 (more correctly M 7 =M ˜ where M ˜ ⊗1 can be generated by applying repeatedly M L2 (Rm−n ) , so m ˜ that M is an operator on L2 (R )) from the set m % 1# & ˜ {Ωsl := exp − 2 p2j ψs,l |s ∈ N, l ∈ ls } .

(III.3.11)

j=1

˜ s,l are mutually orthogonal. (This can be seen Moreover the spaces generated from different Ω if one introduces spherical coordinates. Then the restrictions of the harmonic polynomials to the unit sphere, which are given by the spherical harmonics, are orthogonal in L2 (S m−1 , dωn−1 ) where dωn−1 is the uniform measure on the sphere S m−1 .) So we conclude that the set of orthonormal vectors m # % & {Ωslt := Nsl exp − 12 p2j ψs,l ⊗ φt (pm+1 , . . . , pn ) |s ∈ N, l ∈ ls , t ∈ t}

.

(III.3.12)

j=1

where Nsl are normalization constants and {φt }t∈t is an orthonormal basis of L2 (Rn−m ) fulfills all the demands required above, i.e. L2 (Rn ) =

# s,l,t

⊕

7 slt |p span{p(M)Ω

polynomial}

.

(III.3.13)

Now the vectors Ωslt are tensor products of the form Ωslt = Ωsl ⊗ φt where Ωsl ∈ H1 := L2 (Rm ) 7 =M ˜ ⊗ 1H acts only on the first factor. Hence and φt ∈ H2 := L2 (Rn−m ) and furthermore M 2 we have for the spectral measures µslt (λ): ˜ )Ωsl ) ⊗ φt >H =< Ωsl , Θ(λ − M ˜ )Ωsl >H ||φt ||H µslt := < Ωsl ⊗ φt , (Θ(λ − M 1 2 ˜ = < Ωsl , Θ(λ − M )Ωsl >H1 =: µsl , (III.3.14)

˜ so that we only need to consider the spectral measures µsl with respect to the operator M on H1 . Additionally, using the rotational symmetry of the Master Constraint Operator (and an idea outlined in [51]) one can simplify the calculations even more: The space of harmonic polynomials on Rm of degree s is an irreducible module for the rotation group O(m) under the left regular representation, which acts as U (R) : ψ(5 p) &→ ψ(R−1 · p5)

(III.3.15)

on the space of functions over Rm . Here R is a rotation matrix for Rm . This representation is unitary if considered on the Hilbert space H1 = L2 (Rm ). The irreducibilty of the representation ensures that one can generate all vectors {Ωsl }l∈l by applying rotations U (R) to just one vector, ˜ invariant (since M ˜ say Ωs := Ωs0 . But these rotations leave the spectral projectors of M commutes with the rotations U (R)), so that we have ˜ )Ωsl >=< U (R)Ωs0 , Θ(λ − M ˜ )U (R)Ωs0 > µsl = < Ωsl , Θ(λ − M ˜ )Ωs0 >= µs0 =: µs = < Ωs0 , Θ(λ − M

(III.3.16)

for an appropriate rotation R. Here one can see, that it is very helpful to know the symmetries 7 These may come of the Master Constraint Operator, i.e. unitary operators commuting with M. 88

from (exponentiated) strong Dirac observables, that is operators which commute with all the constraints on the whole Hilbert space H. Examples for these are operators of the type 1H1 ⊗U2 , i.e. unitary operators which act only on H2 = L2 (Rn−m ). We used this kind of symmetry in (III.3.14). However the U (R)’s from above are not of this type: The classical counterparts of their generators (i.e. angular momentum on Rm ) Poisson-commutes with the Master Constraint but they also vanish on the constraint hypersurface {pi = 0, i ≤ m}. Nevertheless the rotation group O(m) is useful in constructing the direct Hilbert space decomposition. Later we will see, that the vanishing of its (classical) generators on the constraint hypersurface corresponds to the fact, that the representation of the rotation group on the induced Hilbert space is trivial. So we just need to select for each s ∈ N a particular homogeneous harmonic polynomial ψs of degree s. We choose ψs (p1 , . . . , pm ) = (p1 + ip2 )s

; s = 0, 1, 2, . . .

(III.3.17)

so that the vectors Ωs become Ωs = √

1

1

π m/2 s!

2

e− 2 rm r2s eisϕ

(III.3.18)

( 2 2 2 2 where we introduced the coordinates rm = m i=1 pi , r2 = p1 +p2 and the angle ϕ = arctan(p1 /p2 ). With this at hand we can compute the spectral measures µs (we will assume that m ≥ 4): µs (λ)

= = =

2 =r 2 +r 2 rm 2 m−2

=

r2 =rm cos φ

rm−2 =rm sin φ

=

2 y=rm

=

˜ )Ωs >H < Ωs , Θ(λ − M 1 ! 1 2 m 2 d p Θ(λ − rm )e−rm r22s π m/2 s! Rm ! 2π ! ! ! 1 2 m−3 −rm 2 dϕ dr2 dωm−3 drm−2 Θ(λ − rm )r22s+1 rm−2 e m/2 π s! 0 R+ S m−3 R+ ! ! π/2 m−3 2πVol(S ) 2 2 2s+m−1 dφ drm Θ(λ − rm )rm (cos φ)2s+1 (sin φ)m−3 e−rm m/2 π s! 0 R+ ! 2πVol(S m−3 ) s! Γ(m/2 − 1) 1 dy Θ(λ − y) y s+m/2−1 e−y 2 Γ(s + m/2) 2 R+ π m/2 s! ! 1 dy Θ(λ − y) y s+m/2−1 e−y (III.3.19) Γ(s + m/2) R+

From the second to the third line we transformed the cartesian (p3 , . . . , pm ) to ( coordinates 2 2 and angels varying over spherical coordinates on Rm−2 with radial coordinate rm−2 = m p i=3 i S m−3 . Vol(S m−3 ) is the euclidian volume of the (m − 3)-dimensional unit sphere. From the third to the fourth line we again introduced spherical coordinates (rm , φ) for the two radii r2 2 = r2 + r2 and rm−2 , so that rm 2 m−2 . Since r2 , rm−2 are positive, φ varies over [0, π/2]. Then we 2 for the remaining integral. In the integrated over φ and used a new integration variable y = rm m−3 m/2−1 last step we used that Vol(S ) = 2π /Γ(m/2 − 1). To obtain the final spectral measure µ(λ) we have to sum over ( all the measures µslt ≡ µs multiplied by constants αslt . We choose the constants such that l,t αslt = 2−s−1 , so that the

89

sum over all the αslt ’s is one. The spectral measure becomes =

µ(λ)

#

αslt µslt (λ) =

s,l,t

= = =

x=r22 /2

= =

2 y=rm−2

∞ #

2−s−1 µs (λ)

s=0

! ! 2−s 2 2 m−3 −rm dr2 drm−2 Θ(λ − rm )r22s+1 rm−2 e m/2 s! R+ 2π R+ s=0 ! ! m−3 1 2 πVol(S ) 2 2 2 r2 r m−3 e−rm dr dr Θ(λ − r )r e 2 m−2 2 m m−2 m/2 π R+ R+ ! ! m−3 πVol(S ) 2 m−3 2 dx drm−2 Θ(λ − rm−2 − 2x)e−x e−rm−2 rm−2 m/2 π R+ R+ ! 1 1 2 % & 2 πVol(S m−3 ) m−3 2 drm−2 Θ(λ − rm−2 ) 1 − e− 2 λ+ 2 rm−2 e−rm−2 rm−2 m/2 π R+ ! 1 1 & % 1 dy Θ(λ − y) 1 − e− 2 λ+ 2 y e−y y m/2−2 (III.3.20) Γ(m/2 − 1) R+

∞ 2πVol(S m−3 ) #

Here we inserted for µs the third line of (III.3.19), exchanged integration and summation, performed a variable transformation x = r22 /2 in the fourth line, and integrated over the new 2 variable x in the fifth line. Finally we changed to the integration variable y = rm−2 in the last m/2−1 m−3 line and used the explicit expression 2π /Γ(m/2 − 1) for Vol(S ). This gives for the derivative of µ(λ): ! 1 1 & % dµ(λ) 1 = dy δ(λ − y) 1 − e− 2 λ+ 2 y e−y y m/2−2 dλ Γ(m/2 − 1) R+ ! 1 1 & % 1 + dy Θ(λ − y) 12 e− 2 λ+ 2 y e−y y m/2−2 Γ(m/2 − 1) R+ ! 1 1 1 = e− 2 λ dy Θ(λ − y)e− 2 y y m/2−2 . (III.3.21) 2 Γ(m/2 − 1) R+ Next we have to calculate the Radon–Nikodym derivatives ρs = ρslt of µs with respect to µ. Since both measures are absolutely continuous with respect to the Lebesgue measure, we can write )! λ *−1 1 dµs (λ) dµs (λ)/dλ 2 Γ(m/2 − 1) s+m/2−1 − 1 λ − y m/2−2 ρs (λ) = = = λ e 2 dy e 2 y (III.3.22) dµ(λ) dµ(λ)/dλ Γ(m/2 + s) 0 For 0 < λ ≤ ∞ the derivatives ρs (λ) will be some strictly positive numbers for all s ∈ N. But for the limit λ → 0 we apply L’hospital’s rule, resulting in lim ρs (λ) =

λ→0

(s + m/2 − 1)λs+m/2−2 − 12 λs+m/2−1 2 Γ(m/2 − 1) lim Γ(m/2 + s) λ→0 λm/2−2

(III.3.23)

which gives ρs (0) = 0 for s > 0 and ρ0 (0) = 2. Now we can construct the induced Hilbert space H⊕ (0). We notize that there is only one linearly independent harmonic polynomial of degree zero, namely ψ00 ≡ 1. Therefore H⊕ (0) has an orthonormal basis {et }t∈t which corresponds to the set of vectors {Ω00t }t∈t. Hence we can 90

identify H⊕ (0) with the Hilbert space L2 (Rn−m ) of functions in the variables (pm+1 , . . . , pn ). Interestingly, because of ρslt (λ) > 0 for λ > 0, the induced Hilbert spaces H⊕ (λ) for λ > 0 are in some sense much bigger, since they have a basis {eslt |s ∈ N, l ∈ l, t ∈ t} corresponding to m−1 all the vectors Ωslt . The latter can be seen as a basis in L2 (S m−1 , λ 2 dωm−1 ) ⊗ L2 (Rn−m ), m−1 therefore we can identify H⊕ (λ) with L2 (S m−1 , λ 2 dωm−1 ) ⊗ L2 (Rn−m ) for λ > 0. In this example we recovered the usual result. This might seem surprising because one would 7 = pˆ2 + pˆ2 = 0 is think, e.g. that the space of solutions to the single quadratic constraint M 1 2 larger than the space of solutions to the individual linear constraints pˆ1 = pˆ2 = 0. Indeed, the general solution of the former is of the form f1 (z) + f2 (¯ z ) where z = x1 + ix2 and f1 , f2 are smooth functions while the general solution to the latter are the constants where we have used the representation pˆj = i!∂/∂xj on H = L2 (R2 , d2 x). However, solutions of the first type do not appear in the spectral resolution of the Master Constraint. Intuitively, this comes about because a physical Hilbert space based on square integrable linear combinations of holomorphic or antiholomorphic functions must be of the form L2 (C, dzd¯ z ρ(|z|)) with a damping factor ρ(|z|). However, this Hilbert space is not a representation space for a self–adjoint representation 7 Indeed, the induced action of the Dirac observables of the Dirac obsrvables with respect to M: pˆj , j = 1, 2 is not self–adjoint in this representation. Thus, this representation is not viable unless we restrict ourselves to the constant functions. These are automatically selected by the spectral analysis of the Master Constraint which in turn is induced by the spectral analysis of the individual constraints Cˆj = pˆj .

III.3.2

A Second Class System

Here we will discuss a simple second class system28 , given by the constraints Ci = xi

and

Bi = pi

i = 1, . . . , m (III.3.24)

on the phase space R2n . (The xi denote configuration variables and the pi their conjugated momenta.) The Poisson brackets are given by {Ci , Bj } = δij and all other Poisson brackets vanish. Because of the Heisenberg Uncertainty relations one cannot expect to find eigenfunctions of the Master Constraint Operator corresponding to the eigenvalue zero. We will therefore alter the Master Constraint and validate whether this gives sensible results. A complete set of observables is given by phase space functions which are independent on the first m configuration variables and momenta. As in the last case we will quantize the system by choosing the auxilary Hilbert space L2 (Rn ) of functions of the configuration variables ψ(x). The operators xˆi act as multiplication operators and the momentum operators pˆi as derivatives. The Master Constraint Operator is defined as the sum of the squares of the constraints but we can as well consider the following slight variation of this prescription:

28

7 = M

m #

1 2 pi 2 (ˆ

+ ωi2 (ˆ xi )2 )

,

(III.3.25)

i=1

i.e. a constrained system where the Poisson brackets between the constraints do not vanish on the constraint hypersurface

91

where ωi are m positive constants. This Master Constraint coincides with the Hamiltonian for m uncoupled harmonic oscillators with different frequencies. ( This Hamiltonian has pure point and positive spectrum, the lowest eigenvalue being λ0 = !2 m i=1 ωi . 7 such Since zero is not in the spectrum of the Master Constraint Operator we have to alter M, that its spectrum includes zero. Of course we have to check in the end, whether this procedure gives a sensible quantum theory. The simplest thing one can do, is to subtract λ0 from the 7" = M 7 −λ0 . This is equivalent to the normal ordering of the Master Constraint to obtain M Master Constraint Operator. This could be done also for the limit m → ∞(if we multiply the individual operators in the sum of (III.3.25) by positive constants Qm with m Qm < ∞. This condition will naturally reappear in the free field theory examples of III.5. 7 " includes zero and we can construct the induced Hilbert space to the Now the spectrum of M eigenvalue zero. To this end we have to find a cyclic basis, which we will choose to be a tensor basis of the following kind: Ωn1 ,...,nm ,k = fn1 ,...,nm (x1 , . . . , xm ) ⊗ hk (xm+1 , . . . , xn )

(III.3.26)

7 (considered on L2 (Rm )) and where fn1 ,...,nm , ni ∈ N are the (normalized) eigenfunctions of M 7 to the vectors Ωn ,...,n ,k {hk | k ∈ N} is an orthonormal basis in L2 (Rn−m ). The application of M m 1 just multiplies them with the corresponding eigenvalue so that we need them all in our cyclic basis. The associated spectral measures are 7 " )Ωn ,...,n ,k >= Θ(λ − λn ,...,n ) µn1 ,...,nm ,k (λ) =< Ωn1 ,...,nm ,k , Θ(λ − M (III.3.27) m m 1 1 ( where λn1 ,...,nm = !2 m i=1 ni ωi . Only measures with n1 = n2 = . . . = nm = 0 have the point zero in their support. The final spectral measure can be defined to be # µ(λ) = 2−(n1 +1) · · · 2−(nm +1) · 2−(k+1) µn1 ,...,nm ,k (λ) = Θ(λ) + ν(λ) (III.3.28) n1 ,...,nm ,k∈N

( where ν(λ) = n1 ,...,nm ≥1,k∈N 2−(n1 +1) · · · 2−(nm +1) ·2−(k+1) µn1 ,...,nm ,k (λ) is a pure point measure which does not have support at zero. The relevant Radon–Nikodym derivatives for the induced Hilbert space to the eigenvalue zero are therefore ρn1 =0,...,nm =0,k (0) = 1 ,

(III.3.29)

hence we can interpret the set {Ωn1 =0,...,nm =0,k | k ∈ N} as a basis in the induced Hilbert space. So as expected for a pure point spectrum the induced Hilbert space is just the (proper) eigenspace to the eigenvalue zero. This eigenspace can be identified with the space L2 (Rn−m ) by mapping Ωn1 =0,...,nm =0,k to the basis vector hk (xm+1 , . . . , xn ) in L2 (Rn−m ). Also the action of Dirac observables (that is quantized phase space functions, which do not depend on the first m config7 " and on L2 (Rn−m ). We uration variables and momenta) coincides on the null eigenspace of M can therefore conclude, that the physical Hilbert space is given by L2 (Rn−m ), which is the result one would expect beforehand.

III.3.3

SU(2) Model with Compact Gauge Orbits

If the constraints generate a (semi-simple) compact Lie group it is in general straightforward to apply the Master Constraint Programme. The Master Constraint Operator coincides in this case 92

with the Casimir of the Lie group and has a pure point spectrum. The Direct Integral Decomposition of the kinematical Hilbert space (which in this case is truly a direct sum decomposition) is equivalent to the reduction of the given representation of the Lie group into irreducible representations and the physical Hilbert space corresponds to the isotypical component29 of the equivalence class of the trivial representation. Here we will consider the configuration space R3 with the three so(3)-generators as constraints: Li = ,kij xj pk ,

{Li , Lj } = ,kij Lk

(III.3.30)

where ,kij = ,ijk is totally antisymmetric with ,123 = 1 and we summed over repeated indices. (In the following, indices will be raised and lowered with respect to the metric gik = diag(+1, +1, +1).) The observable algebra of this system is generated by e− = pi pi

e+ = xi xi

d = xi p i {d, e± } = ∓2e±

{e+ , e− } = 4d

(III.3.31)

.

It constitutes an sl(2, R)-algebra. We have the identity −d2 + e+ e− = Li Li

(III.3.32)

between the Casimirs of the constraint and observable algebra. III.3.3.1

Quantization

We start with the auxilary Hilbert space L2 (R3 ) of square integrable functions of the coordinates. The momentum operators are pˆj = −i(!)∂j and the x ˆj act as multiplication operators. There arises no factor ordering ambiguity for the quantization of the constraints, but for the observable algebra to close, we have to choose: dˆ = 12 (ˆ xi pˆi + pˆi x ˆi ) = x ˆi pˆi − 32 i!

eˆ− = pˆi pˆi

eˆ+ = x ˆi x ˆi

.

(III.3.33)

The commutators between constraints and between observables are then obtained by replacing 5 6 1 the Poisson bracket with i! ·, · . The identity (III.3.32) is altered to: ˆ iL ˆi dˆ2 − 12 (ˆ e+ eˆ− + eˆ− eˆ+ ) − 34 !2 = −L

.

(III.3.34)

For the implementation of the Master Constraint Progamme we have to construct the Direct Integral Decomposition with respect to the Master Constraint Operator " $ 2 2 2 2 − + 1 + − 3 2 ˆ 7 ˆ ˆ ˆ e eˆ + eˆ eˆ ) − 4 ! . (III.3.35) M := L1 + L2 + L3 = − d − 2 (ˆ 29

Compact Lie groups are completely reducible. The isotypical component of a given equivalence class within a reducible representation is the direct sum of all its irreducible representations into which it can be decomposed and which lie in the given equialence class.

93

The Master Constraint Operator is the Casimir of SO(3) on L2 (R3 ). Its spectrum and its (normalized) eigenfunction are well known, the latter are given by the spherical harmonics. To discuss these we make a coordinate transformation to spherical coordinates: 5 5 x1 = r cos(φ) sin(θ) φ ∈ 0, 2π) and θ ∈ 0, π) x2 = r sin(φ) sin(θ)

x3 = r cos(θ)

(III.3.36) d x = r sin(θ)dr dθ dφ 3

2

, 1 1 2 :2 = L ˆ 21 + L ˆ 22 + L ˆ 23 = −!2 ∂ L + ∂ (sin θ ∂ ) θ θ sin2 θ φ sin θ 31 4 1 1 ∂θ (sin θ∂θ ) + 2 2 ∂φ2 pˆ2 = −!2 ∆ = −!2 2 ∂r (r 2 ∂r ) + 2 r r sin θ r sin θ 1 :2 2 1 2 = −! 2 ∂r (r ∂r ) + 2 L r r 2 2 x ˆ = r

(III.3.37)

(III.3.38)

:2 to the eigenvalue !2 l(l+1), l ∈ N are of the form Ylm (θ, φ) R(r) 7 =L The eigenfunctions of M where Ylm , l ∈ N, −l ≤ m ≤ l are the spherical harmonics on the two-dimensional sphere S 2 and R(r) is an arbitrary function in L2 (R+ , r 2 dr). To discuss the Direct Integral Decomposition of the kinematical Hilbert space L2 (R3 ) we have to find a cyclic basis of this Hilbert space. ˆ := eˆ− + eˆ+ , i.e. the three-dimensional harmonic oscillator We will choose the eigenbasis of H Hamiltonian. Its (normalized) eigenfunctions are given by [48] ψnlm (r, θ, φ) = Nnl r l exp(−

r2 ) M (−n, l + 32 , !1 r 2 ) Ylm (θ, φ) 2!

(III.3.39)

where the index n takes values in N, the constant Nnl is a normalization constant and M (−n, l + 3 2 2 , r ) are confluent hypergeometric functions. Since the first argument of these is a whole negative number, the functions M (−n, l+ 32 , r 2 ) reduce to polynomials in r 2 of order n. Therefore, for fixed indices m, l the functions ψnlm are polynomials in r with minimal degree r l . (From this, one can see, that L2 (R3 ) is not equivalent to a tensor product L2 (R+ , r 2 dr) ⊗ L2 (S 2 , sin θdθdφ). If this would be the case the {ψnlm | n ∈ N} should span the whole Hilbert space L2 (R+ , r 2 dr) for arbitrary fixed indices m, l. However, they span only the subspace of polynomials with minimal degree l. We will come back to this point later on.) 7 to the functions ψnlm just multiplies them with !2 l(l + 1), hence we need them Applying M all as cyclic vectors Ωnlm . The associated spectral measures can easily be calculated to be 7 nlm >=< ψnlm , Θ(λ − !2 l(l + 1))ψnlm >= Θ(λ − !2 l(l + 1)) . µnlm (λ) =< ψnlm , Θ(λ − M)ψ (III.3.40) The spectral measures are pure point, they are homogeneous in n and m and have only the point !2 l(l + 1) as support. Hence the kinematical Hilbert space decomposes into eigenspaces 7 in the following way of M L2 (R3 ) =

∞ # l=0

clos(span{ψnlm | n ∈ N, −l ≤ m ≤ l}) 94

.

(III.3.41)

The induced Hilbert space to the eigenvalue λ = 0 is given by clos(span{ψn00 | n ∈ N}). These functions are constant in θ, φ and are polynomials in r 2 of order n (weighted with a gaussian factor). The above set can be taken as an orthonormal basis in L2 (R+ , r 2 dr), therefore we can identify the physical Hilbert space with L2 (R+ , r 2 dr). On this (physical) Hilbert space the observable algebra is given by eˆ− = −!2

eˆ+ = r 2

1 ∂r (r 2 ∂r ) r2

dˆ = −i! (r∂r + 32 )

(III.3.42)

and because of (III.3.34) we have dˆ2 − 12 (ˆ e+ eˆ− + eˆ− eˆ+ ) = 34 !2

III.3.4

(III.3.43)

.

A Model with Structure Functions rather than Structure Constants Linear in the Momenta

As an example for a model with structure functions we will discuss a sort of a deformed SO(3) constraint algebra n C1 = x2 pm 3 − x3 p2

C2 = xn3 p1 − x1 pm 3

{C1 , C2 } = m n p3m−1 xn−1 C3 3

C3 = x1 p2 − x2 p1

{C2 , C3 } = C1

{C3 , C1 } = C2

(III.3.44) (III.3.45)

where n and m are positive natural numbers. For n = m = 1 we recover the SO(3)-algebra. One can do a similar deformation for SO(p, q)-algebras. The Dirac observables (see Example 5 in section II.5) for this system generate an sl(2, R) algebra: 2 p1−m xn+1 3 m+n 3 2 d = x1 p1 + x2 p2 + x3 p 3 m+n

e− = p21 + p22 +

e+ = x21 + x22 +

2 x1−n pm+1 3 m+n 3 (III.3.46)

{d, e± } = ∓2e±

{e+ , e− } = 4d

.

(III.3.47)

For general m, n these observables commute only on the constraint hypersurface with the constraints. We define the Master Constraint as M = C12 + C22 + C32

.

(III.3.48)

The constraints C1 and C2 do not strongly Poisson–commute with M (for n )= 1 or m )= 1), but C3 does. For the rest of this chapter, we will consider the case m = 1 and n odd. For these parameter values the phase space function e+ is non–negative and commutes on the whole phase space R3 × R3 with the Master Constraint. This will be helpful for the spectral analysis of the quantized Master Constraint. The case n = 1 and m odd can be treated in the same way, using Fourier transformation. For m > 1 the observable e+ contains negative powers of p3 , therefore the analysis gets more complicated. 95

III.3.4.1

Quantization

We will quantize this system in the usual way by assigning multiplication operators xˆi to the configuration variables xi and differential operators pˆj = −i!∂j to the momenta pj . The kinematical Hilbert space, we are starting with, is L2 (R3 ). There arises no factor ordering ambiguity for the quantization of the constraints, and since the structure function f = n(ˆ x3 )n−1 commutes with C3 (see (III.3.45)), it is possible to quantize all the constraints as symmetric operators as we will explain in the conclusion section and to have Cˆ3 standing to the right of the structure function: 5 6 5 6 5 6 Cˆ1 , Cˆ2 = i! n(ˆ x3 )n Cˆ3 Cˆ2 , Cˆ3 = i! Cˆ1 Cˆ3 , Cˆ1 = i! Cˆ2 . (III.3.49) We have to analyze the Master Constraint Operator 7 = Cˆ 2 + Cˆ 2 + Cˆ 2 M 1 2 3

(III.3.50)

,

which is a second order partial differential operator. It can be densely defined and is symmetric on the linear span of the Hermite functions. To solve the eigenvalue equation for the Master Constraint Operator we will introduce new coordinates (t, θ, φ) analogous to the coordinates (r, θ, φ) in the SO(3) case. However, for n )= 1 we have to consider the regions x3 ≥ 0 and x3 ≤ 0 seperately: For x3 ≥ 0 we define 5 & 5 6 x1 = t cos φ sin θ t≥0 φ ∈ 0, 2π θ ∈ 0, π2 x2 = t sin φ sin θ 1 2 n + 1 n+1 x3 = ( ) (t cos θ) n+1 2

and for x3 ≤ 0

x3 = −(

(III.3.51)

1 2 n + 1 n+1 ) (−t cos θ) n+1 2

θ∈ n

5π

n+3

2,π

6

(III.3.52)

. −n+1

2 The measure is transformed to dx1 dx2 dx3 = sin θ ( n+1 ) n+1 t n+1 | cos θ| n+1 dtdθdφ. With these coordinates we have 2 x21 + x22 + xn+1 = t2 = e+ (III.3.53) n+1 3 7 does not include derivatives with respect to t. Indeed, for cos θ ≥ 0, and M

7 = −!2 M

3n + 14 2

2n n+1

;

(t cos θ)

2n−2 n+1

∂θ2

+

"

3n−1 $ 3n − 14 2n−2 n−3 (t cos θ) n+1 n+1 n+1 −t (cos θ) sin θ ∂θ + t sin θ n+1 < 4n " 2n $ (t cos θ) n+1 3 2 4 n+1 + ∂φ2 n+1 t2 sin2 θ

(III.3.54)

7 for cos θ ≤ 0 is obtained from the above formula by replacing t with −t. The operator and M 7 M simplifies considerably, if we introduce the coordinate 2

u = (cos θ) n+1

u = −(− cos θ)

for

2 n+1

cos θ ≥ 0

for

cos θ ≤ 0

96

u ∈ [−1, 1]

.

(III.3.55)

The coordinate u is proportional to the coordinate x3 and therefore one gets the same operator for x3 positive and for x3 negative: ; 2n−2 " $< 3 n + 1 4 2n " t 2n−2 3 2 4 2n $ n+1 u2n n+1 2t n+1 n+1 2 2 n+1 7 = −! ∂φ + M + ∂u (1 − u )∂u . 2 1 − un+1 n+1 n+1 (III.3.56)

1

n+3

n+1 t n+1 dtdudφ. The measure is now dx1 dx2 dx3 = ( n+1 2 ) 7 commute, we can diagonalize them simultaneously. Since the operators Cˆ3 = −i!∂φ and M Choosing periodic boundary conditions in φ one obtains spec(Cˆ3 ) = {!k|k ∈ Z} and the null 7 near zero, eigenspace consists of constant functions in φ. We are interested in the spectrum of M 7 ˆ so it suffices to consider M restricted to the null eigenspace of C3 , as it has elsewhere spectrum 7 = Cˆ 2 + Cˆ 2 + Cˆ 2 ). The restriction of M 7 to this space is bounded from below by !2 (since M 1 2 3

" $ 3 4 2n 2n−2 n+1 2 n + 1 n+1 2t n+1 7 M|Cˆ3 =0 = −! ∂u (1 − u )∂u , 2 n+1

(III.3.57)

and can be seen as a product of two commuting operators B1 and B2 : B1 := !

2

3n + 14 2

2n n+1

"

2n−2

2t n+1 n+1

B2 := −∂u (1 − u

,

n+1

)∂u

$

.

(III.3.58)

In the following, we will consider the operator B2 on the Hilbert space L2 ((−1, 1), du) and show that its spectrum is purely discrete. Afterwards we will come back to the product of the operators B1 · B2 . (The product B1 · B2 cannot be seen as a tensor product of B1 and B2 since n+3 L2 (R3 , dx3 ) is not equivalent to L2 (R+ , t n+1 dt) ⊗ L2 ((−1, 1), du) ⊗ L2 ((0, 2π), dφ). Moreover one has to be careful, since it is not guaranteed that B2 has a self–adjoint extension as an operator on L2 (R3 ). This might be the case, because B2 does not preserve the space of Hermite functions in the three variables (x1 , x2 , x3 ), only the combination B1 ·B2 does preserve this space. Nevertheless one can consider the operator B2 defined on the Hilbert space L2 ((−1, 1), du).) The operator B2 is symmetric and positive on the dense domain D(B2 ) = {f ∈ L2 ((−1, 1), du)|f ∈ AC(−1, 1), (1 − un+1 )f " ∈ AC(−1, 1),

B2 · f ∈ L2 ((−1, 1), du), lim (1 − un+1 )f " (u) = 0} u→±1

(III.3.59)

where AC denotes the class of absolutely continuous functions. The positivity can be seen by using integration by parts ! 1 ∂u f (u)(1 − un+1 )∂u f (u) du ≥ 0 (III.3.60) < f, B2 · f >L2 (u) = −1

for functions f ∈ D(B2 ). A positive and symmetric operator has always selfadjoint extensions (see [49]). The operator B2 is an ordinary differential operator, more specifically it is a Sturm–Liouville operator on the interval I := (−1, 1), so we can utilize the theory of Sturm-Liouville operators, see for example [50]. Sturm-Liouville operators can be classified according to the behaviour of 97

eigensolutions at the endpoints of the interval I: A Sturm-Liouville operator A is limit circle at the endpoint a if for one λ ∈ C all solutions to (A − λ)f = 0 are square integrable near a. One can prove, that if this is the case for one λ, then it holds for all λ ∈ C. It therefore suffices to consider the solutions to B2 · f = λf for one λ ∈ C, in particular λ = 0. One solution for λ = 0 is obviously given by f01 (u) ≡ 1, the other solution is given by ! u ! u 1 1 " f02 (u) = du = du" . (III.3.61) "n+1 " 2 "n+1 ) )(f01 (u )) 0 (1 − u 0 (1 − u

Both solutions are square integrable on (−1, 1). The function f02 is integrable because for n > 1 and n odd, the function |f02 (n > 1)| is less than |f02 (n = 1)| on the interval (−1, 1). But f02 (n = 1) = artanh(u) is square integrable on (−1, 1). We are therefore in the limit circle case for both boundaries u± = ±1. If both boundary points are limit circle for an operator A, the following holds (see [50]): The spectrum of any self adjoint extension is purely discrete, the eigenfunctions are simple (multiplicity one) and form an orthonormal basis and the resolvent of A is a Hilbert–Schmidt operator. Notice that Hilbert–Schmidt operators are in particular compact, hence the spectrum of the resolvent Rz (A) for z ∈ R not in the spectrum of A has an accumulation point at most at 7 is unbounded we know that zero. Thus A does not have any accumulation point and since M the eigenvalues actually diverge (when ordered according to size). Hence B2 has a discrete simple spectrum, and since f01 (u) ≡ 1 is a square integrable solution for λ = 0 (which fulfills contrary to f02 the boundary conditions), zero is included in the spectrum. This holds for any selfadjoint extension of B2 . Now B2 with the domain (III.3.59) is positive. We choose in the following any selfadjoint extension such that it is still positive and such that B1 · B2 as an operator in L2 (R3 ) corresponds to the positive self–adjoint extension of 7 we have chosen before (e.g. the Friedrich extension)30 . The eigenvalues of B2 are therefore M, positive (that is λj ≥ 0 and λj = 0 if and oly if j = 0). Since the operator B2 has a null eigenspace consisting of functions constant in u and therefore θ, one would expect that the physical Hilbert space consists of functions functionally independent of θ (and φ as was argued above). However we are dealing not just with the operator B2 but with the product B1 · B2 . The operator B1 is a multiplication operator and has continuous spectrum on the whole positive axis including zero. Therefore we have to discuss the decomposition of the Hilbert space into a direct integral Hilbert space and to calculate the spectral measures. To this end we will consider the restriction of the Master Constraint Operator to the subspace Cˆ3 = 0 as it has elsewhere a spectrum, which is bounded from below by !2 . Since this subspace coincides with functions in L2 (R3 ), which are constant in the angle variable φ, we can identify this subspace with H" := L2 (R+ × R, ρ dρdx3 ) where the variable ρ is defined by ρ2 = x21 + x22 . A dense system of vectors in this Hilbert space is generated by the set {fpq := ρ2q xp3 exp(− 12 t2 )

| q, p ∈ N}

.

(III.3.62)

We need just the even polynomials in ρ since the variable ρ has range only in the positive half 7 We will begin axis31 R+ . Now one has to find a cyclic system of vectors with respect to M. 30

The operator B1 is self–adjoint with the domain DSA (B1 ) = {f ∈ L2 (R3 )|B1 · f ∈ L2 (R3 )}. Indeed, polynomials of any positive power of tr could be used in order to construct a basis of the Hilbert space by the Gram–Schmidt procedure because any square integrable function t %→ f (t) can be written as t %→ fr (tr ) where fr (s) := f (s1/r ). Notice that we do not transform the measure to the variable tr here. 31

98

2 with functions that are polynomials in t2 = (ρ2 + n+1 xn+1 ) weighted with exp(−t2 ). All these 3 7 hence to get a cyclic system, we need all the even powers of functions are annihilated by M, t. (Again, we just need the even powers, since t extends over the positive half axis.) If one performs the Gram-Schmidt procedure for this system {t2k exp(−t2 ) | k ∈ N} one will get an ortho–normal set of vectors {v0k | k ∈ N} which can be identified as a basis in the Hilbert space n+3 L2 (R+ , t n+1 dt). The ascociated spectral measures are given by

µ0k (λ) =< v0k , Θ(λ − B1 · B2 )v0k >=< v0k , Θ(λ − B1 · 0)v0k >= Θ(λ) ,

(III.3.63)

so that the set of vectors {v0k } is associated with a pure point spectral measure. We will call the subspace spanned by these vectors Hpp (anticipating that all vectors orthogonal to this subspace are associated to spectral measures, which are absolutely continuous). Now we have to find the orthogonal complement to Hpp . If one rewrites the functions fpq in terms of the coordinates (t, u) one gets 2p

1

2

fpq = ρ2q xp3 = C(p, q)(1 − un+1 )q up t2q+ n+1 e− 2 t

(III.3.64)

where C(p, q) is a p, q dependent constant. It is important to note here that a power of ua is alway 2a 2 accompanied by a power of at least t n+1 . The underlying reason for this is that u ∼ x3 t− n+1 . Using this fact, it is straightforward to see, that also 1

2p

2

gpq = up t n+1 +2q e− 2 t

p, q ∈ N

(III.3.65)

or 2p

1

2

hpq = Pp (u)t n+1 +2q e− 2 t

(III.3.66)

generate a dense subspace in H" . Here Pp (u) are the Legendre polynomials of order p, that is Pp (u) is a polynomial with degree p. To see this, notice that (III.3.64) is a polynomial in terms of t2 and x := ut2/(n+1) which explains the statement for gpq . Next notice that Pp is a polynomial of order p in u which can be written as a polynomial in x of the same order and t2(p−k)/(n+1) , k = 0, .., p. Now any positive power of t can be expanded in terms of the v0k and thus in terms of the t2q again. Since P0 (u) is the constant function, the space spanned by {h0q |q ∈ N} coincides with the space spanned by {v0k |k ∈ N}. Moreover Pp , p > 0 is orthogonal to P0 (in L2 ((−1, 1), du)), so that the orthogonal complement to Hpp is spanned by {hpq |N ' p > 0, q ∈ N}. We will now show, that the spectral measures associated to vectors from this subspace are absolutely continuous. In order to do this we have to expand the functions Pp (u) ∈ L2 ((−1, 1), du), p > 0 into eigenfunctions of B2 : Pp (u) =

∞ #

apk ψk (u)

(III.3.67)

k=1

where ψk is the normalized k − th eigenfunction of B2 (and we assume that the corresponding eigenvalues λk are ordered and that ψ0 is the constant function). The constant function (i.e. ψ0 ) does not appear in this decomposition since Pp is orthogonal (in L2 ((−1, 1), du)) to the constant function for p > 0. 99

2n

n+1 2 Using this decomposition and abbreviating cn = !2 ( n+1 2 ) n+1 we can write

7 pq > µp! q! pq (λ) := < hp! q! , θ(λ − M)h / ∞ 0 3n + 14 1 ! ! # p! 2(p+p! ) ! )+ n+3 n+1 +2(q+q n+1 = dtdu exp(−t2 ) t n+1 ak! ψ k! (u) 2 R+ [−1,1] k ! =1 / ∞ 0 # p 2n−2 θ(λ − cn t n+1 B2 ) ak ψk (u) =

=

3n + 14

1 n+1

2

3n + 14

!

∞ #

k ! ,k=1 R+

1 n+1

2

!

k=1

dtdu exp(−t2 ) t [−1,1] 2n−2

∞ ! #

2(p+p! ) +2(q+q ! )+ n+3 n+1 n+1

!

θ(λ − cn t n+1 λk ) apk! apk ψ k! (u)ψk (u) dt exp(−t2 ) t

2(p+p! ) +2(q+q ! )+ n+3 n+1 n+1

k=1 R+ 2n−2

!

θ(λ − cn t n+1 λk ) apk apk

(III.3.68)

where in the last line we used the ortho-normality of ψk in L2 ((−1, 1), du). The sum in the last line converges absolutely, since the integral over t can be bounded (k– independently) from above by ignoring the θ–function. Then we are left with the sum over the ! absolute values of apk apk , which can be estimated using the Cauchy-Schwarz inequality and the fact that the apk are the expansion coefficients of Pk ∈ L2 ((−1, 1), du). This shows that all the measures µp! q! pq are absolutely continuous with respect to the Lebesgue measure on R+ . Indeed we have ! ∞ 3n + 14 1 # 2(p+p! ) n+3 2n−2 ! dµp! q! pq (λ) n+1 p! p = ak ak dt exp(−t2 ) t n+1 +2(q+q )+ n+1 δ(λ − cn t n+1 λk ) dλ 2 R+ k=1

=

3n + 14

=: λ

2

1 n+1

∞ #

!

apk apk

k=1

3+p+p! +(q+q ! )(n+1) n−1

%

! ! & % n+1 & % λ & 3+p+p +(q+q )(n+1) n+1 λ n−1 n−1 exp − ( ) 2cn λk (n − 1) cn λk cn λk

fp! q! pq (λ)

(III.3.69)

2n−2

where in the second line we performed a coordinate transformation sk = cn λk t n+1 in order to solve the delta–function. The sum in the second line converges absolutely (uniformly in λ) since the exponential factor can be estimated by 1 and the λk have to be bigger than 1 for some finite k because the spectrum of B2 does not have an accumation point as we showed above. Hence fp! q! pq is well defined, in particular fp! q! pq (0) > 0 for p" = p (since all terms in the sum are then positive). To summarize, our Hilbert space H" can be decomposed into two subspaces Hpp and Hac 7 restricted to Hpp has a pure point spectral measure and M 7 restricted to Hac has an where M absolutely continuous spectral measure. We therefore have to discuss the Direct Integral Decomposition of Hpp and Hac separately. The ‘Direct Integral Decomposition’ for Hpp is straight7 the physical Hilbert forward: Since Hpp is a proper eigenspace to to the null eigenvalue of M, space corresponding to the pure point spectrum coincides with Hpp (which can be identified n+3

with L2 (R+ , t n+1 dt)).

100

To perform the direct integral Hilbert space decomposition of Hac spanned by {hpq |N ' p > 0, q ∈ N} we choose an ortho–normal cyclic system {Ωm |N ' m > 0} in that space, such that Ω1 = N1 h10 , where N1 is a normalization constant. Since {hpq |N ' p > 0, q ∈ N} generates Hac we can always find coefficients Am pq such that ∞ #

Ωm =

Am pq hpq

(III.3.70)

.

p=1,q=0

Consider the spectral measures µm , m > 1 associated to this cyclic system 7 m> µm (λ) := < Ωm , θ(λ − M)Ω ∞ # 7 = < Ωm , θ(λ − M) Am pq hpq > =

7 < Ωm , θ(λ − M)

p=1,q=0 ∞ #

(III.3.71)

Am pq hpq >

p=1,q=0

(p,q),=(1,0)

7 1 >= N1 < Ωm , θ(λ − M)h 7 10 >= 0 due to the defining property of a since < Ωm , θ(λ − M)Ω cyclic system. Hence ∞ #

µm (λ) =

p! =1,q ! =0 ∞ #

=

p! =1,q ! =0

∞ #

p=1,q=0

(p,q),=(1,0) ∞ #

m 7 Ap! q! Am pq < hp! q ! , θ(λ − M)hpq > m

Ap! q! Am pq µp! q ! pq (λ) .

(III.3.72)

p=1,q=0

(p,q),=(1,0)

Differentiating under the sums32 dµm (λ) dλ

∞ #

=

p! =1,q ! =0

=

6

λ n−1

∞ #

p=1,q=0

(p,q),=(1,0) ∞ ∞ # #

p! =1,q ! =0

=: λ

6 n−1

m

Ap! q! Am pq

m

dµp! q! pq (λ) dλ

Ap! q! Am pq λ

p+p! +(q+q ! )(n+1)−3 n−1

fp! q! pq (λ)

p=1,q=0

(p,q),=(1,0)

gm (λ)

(III.3.73)

where gm (λ) is a non–singular function of λ near zero (that is gm (0) is either zero or some finite value). This holds for m > 1; for m = 1 we get 5 dµ1 (λ) = λ n−1 (N1 )2 f1010 (λ) dλ 32

(III.3.74)

Strictly speaking one must verify that we may interchange summation and differentiation. This could be done for instance by verifying that the series for µm (λ) converges absolutely and uniformly in λ at least close to zero (the absolute and uniform convergence of the series for µ(λ) then follows). We have not checked that this is the case but given the fact that the µp! q! pq converge rapidly to zero in the vicinity of λ = 0 this should be true. With the tools given here, one could check this explicitly by a tedious but straightforward calculation.

101

where f1010 (0) = c > 0. The total spectral measure µac (λ) on Hac is given by (if the maximal multiplicity is less than infinity, choose appropriate normalization constants different from 2−m ) dµac (λ) dλ

:= =

∞ 1 dµ1 (λ) # −m dµm (λ) + 2 2 dλ dλ m=2

# 5 6 1 n−1 λ (N1 )2 f1010 (λ) + λ n−1 2−m gm (λ) . 2

(III.3.75)

m=2

Now we can consider the Radon–Nikodym derivatives in the limit λ → 0. Assume that m > 1: 6

dµm (λ)/dλ λ n−1 gm (λ) lim ρm (λ) = lim = lim =0 5 6 ( λ→0 λ→0 dµac (λ)/dλ λ→0 1 λ n−1 (N )2 f −m g (λ) n−1 (λ) + λ 2 1 1010 m m=2 2 (III.3.76) Hence all the ρm , m > 1 vanish at λ = 0. But the same calculation for m = 0 gives ρ1 (0) = 2. Therefore the induced Hilbert space for λ = 0 from Hac is one-dimensional, that is unitarily equivalent to C. Putting the contributions from Hpp and Hac together we get Hphys " n+3

L2 (R+ , t n+1 dt) ⊕ C (where the inner product in C can be rescaled arbitrarily). The first term 7 the second to the continious speccorresponds to the (proper) null subspace with respect to M trum of the multiplication operator in t. Hence our physical Hilbert space is a sum of an L2 –space in the Dirac observable t and a one–dimensional space corresponding to the solution δ(t) to the Master Constraint Operator. Remark: By the standard theory for Sturm–Liouville operators such as B2 we know that the eigenfunction ψk has k zeroes and that the eigenvalues asymptote to λk ∝ k2 . By the Weierstrass theorem, ψk can be approximated in the sup–norm arbitrarily well on the compact set [−1, 1] by polynomials of degree at least k (in order to have k real roots). Thus, the ψk are actually not too different from the standard Legendre polynomials Pk and if they would be really polynomials of degree k we could just use the functions Ωk = ψk t2k/(n+1) as a cyclic system. Then the above calculations would become entirely trivial because the Radon–Nikodym derivative at λ = 0 would be obviously non–vanishing for the lowest order k = 1. Unfortunately the eigenvectors ψk are not polynomials unless n = 1 and so we had to go through this very elaborous analysis.

III.3.5

Another Model with Pure Point and Absolutely Continuous Spectra

Here we will discuss an example, where, similarly to the previous one, the Master Constraint Operator has pure point and absolutely continuous spectrum at zero. The example is simpler without structure functions and serves the purpose to illustrate an important point when choosing the cyclic system. We will start with a kinematical Hilbert space L2 (R2 ) and a Master Constraint Operator 7 = (ˆ M x21 + x ˆ22 )(ˆ x1 pˆ2 − x ˆ2 pˆ1 )2

(III.3.77)

where the x ˆi are multiplication operators and the pˆi := −i!∂i act by differentiation. Apriori one would expect that the physical Hilbert space includes the space of functions with zero angular 102

momentum and a one-dimensional part which corresponds to the generalized eigenfunction δ(x1 )· δ(x2 ). We will now construct the Direct Integral Decomposition of the kinematical Hilbert space 7 First of all, we have to find a cyclic basis for L2 (R2 ). To this end we note, with respect to M. 2 that L2 (R ) is spanned by polynomials in x1 , x2 weighted by a gaussian factor. These can be generated by the set {xn1 1 xn2 2 exp(− 12 (x21 + x22 ) | n1 , n2 ∈ N} (III.3.78)

and also by the set

n 1 {vmn = xm + x− exp(− 2 x+ x− ) | n, m ∈ N}

(III.3.79)

where we defined x± := x1 ± ix2 . In spherical coordinates defined by x1 = r cos φ x2 = r sin φ

r ∈ R+ , φ ∈ [0, 2π)

(III.3.80)

the vectors vnm can be expressed as vnm = r n+m exp(iφ(m − n)) e−r

2 /2

.

(III.3.81)

By introducing new indices N := (n − m) ∈ Z and k := 12 (m + n − |n − m|) ∈ N the set (III.3.79) can also be written as {vN k = r |N |+2k exp(iN φ) exp( 12 r 2 ) | N ∈ Z, k ∈ N} .

(III.3.82)

7 to a vector vN 0 , N )= 0 generates all the other vectors vN k , k ∈ N with Applying repeatedly M 7 the same index N . We therefore need just the vN 0 as cyclic vectors. But for N = 0 applying M to v0k gives zero, so that we also need the whole set {v0k | k ∈ N}. However, notice that this set is not orthonormal, one has therefore to perform a Gram-Schmidt procedure. This will result in an orthonormal set of the form {Ω0k := fk (r 2 ) exp(−r 2 /2)} where the fk are polynomials of order k in r 2 . This set can be taken as a basis for L2 (R+ , rdr). Remark: At this point it is important to draw attention to the following subtlety: Recall the definition of the tensor product of two Hilbert spaces Hj : This is the Hilbert space H1 ⊗H2 consisting of pairs ψ1 ⊗ ψ2 := (ψ1 , ψ2 ) with ψj ∈ Hj equipped with the inner product < ψ1 ⊗ ψ2 , ψ1" ⊗ ψ2" >H1 ⊗H2 := =2 (j) (1) (2) " j=1 < ψj , ψj >Hj . One can show that if bk is a basis for Hj then bk ⊗ bn is a basis for H1 ⊗ H2 . If (Xj , Bj , µj ) are measure spaces and Hj := L2 (Xj , dµj ) are separable Hilbert spaces then one can show by using Fubinis theorem that H1 ⊗ H2 is isometrically isomorphic to L2 (M1 × M2 , dµ1 ⊗ dµ2 ) via ψ1 ⊗ ψ2 &→ ψ1 (x1 )ψ2 (x2 ) where µ1 ⊗ µ2 is the measure on M1 × M2 based on the smallest σ−algebra B1 ⊗ B2 containing the ‘rectangles’ B1 × B2 where Bj ∈ Bj and by definition µ1 ⊗ µ2 (B1 × B2 ) := µ1 (B1 )µ2 (B2 ). Now consider the problem at hand: In polar coordinates we have L2 (R2 , d2 x) = L2 (R2 , rdrdφ) with R2 = [(R+ − {0})× S 1 ]∪ {(0, 0)}. One could now think that since the one point sets {(0, 0)} and {0} respectively have d2 x and dx Lebesgue measure zero respectively that L2 (R2 , d2 x) = L2 (R2 − {(0, 0)}, d2 x) = L2 ((R+ − {0}) × S 1 , d2 x)

= L2 ((R+ − {0}, rdr) ⊗ L2 (S 1 , dφ) = L2 ((R+ , rdr) ⊗ L2 (S 1 , dφ)

(III.3.83)

103

If (III.3.83) would be true then, given some ONB b(1)k (r) for L2 (R+ , rdr) consisting of polynomi√ 2 (2) als in r times e−r /2 obtained via the Gram–Schmidt procedure and an ONB bn (φ) = einφ / 2π (1) (2) for L2 (S 1 , dφ) we would obtain a basis bk ⊗ bn for L2 (R2 , d2 x). However, in the tensor product 2 we then obtain a dense set of vectors of the form r k einφ e−r /2 where the pair (k, n) ∈ N0 × Z is unrestricted. On the other hand the Hermite polynomial basis for L2 (R2 , d2 x) provide a dense 2 set consisting of vectors of the form r |n|+2l einφ er /2 as we have just seen. We conclude that the (1) (2) basis bk ⊗ bl for L2 ((R+ , rdr) ⊗ L2 (S 1 , dφ) is overcomplete for L2 (R2 , d2 x) and hence these 2 Hilbert spaces are not isometrically isomorphic. In other words, the functions r k einφ e−r /2 such that k − |n| is not a non–negative and even integer could be expanded in terms of Hermite polynomials because they are obviously square integrable. What has gone wrong in (III.3.83) is that on R2 the coordinates (r, φ) are singular at r = 0. The coordinates x1 , x2 are globally defined which results in the fact that there is the restriction k − |n| = 2l, l = 0, 1, 2, .. in order that the functions r k einφ are regular at r = 0 when expressed in terms of x1 , x2 . This topological subtlety that R2 is a plane while R+ × S 1 is a half – infinite cylinder with different orthonormal bases is of outmost importance for the Direct Integral Decomposition (DID) because neglecting this difference would result in a much bigger physical Hilbert space as we will see shortly. This ends our remark. The normalization of the vectors vN 0 gives ΩN 0 = (π |N !|)−1/2 r |N | exp(iN φ) exp(−r 2 /2)

.

(III.3.84)

Their associated spectral measure are 7 N0 > = µN 0 =< ΩN 0 , Θ(λ − M)Ω =

! ∞ 2 r 2|N | exp(−r 2 ) Θ(λ − r 2 N 2 ) rdr |N |! 0 ! ∞ 1 x|N | exp(−x) Θ(λ − N 2 x) dx (III.3.85) |N |! 0

which shows, that these measures are absolutely continuous with respect to the Lebesgue measure on R+ . The spectral measures associated to the Ω0k can be calculated to 7 0k >= Θ(λ) < Ω0k , Ω0k >= Θ(λ) µ0k =< Ω0k , Θ(λ − M)Ω

(III.3.86)

which shows, that these measures are of pure point type. Hence the kinematical Hilbert space L2 (R2 ) decomposes into a direct sum of two Hilbert 7 has either absolutely continuous or pure point spectrum. Hac spaces Hac and Hpp , on which M 7 k ΩN 0 | N )= 0, N ∈ Z, k ∈ N} and Hpp is the closure is defined to be the closure of the span of {M

of the span of {Ω0k | k ∈ N}. We will now discuss the Direct Integral Decomposition on each of these two spaces seperately. The Direct Integral Decomposition of Hpp (which can be identified with L2 (R+ , rdr) is easy to obtain, since all the measures µ0k are the same and have just the point zero in their support. 7 – the null eigenspace of M 7 on Hpp coincides Hence Hpp is alredy decomposed with respect to M with Hpp . The contribution from the pure point part of the spectrum to the physical Hilbert space is therefore given by Hpp . 104

For the Direct Integral Decomposition of Hac we have to calculate first the total spectral measure µac (λ) =

1 2

=

!

∞ #

N =1 ∞

0

2−N (µN 0 (λ) + µ−N 0 (λ))

/

0 ∞ # (x/2)N 2 Θ(λ − N x) exp(−x) dx N!

.

(III.3.87)

N =1

This gives for the Radon–Nikodym derivative with respect to the Lebesgue measure " $ ∞ # 1 λ N dµac = exp(−λ/N 2 ) . (III.3.88) dλ N 2 N ! 2N 2 N =1

The Radon–Nikodym derivative of the measures µN 0 with respect to the Lebesgue measure is dµN 0 λ|N | = 2 exp(−λ/N 2 ) dλ N |N |!

(III.3.89)

so that the Radon–Nikodym derivatives of µN 0 with respect to µac can be calculated to ρN 0 (λ) =

dµN 0 /dλ (N 2 |N |!)−1 exp(−λ/N 2 ) = λ|N | ( % λ &M ∞ 1 dµac /dλ exp(−λ/M 2 ) 2 2 M =1 M M !

.

(III.3.90)

2M

In the limit λ → 0 most of the ρN 0 vanish:

2 2 for |N | = 1, lim ρN 0 (λ) = λ→0 0 for |N | > 1.

(III.3.91)

Hence the contribution from the absolutely continuous part of the spectrum to the physical Hilbert space consists of just two vectors. This contribution corresponds to the fact that the 7 delta function δ(x1 )δ(x2 ) is a generalized eigenvector of M. The total physical Hilbert space is the sum of the contributions from Hpp and Hac , i.e the sum of Hpp which can be identified with L2 (R+ , rdr) and two vectors {e10 , e−10 }. Notice that if we had made the wrong identification of L2 (R2 , d2 x) with L2 (R+ , rdr) ⊗ L2 (S 1 , dφ) discussed above 2 then we would have had to use the vectors Ω"N 0 ∝ einφ e−r , n )= 0 to get a cyclic system. The spectral measures of these vectors all coincide and therefore the corresponding Radon–Nikodym derivatives would all be non vanishing at λ = 0 and the contribution to the physcal Hilbert space from the continuous spectrum would be infinite dimensional which would be physically wrong because the constraint x21 + x22 = 0 corresponds to only one point in phase space and the correponding physical Hilbert space should be finite dimensional.

III.4

SL(2, R) Models

In this section we will analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in section III.3: These are systems with an SL(2, R) gauge symmetry and the complications arise because non–compact semisimple Lie groups are not amenable (have no finite translation invariant measure). 105

We will apply the Master Constraint Progamme to two examples – in the first example we are concerned with the gauge group SO(2, 1) and in the second example with the gauge group SL(2, R), which is the double cover of SO(2, 1). We will see that both examples share the same problem – the spectrum of the Master Constraint Operator does not include the value zero. The reason for this is the following: The value zero in the spectrum of the Master Constraint Operator corresponds to the appearence of the trivial representation in a Hilbert space decomposition of the given unitary representation of the gauge group on the kinematical Hilbert space. Now, the groups SO(2, 1) and SL(2, R) (and all groups which have these two groups as subgroups, e.g. symplectic groups and SO(p, q) with p, q > 1 and p + q > 2 ) are non–amenable groups, see [52]. One characteristic of non–amenable groups is, that the trivial representation does not appear in a Hilbert space decomposition of the regular representation into irreducible unitary subrepresentations. Since the decomposition of the regular decomposition is often used to decompose tensor products, it will often happen, that a given representation of a non–amenable group does not include the trivial representation in its Hilbert space decomposition. 7 Since the value zero is not included in the spectrum of the Master Constraint Operator M " 7 7 we will use a redefined operator M = M −λmin as proposed in section III.2, where λmin is the

minimum of the spectrum. One can interprete this procedure as a quantum correction (which is proportional to !2 ). Nevertheless the redefinition of the Master Constraint Operator has to be treated carefully, since it is not guaranteed that all relations between the observables implied by the constraints are realized on the resulting physical Hilbert space. This phenomenon will occur in the second example. However we will show that it is possible to alter the Master Constraint Operator again and to obtain a physical quantum theory which has the correct classical limit. In both examples we will use the representation theory of SL(2, R) and its covering groups to find the spectra and the Direct Integral Decompositions with respect to the Master Constraint Operator. As we will see, the diagonalization of the Master Constraint Operator is equivalent to the diagonalization of the Casimir Operator of the given gauge group representation, which in turn is equivalent to the decomposition of the given gauge group representation into a direct sum and/or direct integral of irreducible unitary representations. Furthermore, we will see that our examples exhibit the structure of a dual pair, see [53]. These are defined to be two subgroups in a larger group, where one subgroup is the maximal commutant of the other and vice versa. In our examples one subgroup is the group generated by the observables and the other is the group generated by the constraints. Now, given this structure of dual pairs, one can show that the decomposition of the representation of the gauge group is equivalent to the reduction of the representation of the observable algebra (on the kinematical Hilbert space). This is explained in further detail in appendix III.7.1.2. In our examples this fact will help us to determine the induced representation of the observable algebra on the physical Hilbert space. Moreover we can determine in this way the induced inner product on the physical Hilbert space, so that it will not be necessary to perform all the steps of the Direct Integral Decomposition in order to find the physical inner product. We summarized the representation theory of the sl(2, R) algebra in appendix III.7.1.1. Appendix III.7.1.2 explains the theory of oscillator representations for sl(2, R), which is heavily used in the two examples. It also contains a discussion how the representation theory of dual pairs can be applied to our and similar examples.

106

III.4.1

A Model with Gauge Group SO(2, 1)

Here we consider the configuration space R3 with the three so(2, 1)-generators as constraints: Li = ,kij xj pk ,

{Li , Lj } = ,kij Lk

(III.4.1)

where ,kij = gkm ,mij , ,ijk is totally antisymmetric with ,123 = 1 and gik is the inverse of the metric gik = diag(+, +, −). Indices are raised and lowered with gik resp. gik and we sum over repeated indices. The gauge group SO(n, 1) was previously discussed in [18], where group averaging was used to construct the physical Hilbert space. We will compare the results of [18] and the results obtained here at the end of the section. The observable algebra of the system above is generated by d = xi p i

e+ = xi xi

e− = pi pi

(III.4.2)

.

This set of observables exhibits the commutation relations of the generators of the sl(2, R)algebra (which coincides with so(2, 1)): {d, e± } = ∓2e±

{e+ , e− } = 4d

(III.4.3)

.

We have the identity d2 − e+ e− = Li Li

(III.4.4)

between the Casimirs of the constraint and observable algebra. III.4.1.1

Quantization

We start with the auxilary Hilbert space L2 (R3 ) of square integrable functions of the coordinates. The momentum operators are pˆj = −i(!)∂j and the x ˆj act as multiplication operators. There arises no factor ordering ambiguity for the quantization of the constraints, but to ensure a closed observable algebra, we have to choose: dˆ = eˆ+ =

1 i x pˆi 2 (ˆ i x ˆx ˆi

+ pˆi x ˆi ) = x ˆi pˆi − 32 i! eˆ− = pˆi pˆi

(III.4.5)

The commutators between constraints and between observables are then obtained by replacing 5 6 1 the Poisson bracket with i! ·, · . The identity (III.4.4) is altered to be: ˆ iL ˆi dˆ2 − 12 (ˆ e+ eˆ− + eˆ− eˆ+ ) − 34 !2 = L

(III.4.6)

.

(From now on we will skip the hats and set ! to 1.) For the implementation of the Master Constraint Programme we have to construct the 7 spectral resolution of the M 7 := L2 + L2 + L2 = Li Li + 2L2 = d2 − 1 (e+ e− + e− e+ ) − M 1 2 3 3 2

3 4

+ 2L23

.

(III.4.7)

7 and L3 commute, so we can To this end we will use the following strategy: The operators M diagonalize them simultaneously. The diagonalization of L3 is easy to achieve, its spectrum 107

7 on each eigenspace of being purely discrete, namely spec(L3 ) = Z. Now we can diagonalize M 7 L3 seperately. On these eigenspaces the diagonalization of M is equivalent to the diagonalization of the so(2, 1)-Casimir Li Li and because of identity (III.4.6) equivalent to the diagonalization of the sl(2, R)-Casimir C = − 14 (d2 − 12 (e+ e− + e− e+ )). As we will show below the sl(2, R)representation given by (III.4.5) is a tensor product of three representations, which are known as oscillator and contragredient oscillator representations. To obtain the spectral resolution of the Casimir C, we will reduce this tensor product into its irreducible components. III.4.1.2

The Oscillator Representations and its Reduction

For the reduction process it will be very convenient to work with the following basis of the sl(2, R)-algebra: h = 14 (e+ + e− ) n+ = − 12 (i d − 12 (e+ − e− )) n− = 12 (i d + 12 (e+ − e− ))

= 12 (a†1 a1 + a†2 a2 − a†3 a3 + 12 ) = 12 (a†1 a†1 + a†2 a†2 − a3 a3 ) = 12 (a1 a1 + a2 a2 − a†3 a†3 )

C = 14 (d2 − 12 (e+ e− + e− e+ )) = −h2 + 12 (n+ n− + n− n+ )

with commutation and adjointness relations 5 6 5 + −6 h, n± = ±n± n , n = −2h

(n+ )† = n−

.

,

(III.4.8)

(III.4.9)

Here we introduced the anihilation and creation operators ai =

√1 (xi 2

+ ipi ) and a†i =

√1 (xi 2

− ipi ) .

(III.4.10)

Now it is easy to see that this representation is a tensor product of the following three sl(2, R)representations: hi = 12 (a†i ai + 12 ) for i = 1, 2 1 † † n+ i = 2 ai ai

and h3 = − 12 (a†3 a3 + 12 ) 1 n+ 3 = − 2 a3 a3 1 † † n− 3 = − 2 a3 a3

1 n− i = 2 ai ai

(III.4.11)

These representations are known as oscillator representation ω (for i = 1, 2) and contragredient oscillator representation ω ∗ (for i = 3), see [51] and III.7.1.2, where these representations are explained. The oscillator representation is the sum of two irreducible representations, which are the representations D(1/2) and D(3/2) from the positive discrete series of the double cover of Sl(2, R) (corresponding to even and odd number Fock states). Similarly ω ∗ " D∗ (1/2)⊕D∗ (3/2), where D∗ (1/2) and D∗ (3/2) are from the negative discrete series. As mentioned before we will reduce this tensor product to its irreducible subrepresentations in order to obtain the spectrum of the Casimir (and with it the spectrum of the Master Constraint Operator). In appendix III.7.1.2 one can find the general strategy and some formulas to reduce such tensor products. Furthermore appendix III.7.1.1 reviews the sl(2, R)-representations, which will appear below. To begin the reduction of ω ⊗ ω ⊗ ω ∗ we will reduce ω ⊗ ω. This we can achieve by utilizising the observable L3 . It commutes with the sl(2, R)-algebra (III.4.8), therefore according to 108

Schur’s Lemma its eigenspaces are left invariant by the sl(2, R)-algebra, i.e. its eigenspaces are subrepresentations of sl(2, R). To diagonalize L3 and reduce the tensor product ω ⊗ ω we will employ the “polarized” anihilation and creation operators 1 A± = √ (a1 ∓ ia2 ) 2

1 A†± = √ (a1 ± ia2 ). 2

(III.4.12)

With the help of these, we can write † † † 1 2 (A+ A+ + A− A− − a3 a3 A†+ A†− − 12 a3 a3 A+ A− − 12 a†3 a†3 A†+ A+ − A†− A− .

h = n+ = n− = L3 =

+ 12 ) (III.4.13) (III.4.14)

In the following we will denote by |k+ , k− , k3 > Fock states with respect to A†+ , A†− and a†3 . The operator L3 acts on them diagonally. Its eigenspaces V (±j) corresponding to the eigenvalue ±j, j ∈ N are generated by {|j, 0, k3 >, k3 ∈ N} and {|0, j, k3 >, k3 ∈ N} respectively, i.e. V (j) is (the closure of) the linear span of the |j, 0, k3 >’s resp. |0, j, k3 >’s and all vectors are obtained by applying repeatedly n+ to them. These eigenspaces are invariant subspaces of the representation (III.4.13). This representation restricted to V (j) is still a tensor product representation, namely the representation D(|j| + 1) ⊗ ω ∗ . Its factors are given by h12 |V (j) = 12 (A†+ A+ + A†− A− + 1)|V (j)

and

n− 12 |V (j) = A+ A− |V (j)

and

† † n+ 12 |V (j) = A+ A− |V (j)

and

h3 |V (j) = − 12 (a†3 a3 + 12 )|V (j) 1 n+ 3 |V (j) = − 2 a3 a3 1 † † n− 3 |V (j) = − 2 a3 a3

.

(III.4.15)

Since h12 has a smallest eigenvalue 12 (|j| + 1) on the subspace V (j), this subspace carries a D(|j| + 1)–representation from the positive discrete series (of SL(2, R)) with lowest weight 1 2 (|j| + 1) (see III.7.1.1). ( ∗ So far we have achieved the reduction ω ⊗ ω ⊗ ω ∗ " (D(1) ⊕ ∞ j=0 2D(j + 1)) ⊗ ω . To reduce the representation (III.4.13) completely, we have to consider tensor products of the form D(|j|+ 1)⊗ D∗ (1/2) and D(|j|+ 1)⊗ D∗ (3/2). We take this reduction from [51], see also III.7.1.2 and III.7.1.1 for a description of the sl(2, R)-representations, appearing below: For j even, we have ! 1 +i∞ # 2 ∗ D(|j| + 1) ⊗ D (1/2) " P (t, 1/4)dµ(t) ⊕ D(|j| + 1/2 − 2l) 1 2

l

with 0 ≤ 2l < |j| − 1/2, l ∈ N

(III.4.16)

and for j odd we get ∗

D(|j| + 1) ⊗ D (1/2) "

!

1 2 +i∞ 1 2

P (t, −1/4)dµ(t) ⊕

# l

D(|j| + 1/2 − 2l)

with 0 ≤ 2l < |j| − 1/2, l ∈ N . 109

(III.4.17)

In particular, we have for j = 0 ∗

D(1) ⊗ D (1/2) "

!

1 2 +i∞ 1 2

P (t, 1/4)dµ(t)

(III.4.18)

.

The remainig tensor products are ∗

D(1) ⊗ D (3/2) "

!

1 2 +i∞ 1 2

P (t, −1/4)dµ(t)

D(|j| + 1) ⊗ D ∗ (3/2) " D(|j|) ⊗ D∗ (1/2)

for j > 0 .

(III.4.19)

P (t, ,), , = 14 , − 14 is the principal series (of the metaplectic group, i.e. the double cover of SL(2, R)) characterized by an h-spectrum spec(h) = {, + z, z ∈ Z} and a Casimir C(P (t, ,)) = t(1 − t)Id. The measure dµ(t) is the Plancherel measure on the unitary dual of the metaplectic goup. The representations D(l + 1/2), l ∈ N − {0} are positive discrete series representations of the metaplectic group. The h-spectrum in these representations is given by { 12 (l + 1/2) + n, n ∈ N} and the Casimir by C(D(l + 1/2)) = − 14 (l + 1/2)2 + 12 (l + 1/2). The spectrum of the Casimir C is non-degenerate on each tensor product D(k) ⊗ D∗ (l), l = 1 3 2 , 2 , i.e. the Casimir discriminates the irreducible representations, which appear in this tensor product and the irreducible representations have multiplicity one. The spectrum of h is nondegenerate in each irreducible representation of the metaplectic group . This implies that we can find a (generalized) basis |j, ,, c, h >, which is labeled by the L3 -eigenvalue j, the values , = 14 , − 14 , the Casimir eigenvalue c and the h-eigenvalue h. Summarizing, we have for the (highly degenerate) spectrum of the Casimir C = −h2 + 1 3 + − − + 2 (n n + n n ) on L2 (R ): spec(C) = { 14 + x2 , x ∈ R, x ≥ 0} ∪ {− 14 (l + 1/2)2 + 12 (l + 1/2), l ∈ N − {0}}

.

(III.4.20)

The continuous part of the spectrum originates from the principal series P (t, 1/4) and P (t, −1/4) and the discrete part from those positive discrete series representations D(l), which appear in the decompositions above. This results in the following expression for the spectrum of the so(2, 1)-Casimir Li Li = (4C − 34 ): spec(Li Li ) = { 14 + s2 , s ∈ R s ≥ 0} ∪ {−q 2 + q, q ∈ N − {0}}

.

(III.4.21)

As explained in appendix III.7.1.1 these values correspond to the principal series P ( 12 + s, 0) of SO(2, 1) and the positive or negative discrete series D(2q) resp. D∗ (2q) for q ∈ N − {0}. In the SO(2, 1)-principal series the spectrum of L3 is given by spec(L3 ) = {Z}. In the discrete series D(2q) we have spec(L3 ) = {q + n, n ∈ N} and in D∗ (q) spec(L3 ) = {−q − n, n ∈ N}. III.4.1.3

The Physical Hilbert space

7 = Li Li + 2L2 on the Now we can determine the spectrum of the Master Constraint Operator M 3 j–eigenspaces V (j) of L3 . From the principal series we get the continous part of the spectrum 7 |V (j) ) = { 1 + s2 + 2j 2 , s ∈ R, s ≥ 0} speccont (M 4 110

(III.4.22)

and from the discrete series the discrete part (for j ≥ 1, since there is no discrete part for j = 0) 7 |V (j) ) = {(−q 2 + q) + 2j 2 , q ∈ N, q ≤ |j|} ≥ 2 . specdiscr (M

(III.4.23)

(The inequality q ≤ |j| follows from the fact, that in a representation D(2q) or D∗ (2q) we have |j| ≥ q for the L3 -eigenvalues j.) One can see immedeatily, that zero is not included in the spectrum of the Master Constraint, 7" = the lowest generalized eigenvalue being 14 . Therefore we alter the Master Constraint to M 7 − 1 (!2 ) where appropriate powers of ! have been restored. The generalized null eigenspace M 4 7 " is given by the linear span of all states |j = 0, ,, c = 1 , h >. The spectral measure of of M 4

7 " induces a scalar product on this space, which can then be completed to a Hilbert space. In M particular with this scalar product one can normalize the states |j = 0, ,, c = 14 , h >, obtaining an ortho–normal basis ||,, h >>. This Hilbert space has to carry a unitary representation of the metaplectic group. Actually, it carries a sum of two irreducible representations P (t = 1/2, 1/4) and P (t = 1/2, −1/4), corresponding to the labels , = 1/4 and , = −1/4 of the basis {||,, h >>}. States in these representations are distinguished by the transformation under the reflection R3 : x3 &→ −x3 , which is a group element of O(2, 1). As an operator on L2 (R3 ) it acts as: ˆ 3 : ψ(x1 , x2 , x3 ) &→ ψ(x1 , x2 , −x3 ) . R

(III.4.24)

ˆ 3 acts on states with , = 1 as the identity operator (since these states are linear combinations R 4 of even number Fock states with respect to a†3 ) and on states with , = − 14 by multiplying them with (−1) (since these states are linear combinations of odd number Fock states). It seems natural, to exclude the states with nontrivial behaviour under this reflection. This leaves us with the unitary irreducible representation P (t = 1/2, 1/4). As explained in III.7.1.1 the action of the observable algebra sl(2, R) on the states ||h >>:= ||1/4, h >> is determined (up to a phase, which can be fixed by adjusting the phases of the states ||h >>) by this representation to be: h||h >> = h||h >>

(h ∈ { 14 + Z})

n+ ||h >> = (h + 12 )||h + 1 >>

n− ||h >> = (h − 12 )||h − 1 >>

(III.4.25)

.

This gives for matrix elements of the observables d and e± (see III.4.5) = 2h δh! ,h ± (h +

1 ! 2 )δh ,h+1

± (h −

1 ! 2 )δh ,h−1

(III.4.26) .

(III.4.27)

The operators e+ = x ˆi x ˆi and e− = pˆi pˆi are indefinite operators, i.e. their spectra include positive and negative numbers. To sum up, we obtained a physical Hilbert space, which carries an irreducible unitary representation of the observable algebra. In contrast to these results the group averaging procedure in the framework of Refined Algebraic Quantization in [18] leads to (two) superselection sectors and therefore to a reducible representation of the observables. These sectors are functions with compact support inside the light cone and functions with compact support outside the light 111

cone. Hence the observable e+ = x ˆi x ˆi is either strictly positive or strictly negative definite on theses superselection sectors. From that point of view our physical Hilbert space is preferred because physically e+ should be indefinite. However, as mentioned in [18] the appearence of superselection sectors may depend on the choice of the domain Φ, on which the group averaging procedure has to be defined and thus other choices of Φ may not suffer from this superselection problem. We see, at least in this example, that the Direct Integral Decomposition method gives a more natural and unique result. However, as the given system lacks a realistic interpretation anyway, this difference may just be an artefact of a pathological model.

III.4.2 III.4.2.1

A Model with Two Hamiltonian Constraints and Non – Compact Gauge Orbits Introduction of the Model

Here we consider a reparametrization invariant model introduced by Montesinos, Rovelli and Thiemann in [54]. It has an Sl(2, R) gauge symmetry and a global O(2, 2) symmetry and has attracted interest because its constraint structure is in some sense similar to the constraint structure found in general relativity. Further work on this model has appeared in [55, 56, 57, 58] and references therein. We will shortly summarize the classical (canonical) theory (see [54] for an extended discussion). The configuration space is R4 parametrized by coordinates (u1 , u2 ) and (v1 , v2 ) and the canonically conjugated momenta are (p1 , p2 ) and (π1 , π2 ). The system is a totally constrained (first class) system. The constraints form a realization of an sl(2, R)-algebra: H1 = 12 (5 p2 − 5v 2 )

H2 = 12 (5π 2 − 5u2 )

{H1 , H2 } = D

{H1 , D} = −2H1

D = 5u · p5 − 5v · 5π

(III.4.28)

{H2 , D} = 2H2

(III.4.29)

The canonical Hamiltonian governing the time evolution (which is pure gauge) is H = N H1 + M H2 + λ D where N, M and λ are Lagrange multipliers. Since H1 and H2 are quadratic in the momenta and their Poisson bracket gives a constraint which is linear in the momenta, one could say that this model has an analogy with general relativity. There, one has Hamiltonian constraints H(x) quadratic in the momenta and diffeomorphism constraints D(x) linear in the momenta which have the Poisson structure {H(x), H(y)} ∼ δ(x − y)D(x) and {H(x), D(y)} ∼ δ(x − y)H(x). However, one can make the following canonical transformation to new canonical coordinates (Ui , Vi , Pi , Πi ), i = 1, 2 that transforms the constraint into phase space functions which are linear in the momenta: ui = pi = H1 = −P1 V1 − P2 V2

√1 (Ui + Πi ) 2 √1 (−Vi + Pi ) 2

vi = πi =

H2 = −U1 Π1 − U2 Π2

√1 (Vi + Pi ) 2 √1 (−Ui + Πi ) 2

(III.4.30)

D = P1 U1 + P2 U2 − V1 Π1 − V2 Π2 (III.4.31) .

These coordinates have the advantage, that the constraints act on the configuration variables (U1 , V1 ) and (U2 , V2 ) in the defining two-dimensional representation of sl(2, R) (i.e. by matrix multiplication). 112

For reasons that will become clear later, it is easier for us to stick to the old coordinates (ui , vi , pi , πi ). Now we will list the Dirac observables of this system. They reflect the global O(2, 2)symmetry of this model and are given by (see [54]) O12 = u1 p2 − p1 u2

O23 = u2 v1 − p2 π1

O14 = u1 v2 − p1 π2

O34 = π1 v2 − v1 π2

O13 = u1 v1 − p1 π1

O24 = u2 v2 − p2 π2

(III.4.32)

They constitute the Lie algebra so(2, 2) which is isomorphic to so(2, 1)×so(2, 1). A basis adapted to the so(2, 1) × so(2, 1)-structure is (see [57]) Q1 = 12 (O23 + O14 )

P1 = 12 (O23 − O14 )

Q2 = 12 (−O13 + O24 )

P2 = 12 (−O13 − O24 )

Q3 = 12 (O12 − O34 )

P3 = 12 (O12 + O34 )

(III.4.33)

The Poisson brackets between these observables are {Qi , Qj } = ,ij k Qk

{Pi , Pj } = ,ij k Pk

{Qi , Pj } = 0

(III.4.34)

where ,ij k = glk ,ijk , with glk being the inverse of the metric glk = diag(+1, +1, −1). The LeviCivita symbol ,ijk is totally antisymmetric with ,123 = 1 and we sum over repeated indices. Lateron the (ladder) operators Q± := √12 (Q1 ± iQ2 ) and P± := √12 (P1 ± iP2 ) will be usefull. One can find the following identities between observables and constraints (see [57]): Q21 + Q22 − Q23 = P12 + P22 − P32 = 14 (D2 + 4H1 H2 )

(III.4.35)

4Q3 P3 = (5u − 5v )(H1 + H2 ) − (5u · p5 + 5v · 5π )D + (5u + 5v )(H1 − H2 ) (III.4.36) 2

2

2

2

They imply that on the constraint hypersurface we have Qi = 0 ∀i or Pi = 0 ∀i. certain submanifold of the phase space R8 . Notice that the constraint hypersurface consists of the disjoint union of the following five varieties: {Qj = Pj = 0, j = 1, 2, 3}, {±Q3 > 0, P3 = 0}, {Q3 = 0, ±P3 > 0}. III.4.2.2

Quantization

For the quantization we will follow [54] and choose the coordinate representation where the momentum operators act as derivative operators and the configuration operators as multiplication operators on the Hilbert space L2 (R4 ) of square integrable functions ψ(5u, 5v ): 5 u ψ(5u, 5v ) p5ˆψ(5u, 5v ) = −i!∇

ˆ ψ(5u, 5v ) = −i!∇ 5 v ψ(5u, 5v ) 5π

u ˆi ψ(5u, 5v ) = ui ψ(5u, 5v )

vˆi ψ(5u, 5v ) = vi ψ(5u, 5v ) .

(III.4.37)

In the following we will skip the hats and set ! = 1. For the constraint algebra to close we have to quantize the constraints in the following way: H1 = − 12 (∆u + 5v 2 ) 5 6 H1 , H2 = iD

H2 = − 12 (∆v + 5u2 ) 5 6 D, H1 = 2iH1 113

5 u − 5v · ∇ 5 v) D = −i(5u · ∇ 5 6 D, H2 = −2iH2 .

(III.4.38)

There arises no factor ordering ambiguity for the quantization of the observable algebra. The algebraic properties are preserved in the 5quantization process, i.e. Poisson brackets between 6 observables Oij are simply replaced by −i ·, · . We introduce a more convenient basis for the constraints: H+ = H1 + H2

H− = H1 − H2

D=D

(III.4.39)

.

H− is just the sum and difference of Hamiltonians for one-dimensional harmonic oscillators. (It is the generator of the compact subgroup SO(2) of Sl(2, R) and has discrete spectrum in Z). The commutation relations are now: 5 6 5 6 5 6 H+ , D = −2iH− H+ , H− = −2iD . (III.4.40) H− , D = −2iH+ The operator

2 2 − H− ) C = 14 (D2 + H+

(III.4.41)

commutes with all three constraints (III.4.39), since it is the (quadratic) Casimir operator for sl(2, r) (see Appendix III.7.1.1). According to Schur’s lemma, it acts as a constant on the irreducible subspaces of the sl(2, R) representation given by (III.4.39). The quantum analogs of the classical identities (III.4.35) are 2 2 Q21 + Q22 − Q23 = P12 + P22 − P32 = 14 (D2 + H+ − H− )=C

4Q3 P3 = (5u − 5v )(H+ ) − (5u · p5 + 5v · 5π )D + (5u + 5v )(H− ) 2

III.4.2.3

2

2

2

(III.4.42) .

(III.4.43)

The Oscillator Representation

We are interested in the spectral decomposition of the Master Constraint Operator, which we define as 7 = D 2 + H 2 + H 2 = 4C + 2H 2 . M (III.4.44) + − −

The Master Constraint Operator is the sum of (a multiple of) the Casimir operator and H− , which commutes with the Casimir. Therefore we can diagonalize these two operators simulta7 We can achieve a diagonalization of the Casimir by neously, obtaining a diagonalization of M. looking for the irreducible subspaces of the sl(2, R)-representation given by (III.4.39), since the Casimir acts as a multiple of the identity operator on these subspaces. Hence we will attempt do determine the representation given by (III.4.39). By introducing creation and annihilation operators 1 ai = √ (ui + ∂ui ) 2 1 bi = √ (vi + ∂vi ) 2 we can rewrite the constraints as H− =

#

(a†i ai − b†i bi )

(III.4.45) (III.4.46)

(III.4.47)

i=1,2

H+ = − D =

1 a†i = √ (ui − ∂ui ) 2 1 † bi = √ (vi − ∂vi ) 2

1 # 2 (ai + (a†i )2 + b2i + (b†i )2 ) 2

(III.4.48)

i=1,2

i # (−a2i + (a†i )2 + b2i − (b†i )2 ) 2 i=1,2

114

.

(III.4.49)

This sl(2, R) representation is a tensor product of the following four representations (with i ∈ {1, 2}): (h− )ui = a†i ai +

1 2

1 (h+ )ui = − ((a†i )2 + a2i ) 2 i † 2 dui = ((ai ) − a2i ) 2

(h− )vi = −b†i bi −

1 2

1 (h+ )vi = − ((b†i )2 + b2i ) 2 i dvi = (−(b†i )2 + b2i ) . 2

(III.4.50)

The ui –representations are known as oscillator representations ω and the vi –representations as contragredient oscilalator representations ω ∗ , see appendix III.7.1.2 for a discussion of these representations. As is also explained there these representation are reducible into two irreducible representations D(1/2) and D(3/2) for the oscillator representation ω and D∗ (1/2) and D ∗ (3/2) for the contragredient oscillator representation ω ∗ . The representation D(1/2) respectively D ∗ (1/2) acts on the space of even number Fock states, whereas D(3/2) respectively D ∗ (3/2) acts on the space of uneven Fock states. The representations D(1/2) and D(3/2) are members of the positive discrete series (of the two–fold covering group of Sl(2, R)), D∗ (1/2) and D ∗ (3/2) are members of the negative discrete series. (We have listed all sl(2, R)-representations in appendix III.7.1.1.) Our aim is to reduce the tensor product ω ⊗ ω ⊗ ω ∗ ⊗ ω ∗ into its irreducible components. The isotypical component with respect to the trivial representation would correspond to the physical Hilbert space. To begin with we consider the tensor product ω ⊗ ω. The discussion for ω ∗ ⊗ ω ∗ is analogous. To this end we utilize the observable O12 (and O34 for the tensor product ω ∗ ⊗ ω ∗ ). Since O12 commutes with the sl(2, R)-generators the eigenspaces of O12 are sl(2, R)-invariant. The observable O12 is diagonal in the ‘polarized’ Fock basis, which is defined as the Fock basis with respect to the new creation and annihilation operators 1 A± = √ (a1 ∓ ia2 ) 2

1 A†± = √ (a†1 ± ia†2 ). 2

(III.4.51)

The ‘polarized’ creation and annihilation operators for the v-coordiantes are 1 B± = √ (b1 ∓ ib2 ) 2

1 † B± = √ (b†1 ± ib†2 ). 2

(III.4.52)

With help of these operators we can write the sl(2, R)-generators for the ω ⊗ ω representation and for the ω ∗ ⊗ ω ∗ representation as h−A = A†+ A+ + A†− A− + 1

† † h−B = −B+ B+ − B− B− − 1

hA + = −(A+ A− + A†+ A†− )

† † h+B = −(B+ B− + B+ B− )

dA = i(A†+ A†− − A+ A− )

† † dB = i(B+ B− − B+ B− )

(III.4.53)

and the observables O12 and O34 as O12 = u1 p2 − p1 u2 = A†+ A+ − A†− A−

† † O34 = π1 v2 − v1 π2 = −B+ B+ + B− B−

115

.

(III.4.54)

The (common) eigenspaces (corresponding to the eigenvalues j, j " ∈ Z) for these observables " , k " >; k − k = j and k " − k " = j " ; k , k , k " , k " ∈ N}, where are spanned by {|k+ , k− , k+ + − + − + − − − + " " |k+ , k− , k+ , k− > denotes a Fock state with respect to the annihilation operators A+ , A− , B+ , B− . A closer inspection reveals that these eigenspaces are indeed invariant under the sl(2, R)-algebra. The action of the sl(2, R)-algebra on each of the above subspaces is a realization of the tensor product representation D(|j|+1)⊗D∗ (|j " |+1) (see III.7.1.1). This can be verified by considering the h−A – and the h−B –spectrum on these subspaces. The h−A –spectrum is bounded from below by (|j| + 1), whereas the h−B –spectrum is bounded from above by −(|j " | + 1). This characterizes D(|j| + 1)– and D∗ (|j " | + 1)–representations respectively. Up to now we have achieved ) * ) * ∞ ∞ # # ∗ ∗ ω ⊗ ω ⊗ ω ⊗ ω = D(1) ⊕ 2D(k) ⊗ D (1) ⊕ 2D (k) ∗

∗

k=2

.

(III.4.55)

k=2

For a complete reduction of ω ⊗ ω ⊗ ω ∗ ⊗ ω ∗ we have to reduce the tensor products D(|j| + 1) ⊗ D ∗ (|j " | + 1). III.4.2.4

The Spectrum of the Master Constraint Operator

In [59] the decomposition of all possible tensor products between unitary irreducible representations of SL(2, R) was achieved. ( Actually [59] considers only representations of SL(2, R)/ ± Id, i.e. representations with uneven j and j " . However the results generalize to representations with even j or j " . See [60] for a reduction of all tensor products of SL(2, R), using different methods.) The strategy in this article is to calculate the spectral decomposition of the Casimir operator. Since the Casimir commutes with H− one can consider the Casimir operator on each eigenspace of H− . 7 = 4C + 2H− we can easily adapt the Since the Master Constraint Operator is the sum M results of [59] for the spectral decomposition of the Master Constraint Operator. In the following we will shortly summarize the results for the spectrum of the Master Constraint Operator. The explicit eigenfunctions are constructed in appendix III.7.1.4. To this end we define the subspaces V (k, j, j " ), k ∈ Z, |j| ∈ N by H− |V (k,j,j ! ) = k

and

O12 |V (k,j,j !) = j

and

O34 |V (k,j,j !) = j " .

(III.4.56)

V (k, j, j " ) is the H− -eigenspace corresponding to the eigenvalue k of the tensor product representation D(|j| + 1) ⊗ D∗ (|j " | + 1). Since the H− -spectrum is even for (j − j " ) even and uneven for (j − j " ) uneven these subspaces are vacuous for k + j − j " uneven. One result of [59] is, that the spectrum of the Casimir operator is non-degenerate on these subspaces, which means that there exists a generalized eigenbasis in L2 (R4 ) labeled by (k, j, j " ) and the eigenvalue λC of the Casimir. The spectrum of the Casimir C on the subspace V (k, j, j " ) has a discrete part only if k > 0 for |j| − |j " | ≥ 2 or k < 0 for |j| −| j " | ≤ 2. There is no discrete part if ||j| − |j " || < 2. The discrete part is for (j − j " ) and k even λC = t(1 − t) with

= 0, −2, −6, . . . .

t = 1, 2, . . . , 12 min(|k|, ||j| − |j " ||) 116

(III.4.57)

For (j − j " ) and k odd we have λC = t(1 − t) with t = 32 , 52 , . . . , 12 min(|k|, ||j| − |j " ||) 35 = − 34 , − 15 4 ,− 4 ... .

The continuous part is in all cases the same and given by: 5 & λC = 14 + x2 with x ∈ 0, ∞

(III.4.58)

(III.4.59)

.

The discrete part corresponds to unitary irreducible representations from the positive and negative discrete series of SL(2, R), the continous part corresponds to the (two) principal series of SL(2, R) (see appendix III.7.1.1). For the spectrum of the Master Constraint Operator we have to multiply with 4 and add 2 2k : 2 2 2 λM b = 4t(1 − t) + 2k ≥ 2k − k + 2|k|

with

t = 1, 2, . . . , 12 min(|k|, ||j| − |j " ||) for even k

with t = 32 , 52 . . . , 12 min(|k|, ||j| − |j " ||) for odd k and

λM b =1+

x2

+

2k2

(III.4.60)

>0

(III.4.61)

As one can immediately see, the spectrum does not include zero. Since we have no discrete spectrum for k = 0 the lowest generalized eigenvalue for the Master Constraint is 1 from the continuous part. We will attempt to overcome this problem by introducing a quantum correction to the Master Constraint Operator. Since 1 is the minimum of the spectrum we substract 1 (!2 if units are restored) from the Master Constraint Operator. For the modified Master Constraint Operator we get one solution appearing in the spectral decomposition for each value of j and j " . We call this solution |λC = 14 , k = 0, j, j " > and the linear span of these soltutions SOL" .(The above results show, that these quantum numbers are sufficient to label uniquely vectors in the kinematical Hilbert space.) At the classical level we have several relations between observables, which are valid on the constraint hypersurface. For a physical meaningful quantization we have to check, whether these relations are valid or modified by quantum corrections. For our modified Master Constraint Operator this seems not to be the case: At the classical level we have Q3 = 0 or P3 = 0. But on SOL" , these observables evaluate to: Q3 |λC = 14 , k = 0, j, j " > = P3 |λC = 14 , k = 0, j, j " > =

1 2 (j 1 2 (j

− j " ) |λC = 14 , k = 0, j, j " > + j " ) |λC = 14 , k = 0, j, j " >

.

(III.4.62)

Since j and j " are arbitrary whole numbers, both Q3 and P3 can have arbitrary large eigenvalues (on the same eigenvector) in SOL" . To solve this problem, we will modify the Master Constraint Operator again, by adding a constraint, which implements the condition Q3 = 0 or P3 = 0. Together with the identities (III.4.42) this would ensure that Qi = 0 ∀i or Pi = 0 ∀i modulo quantum corrections. 7 "" = M 7 −1 + (Q3 P3 )2 .(This operator is herOne possibility for the modified constraint is M mitian, since Q3 and P3 commute.) Because of the last relation of (III.4.42) this modification 117

can be seen as adding the square (of one quarter) of the right hand side of this relation, i.e. the added part is the square of a linear combination of the constraints. 7 "" , since we used Q3 and P3 (or O12 and O34 ) in We already know the spectral resolution of M the reduction process for the Master Constraint Operator. Solutions to the Master Constraint 7 "" are the states |λC = 1 , k = 0, j, j " > with |j| = |j " |. We call this solution space Operator M 4 SOL"" . (Up to now this spase is just the linear span of states |λC = 14 , k = 0, j, j " > with |j| = |j " |. Later we will specify a topology for this space.) Now the observable algebra (III.4.33) does not leave this solution space invariant, since not all observables commute with the added constraint Q3 P3 . However, the observables (III.4.33) are redundant on SOL"" , since they obey the relations (III.4.42). So the question is, whether one can find enough observables, which commute with Q3 P3 (and with the constraints, we started with) to carry all relevant physical information. Apart from Q3 and P3 the operators Q21 + Q22 and P12 + P22 commute with the added constraint. But the latter do not carry additional information about physical states, because of the first relation in (III.4.42). Operators of the form p1 (Q)Q3 +p2 (P )P3 , where p1 (Q) (resp. p2 (P )) represents a polynomial in the Q-observables (P -observables), commute with Q3 P3 on the subspace defined by Q3 P3 = 0. Likewise operators of the form p1 (Q)|sgn(Q3 )| + p2 (P )|sgn(P3 )|, where sgn has values 1, 0 and −1 (and is defined by the spectral theorem) leave SOL"" (formally) invariant. In the following we will take as observable algebra the algebra generated by the elementary operators |sgn(Q3 )|Qi |sgn(Q3 )| and |sgn(P3 )|Pi |sgn(P3 )|. This algebra is closed under taking adjoints. Notice, however, that we may add operators such as |sgn(Q3 )|Q+ Q+ |sgn(Q3 )| which does not leave the sectors invariant and thus destroy the superselection structure which is a physical difference from the results of [55]. The next section shows that the latter operator transforms states from the sector {sgn(Q3 ) = −1, sgn(P3 ) = 0} to the sector {sgn(Q3 ) = +1, sgn(P3 ) = 0} (since Q± , P± are ladder operators, which raise or lower the Q3 , P3 eigenvalues by 1 respectively). Likewise one can construct operators wich transform from the sector {sgn(P3 ) = 0} to {sgn(Q3 ) = 0} and vice versa: For instance |sgn(P3 )| P+ · · · P+ (1 − |sgn(Q3 )|)(1 − |sgn(P3 )|) Q+ · · · Q+ |sgn(Q3 )|

(III.4.63)

has this property and leaves the solution space to the modified Master Constraint Operator invariant. Its adjoint is of the same form, transforming from the sector {sgn(Q3 ) = 0} to the sector {sgn(P3 ) = 0}. Thus we may map between all five sectors mentioned before except for the origin. There seems to be no natural exclusion principle for these operators from the point of view of DID and thus we should take them seriously. III.4.2.5

The Physical Hilbert space

Now one can use the spectral measure for the Master Constraint Operator and construct a scalar product in SOL" and SOL"" and then complete them into Hilbert spaces H" and H"" . This is done explicitly in Appendix III.7.1.4, here we only need that this can be done in principle. The so achieved Hilbert space H" has to carry a unitary representation of the observable algebra sl(2, R) × sl(2, R) (since these observables commute with the constraints). In particular we already know the spectra of Q3 and P3 to be the integers Z, since we diagonalized them simultaneously with the Master Constraint Operator. These spectra are discrete, which means that in the constructed scalar product the states |λC = 14 , k = 0, j, j " > (which are eigenstates 118

for Q3 and P3 ) have a finite norm. So we can normalize them to states ||j, j " >> and in this way obtain a basis of H" . Now, because of the identity (III.4.42) we also know the value of the sl(2, R) Casimirs 2 Q1 + Q22 − Q23 and P12 + P22 − P32 on H" to be 14 . Together with the fact that H" has a normalized eigenbasis {||j, j " >>, j, j " ∈ Z} (with respect to Q3 and P3 ) we can determine the unitary representation of sl(2, R) × sl(2, R) to be P (t = 1/2, , = 0)Q ⊗ P (t = 1/2, , = 0)P (see Appendix III.7.1.1). This fixes the action of the (primary) observable algebra to be (modulo phase factors, which can be made to unity by adjusting phases of the basis vectors): Q+ ||j, j " >> = Q− ||j, j " >> = Q3 ||j, j " >>

=

P+ ||j, j " >>

=

P− ||j, j " >>

=

P3 ||j, j " >>

=

1 √ ((j − j " ) + 1) ||j 2 2 1 √ ((j − j " ) − 1) ||j 2 2 " " 1 2 (j − j ) ||j, j >>

+ 1, j " − 1 >>

− 1, j " + 1 >>

1 √ ((j + j " ) + 1) ||j + 1, j " + 1 >> 2 2 1 √ ((j + j " ) − 1) ||j − 1, j " − 1 >> 2 2 " " 1 2 (j + j )| |j, j >>

(III.4.64)

From these results we can derive the action of the altered observable algebra on H"" , i.e. on states ||j, ,j >> with , = ±1: Θ(Q3 )Q+ Θ(Q3 ) ||j, ,j >>

Θ(Q3 )Q− Θ(Q3 ) ||j, j " >> Q3 ||j, ,j >>

"

= δ−1,2 (1 − δ−1,j ) √12 (j + 12 ) ||j + 1, ,(j + 1) >>

= δ−1,2 (1 − δ+1,j ) √12 (j − 12 ) ||j − 1, ,(j − 1) >> = δ−1,2 j ||j, ,j >>

Θ(P3 )P+ Θ(P3 ) ||j, j >>

= δ1,2 (1 − δ−1,j ) √12 (j + 12 ) ||j + 1, ,(j + 1) >>

P3 ||j, ,j >>

= δ1,2 j ||j, ,j >>

Θ(P3 )P− Θ(P3 ) ||j, j " >>

= δ1,2 (1 − δ+1,j ) √12 (j − 12 ) ||j − 1, ,(j − 1) >>

(III.4.65)

where we abbreviated |sgn(O)| by Θ(O). The state |0, 0 > is annihilated by all (altered) observables. III.4.2.6

Algebraic Quantization

In [55] the SL(2, R)-model has been quantized in the Algebraic and Refined Algebraic Quantization framework. Both frameworks resulted in the same representation of the observable algebra. We will shortly review the methods and results of the Algebraic Quantization scheme in order to compare them with the Master Constaint Programme. In this scheme one starts with the auxilary Hilbert space L2 (R4 ), the constraints (III.4.38) and a ∗ –algebra of observables A∗ . One looks for a solution space for the constraints which carries an irreducible representation of A∗ and for a scalar product on this space in which the star–operation becomes the adjoint operation. The solution space V˜ , which was found in [55] is the linear span of states |j, ,j > where j is in Z and , ∈ {−1, +1}. These states are expressible as smooth functions on the (5u, 5v ) configuration space R4 and they solve the constraints (III.4.38). 119

The solution states can be expressed in our ‘polarized’ Fock basis as follows # |j, ,j >= (−1)m |m + 12 (j + |j|), m + 12 (−j + |j|) > ⊗ m=0

|m + 12 (−,j + |j|), m + 12 (,j + |j|) > .

(III.4.66)

(These states are the solutions f (t = 1; k = 0, j, j " = ,j), see (III.7.1.4).) Clearly, the states 7 However there are much more solutions to the |j, ,j > solve the Master Constraint Operator M. Master Constraint Operator (which do not necessarily solve the three constraints (III.4.38)). The algebra A∗ used in [55] is the algebra generated by the observables (III.4.33). The star–operation is defined by Q∗i = Qi , Pi∗ = Pi and extended to the full algebra by complex anti–linearity. This algebra is supplemented to the algebra A∗ext by the operators R21 ,22 = R2∗1 ,22 , which permute between the four different sectors of the classical constraint phase space: R21 ,22 : (u1 , u2 , v1 , v2 , p1 , p2 , π1 , π2 ) &→ (u1 , ,1 u2 , v1 , ,1 ,2 v2 , p1 , ,1 p2 , π1 , ,1 ,2 π2 ) .

(III.4.67)

The algebra A∗ has the following representation on V˜ : Q3 |j, ,j > = δ−1,2 j |j, ,j >

∓i Q± |j, ,j > = δ−1,2 ( √ |j|) |(j ± 1), , (j ± 1) > 2

P3 |j, ,j > = δ+1,2 j |j, ,j >

∓i P± |j, ,j > = δ+1,2 ( √ |j|) |(j ± 1), , (j ± 1) > 2

.

(III.4.68)

The state |j = 0, j " = 0 > is annihilated by all operators in A∗ , in particular, it generates an invariant subspace for A∗ . Now, it is not possible to introduce an inner product on V˜ , in which the star-operation becomes the adjoint operation (because the SO(2, 2)-representation defined by (III.4.68) is non-unitary). However, since |0, 0 > generates an invariant subspace one can take 5 6 ˜ the quotient V /{c|0, 0 >, c ∈ C}, consisting of equivalence classes v = {v + c |0, 0 >, c ∈ C} 5 6 5 6 where v ∈ V˜ . (In particular |0, 0 > is the null vector 0 .) The so(2, 2)-representation on V˜ then defines a representation on this quotient space by O([v]) = [O(v)], where O is an so(2, 2)-operator. (This representation is well defined because we are quotienting out an invariant 5 6 subspace.) A basis in this quotient space is { |j, ,j > , j ∈ N − {0}}. In the following we will drop the equivalence class brackets [·]. The quotient representation is the direct sum of four (unitary) irreducible representations of so(2, 2), labeled by sgn(j) = ±1 and , = ±1. The inner product, which makes these representations unitary is < j1 , ,1 j1 |j2 , ,2 j2 >= c(sgn(j), ,) δj1 ,j2 δ21 ,22 |j| (III.4.69) where c(sgn(j), ,) are four independent positiv constants. By taking the reflections R21 ,22 ∈ O(2, 2) into account, we can partially fix these constants. Their action on states in L2 (R4 ) is (R21 ,22 ψ)(u1 , u2 , v1 , v2 ) = ψ(u1 , ,1 u2 , v1 , ,1 ,2 v2 ) .

(III.4.70)

States with angular momenta j and j " are mapped to states with angular momenta ,1 j and ,1 ,2 j " . Therefore the R21 ,22 ’s effect, that the quotient representation of the observable algebra becomes an irreducible one. Since R21 ,22 is in O(2, 2), it is a natural requirement for them to act 120

by unitary operators. This fixes the four constants c(sgn(j), ,) to be equal (and in the following we will set them to 1). This gives for the action of the algebra A∗ on the normalized basis vectors |j, ,j >N := √1 |j, , j >, j ∈ Z − {0}: |j|

Q3 |j, ,j >N

Q± |j, ,j >N P3 |j, ,j >N

P± |j, ,j >N

= δ−1,2 j |j, ,j >N . ∓i = δ−1,2 ( √ |j(j ± 1)|) |(j ± 1), , (j ± 1) >N 2 = δ+1,2 j |j, ,j >N . ∓i = δ+1,2 ( √ |j(j ± 1)|) |(j ± 1), , (j ± 1) >N 2

.

(III.4.71)

In the limit of large j the right hand sides of (III.4.65) and (III.4.71), ie. the matrix elements of the observables in the two quantizations, coincide except for phase factors. These can be made equal by adjusting the phase factors of the respective basic vectors. Therefore both quantization programs lead to the same semi–classical limit. A first crucial difference in the results of the two quantization approaches is that in the Master Constraint Programme the vector ||j = 0, j " = 0 >> is included in the physical Hilbert space whereas it is excluded during the Algebraic Quantization process. If we exclude the sector changing operators mentioned above by hand, then ||j = 0, j " = 0 >> is annihilated by the altered observable algebra and likewise cannot be reached by applying observables to other states in the physical Hilbert space. If we include the sector changing operators then |j = 0, j " = 0 > is still not in the range of any observable because the observables are sandwiched beween operators of the form |sgn(Q3 )|, |sgn(Q3 )|. However, one can map between all the remaining sectors which thus provides a second difference with [55].

III.5

Free Field Theories

We now move on to free field theories with constraints, namely Maxwell theory and linearized gravity. Since the Master Constraint involves squares of operator valued distributions, one has to be very careful in defining the Master Constraint and we will see that the full flexibility of the Master Constraint Programme must be exploited in order to arrive at sensible results. A further complication arises because for these examples it is very difficult to give an explicit cyclic basis. This is caused by the fact that we have to deal with infintely many dgrees of freedom. We will therefore argue that one can circumvent the problem by using a limiting procedure.

III.5.1 III.5.1.1

Maxwell Theory Definition of the Master Constraint

The canonical formulation of Maxwell Theory on R4 consists of an infinite – dimensional phase space M with canonically conjugate coordinates (Aa , E a ) and symplectic structure {Aa (x), Ab (y)} = {E a (x), E b (y)} = 0, {E a (x), Ab (y)} = e2 δba δ(x, y)

(III.5.1)

where e is the electric charge and we are using units so that α = !e2 is the Feinstrukturkonstante. In particular, the U (1) connection A has units of cm−1 while the electric field E has units of cm−2 . 121

The clean mathematical description of M models M on a Banach space E [61]. Here we will not need all the details of this framework and it suffices to specify the fall – off conditions of A, E at spatial infinity. Namely, in order that the canonical classical action principle be well–defined, that is, the action be functionally differentiable, the fields A, E respectively must fall off at spatial infinity at least as r −1 , r −2 respectively. Consider now the infinite number of Maxwell – Gauss Constraints ! d3 xΛ(∂a E a ) =:< Λ, ∂ · E >h (III.5.2) G(Λ) = R3

where Λ is a smooth test function of rapid decrease and h = L2 (R, d3 x). Consider a positive definite operator K on h independent of M and define the associated Master Constraint by M :=

1 < ∂ · E, K · ∂ · E >h 2

(III.5.3)

.

Obviously M = 0 if and only if ∂ ·E = 0 a.e., that is, if and only if G(Λ) = 0 for all test functions Λ of rapid decrease. For the same reason for a twice differentiable function O on M we have {{M, O}, O}M=0 = 0 if and only if {∂ · E, O}M=0 = 0 a.e, hence if and only if {G(Λ), O}M=0 for all test functions of rapid decrease. Recall that the Maxwell – Hamiltonian is given by (" means equal to on the constraint surface) ! ! 1 3 a b a b ⊥ b H= 2 d x δab (E E + B B ) " ! d3 x δab z a Pab z (III.5.4) 2e where the tranversal projector is given by

and

(P⊥ · f )a = fa − ∂a ∆−1 ∂ b fb

(III.5.5)

√ 1 √ 1/2 −1/2 a z a = √ [ −∆ Aa − i −∆ E ] . 2α

(III.5.6)

In order to solve the Master Constraint for Maxwell Theory on Fock space it is mandatory 7 becomes a densely defined operator defined to choose a nontrivial operator K in order that M via annihilation and creation operators. Let bI be any orthonormal basis of h consisting of real valued, smooth functions of rapid decrease. The index set I in which the I takes values is countable and we could for instance choose the bI to be Hermite functions. Next, consider the Hilbert space h3 := h3 with scalar ' product < f, f " >h3 := d3 x δab fa fb" . Let us define the functions b"I

(3)" bIa (3) bIa

:= := :=

√ √ √

3/2

−∆

bI

−∆

∂a bI

−∆

∂a bI

1/2 −1

(III.5.7)

.

Notice the relations (3)"

(3)"

(3)

(3)

< bI , bJ

>h3 = < b"I , b"J >h

< bI , bJ >h3 = < bI , bJ >h= δIJ 122

.

(III.5.8)

(3)

Let us complete the bI to an orthonormal basis of h3 by choosing some smooth, transversal and (1) (2) (j) orthonormal system bI , bI of rapid decrease, that is, ∂ a bIa = 0, j = 1, 2. Now from (III.5.8) (3)" it is clear that the longitudinal vectors bI do not form an orthonormal system, but they can (1) (2) (3) be, as elements of h3 , expanded in terms of the orthonormal basis bI , bI , bI where only the (3) expnasion coefficients for the bI are non-vanishing. We thus find, using the completeness relation several times M = = = =

1 < ∂ · E, K · ∂ · E >h 2 1# < ∂ · E, bI >h < bI , K · bJ >h < bJ , ∂ · E >h 2 I,J α# (3)" (3)" < [z − z¯], bI >h3 < bI , K · bJ >h < bJ , [z − z¯] >h3 4 I,J α # # (3) (3)" (3)" (3) [ < bM , bI >h3 < bI , K · bJ >h < bJ , bN >h3 ] × 4 M,N

I,J

(3)

(3)

(3)

(3)

× < [z − z¯], bM >h3 < bN , [z − z¯] >h3 √ √ α # # 3/2 3/2 = [ < −∆ bM , bI >h < bI , K · bJ >h < bJ , −∆ bN >h] × 4 M,N

I,J

× < [z − z¯], bM >h3 < bN , [z − z¯] >h3 √ α# 3/2 √ 3/2 (3) (3) = < bJ , −∆ K −∆ bK >h < [z − z¯], bJ >h3 < bK , [z − z¯] >h3 4

.

J,K

(III.5.9)

√ −3 If we would choose K = −∆ then (III.5.9) would simplify very much, however, this is impossible due to the boundary conditions on E which would imply that for this choice of K the integral (III.5.3) diverges logarithmically. However, notice that the operator Q on h defined by √ 3/2 √ 3/2 Q = −∆ K −∆ (III.5.10)

is positive definite. We now come to the quantization of the system. We consider the kinematical operator algebra A generated from annihilation operators " (j) (j) zˆJ := < bJ , z >h3

(III.5.11)

and the corresponding creation operators given by their adjoints which are subject to the usual commutation relations k k † k † [ˆ zJj , zˆK ] = [(ˆ zJj )† , (ˆ zK ) ] = 0, [ˆ zJj , (ˆ zK ) ] = αδjk δJK

.

(III.5.12)

We represent this algebra on the usual kinematical Hilbert space HKin of Maxwell theory given by the Fock space F generated from the cyclic vacuum vector Ω defined by zˆJj Ω =0 123

.

(III.5.13)

In terms of creation and annihilation operators, the Master Constraint Operator becomes # 3 3 † 7 =α QJK [ˆ zJ3 − (ˆ zJ3 )† ]† [ˆ zK − (ˆ zK )] . (III.5.14) M 4 J,K

Notice that (III.5.13) is not normal ordered, hence it is a non-trivial question whether (III.5.13) is even densely defined. Theorem III.5.1. The Master Constraint Operator (III.5.14) is densely defined if and only if Q is a trace class operator. Proof of theorem III.5.1: Since the Fock space is the closure of the finite linear span of finite excitations of the vacuum 7 is densely defined if and only if M 7 Ω has finite norm. We compute Ω, M # 3 † 3 3 † 3 † 7 Ω||2 = ( α )2 || M QJK QM N < [ˆ zJ3 − (ˆ zJ3 )† ](ˆ zK ) Ω, [ˆ zM − (ˆ zM ) ](ˆ zN ) Ω> 4 J,K,M,N # α 3 † 3 † 3 † = ( )2 QJK QM N < αδJK − (ˆ zJ3 )† (ˆ zK ) ]Ω, αδM N − (ˆ zM ) (ˆ zN ) ]Ω > 4 J,K,M,N # α 3 3 3 † 3 † = ( )2 QJK QM N [α2 δJK δM N + < Ω, zˆK zˆJ (ˆ zM ) (ˆ zN ) Ω > 4 J,K,M,N # α = ( )4 QJK QM N [δJK δM N + δJM δKN + δJN δKM ] 2 J,K,M,N

α = ( )4 [2Tr(Q2 ) + (Tr(Q))2 ] . 2

(III.5.15)

Both terms in the last line of (III.5.15) must be finite since Q is symmetric (even positive). The first term in the last line of (III.5.15) is finite if Q is a Hilbert – Schmidt (or nuclear) operator, the second if Q is trace class. Since every trace class operator is nuclear, it is necessary and sufficient that Q be trace class. ! Now the algebra of trace class operators comprises an ideal within the compact operators. Compact operators are bounded, have pure point spectrum and every non-zero eigenvalue has finite multiplicity, hence zero is an accumulation point in their spectrum unless Q is a finite rank operator. A possible choice for K would therefore be for example K=

√

−∆

−3/2

e

l2 ∆−||x||2 /l2 2

√

−3/2

−∆

(III.5.16)

where l is an arbitrary finite length scale. The exponential in (III.5.16) is nothing else than minus a three dimensional harmonic oscillator operator, so its eigenvalues are λn := exp(−[ 32 + n1 + n2 + n3 ]), nj ∈ N0 , hence its trace is explicitly e−3/2 (1 − e−1 )−3 and that of its square e−3 (1 − e−2 )−3 . Theorem III.5.2. The choice (III.5.16) makes the integral (III.5.3) converge. 124

Proof of theorem III.5.2: We expand (III.5.3) in terms of coherent states ψz for the Hamiltonian H defined by ψz = e−||z|| where

2 /2

ez

az ˆa†

|0 >

1 1 z a = √ [xa /l − ilpa ], zˆa = √ [xa /l + l∂/∂xa ] 2 2

Using the overcompleteness relation for coherent states ! d3 zd3 z¯ ψz < ψz , . >h= idh π3 C3

(III.5.17) (III.5.18)

(III.5.19)

we find M =

≤

≤

=

! d3 zd3 z¯ d3 z " d3 z¯" × π3 π3 C3 C3 √ √ −3/2 −3/2 ∂ · E, ψz >h < ψz , e−H ψz ! >h < ψz ! , −∆ ∂ · E >h × < −∆ ! ! 3 3 3 " 3 " ¯ 1 d zd z¯ d zd z × 3 2 C3 π π3 C3 √ √ −3/2 −3/2 d · ψz >h3 | [| < ψz , e−H ψz ! >h |] | < −∆ d · ψz ! , E >h3 | ×| < E, −∆ ! ! ||E||2h3 d3 zd3 z¯ d3 z " d3 z¯" × 2 π3 π3 C3 C3 √ √ −3/2 −3/2 ×|| −∆ d · ψz ||h3 [| < ψz , e−H ψz ! >h |] || −∆ d · ψz ! ||h3 ! ! 2 3 3 3 " 3 " ¯ ||E||h3 d zd z¯ d zd z × 3 2 π π3 C3 C3 + + √ √ −1 −1 × < ψz , −∆ ψz >h [| < ψz , e−H ψz ! >h |] < ψz ! , −∆ ψz ! >h 1 2

!

(III.5.20)

where in the third step we have used the Schwartz inequality. ||E||2

Since the classical electric field energy 2 h 3 converges, it suffices to estimate the integrals in (III.5.20). Using the expansion of the coherent states into energy eigenfunctions it is easy to see that 1 | < ψz , e−H ψz ! >h | = e−3/2 | exp(− [za z¯a + za" z¯a" − 2¯ za za" /e])| 2 1 # 1 1 = e−3/2 exp(− |za − za" |2 ) exp(− (1 − )[za z¯a + za" z¯a" ]) 2e a 2 e

1 1 ≤ e−3/2 exp(− (1 − )[za z¯a + za" z¯a" ]) 2 e

Next, using the Fourier transform we get (up to a constant phase) ! √ ˜ ψz (k) = d3 xe−ikx ψz (x) = (2l π)3/2 exp(−||p||2 l2 /2 + ||k||2 l2 /2 − ilk · z) 125

(III.5.21)

(III.5.22)

and have by using polar coordinates

= = ≤

!

d3 k 2 2 2 ψz >h= (2l π) e e−||k|| l −2l k·p 3 ||k||(2π) ! ∞ ! 1 √ −3 2 2 2 2πl π e−||p|| l rdre−r dte−2l||p||rt 0 −1 ! ∞ √ −3 1 2 2 π π dr[e−(r−l||p||) − e−(r+l||p||) ] ||p|| 0 ! l||p|| dr 1 2 √ e−r ||p|| −l||p|| π 1 . ||p|| < ψz ,

=

√

√

−1

−∆

3 −||p||2 l2

Thus we can finish our estimate by ! ! ||E||2h3 e−3/2 d3 p − l2 (1− 1 )||p||2 2 3 − 2l12 (1− 1e )||x||2 2 e . M ≤ [ d xe ] [ e 2 ] 2 π6 ||p|| 1 −11 ! ! ||E||2h3 le−3/2 1 d3 k − 1 ||p||2 2 3 − 12 ||x||2 2 . = 1 − [ d xe ] [ e 2 ] 2 π6 e ||k|| 1 −11 ! ∞ ||E||2h3 e−3/2 1 2 1 3 2 = l 1− (2π) (4π) [ r 3/2 dre− 2 r ]2 6 2 π e 0 1 √ −11 2 ||E||h3 e−3/2 227 1 = l 1− Γ(5/4)2 2 π e where the last integral resulted in a Γ−function. !

(III.5.23)

(III.5.24)

In what follows we will not need the specifics of Q, we choose any trace class operator such that (III.5.9) converges. Having made sure that both the classcial and quantum Master Constraint are well-defined we can proceed to the solution of both the classical and the quantum problem. III.5.1.2

Physical Hilbert Space

The full Fock space F can be conveniently written in the form F = F ⊥ ⊗ F / where F / = F (3) contains the longitudinal excitations while F ⊥ = F (1) ⊗ F (2) constains the transversal ones. The Master Constraint Operator acts as an identity on the Hilbert space of transversal modes so that we need only to consider the action of the Master Constraint Operator on the longitudinal Hilbert space. This longitudinal Hilbert space in turn acquires the direct sum structure F / = ⊕{n(3) } I

I∈I ;

P

I

(3)

nI ( = < ψ, (M F 2 7 7 = < Ω, (M − MJ ) Ω >F ( # # α = ( )4 {2 Q2I + [ QI ]2 } 2 I∈I−J

(III.5.28)

I∈I−J

7 −M 7 J commutes with all the annihilation where in the second step we used that the operator M operators which create ψ from the vacuum Ω together with ||ψ|| = 1. In the last line we used the previous computation (III.5.15). Now since Q is trace class, given , > 0, ψ ∈ F / we find J1 such that Jψ ⊂ J1 and such that (III.5.28) is smaller than ,. Since the series in (III.5.28) is monotonously decreasing as J → I, this holds also for all J1 ⊂ J which establishes the proof. We note that the label J1 depends on , but not on ψ itself but only on Jψ , hence the limit is partly uniform. ! The idea to solve the constraint now rests on the following observation. 127

Theorem III.5.4. Let An be a sequence of self – adjoint operators with common dense domain D such that i) An ψ → Aψ for all ψ ∈ D where A is another self – adjoint operator also defined densely on D. Let x, y ∈ R and suppose ii) that x, y are not in the pure point spectrum σpp (A) of A. Then s − limn→∞ En ((x, y)) = E((x, y)) where En (·), E(·) are the spectral projectors of An , A respectively. Proof of theorem III.5.4: Recall that An → A in the strong resolvent sense provided that Rz (An ) → Rz (A) strongly for any (and therefore all, see theorem VIII.19 of [35]) z ∈ C with :(z) )= 0. Here Rz (A) = (A−z)−1 is the resolvent of A. By i) and [35], theorem viii.25a), An → A in the strong resolvent sense. By ii) and [35] theorem viii.24b) the claim follows. ! The theorem applies in particular when An , A have purely continuous spectrum in which case the convergence holds for all measurable sets. We can apply this theorem to our case because both assumptions i) and ii) are satisfied. Namely, 7 J are densely defined on the dense set of finite linear by theorem III.5.3) all the operators M 7 strongly. Furthermore, the operators combinations of Fock sates and converge there to M (3) (3) † pˆI := i[ˆ zI − (ˆ zI ) ] are mutually commuting, self – adjoint and have absolutely continuous 7 J,M 7 are mutually commuting and self adjoint (by the spectral spectrum. Hence also the M theorem) and have absolutely continuous sectrum. It follows that s − limJ →I EJ (B) = E(B) 7 J and M 7 respectively. for any measurable set B where EJ , E denote the p.v.m. of M Let Ω0 be any vector such that the spectral measure µΩ0 associated to Ω0 is of maximal type33 (not to be confused with the Fock vacuum). For instance it may be the vector which actually defines the spectral measure µ underlying a direct integral representation of F / subordinate to E. For the spectral measure µΨ,Ψ! (x) :=< Ψ, E((−∞, x])Ψ" > we then have dµΨ,Ψ! (0) dµΩ0 (0)

=

µΨ,Ψ! (x) x→0+ µΩ0 (x)

=

< ψ(0), ψ " (0) >H⊕ < Ψ, E((−∞, x])Ψ" > 0 = x→0+ < Ω0 , E((−∞, x])Ω0 > < ω0 (0), ω0 (0) >H⊕

lim

lim

(III.5.29)

0

where ψ, ψ " , ω0 are the direct integral representations of Ψ, Ψ" , Ω0 respectively. In the following we will show that one can approximate the left hand side of (III.5.29) arbitrary well by quantities involving only Fock states with finitly many excitations. Since I is countable it is in bijection with N and thus we take I = 1, 2, .. for simplicity of notation. Consider any n, m and let Ψm , Ψ"m , Ω0m be unit vectors which are finite linear combinations of Fock states with excitations at most up to label I = m and let En be the p.v.m. 33

i.e. µΩ0 ≥ µΩ for any Ω ∈ F (

128

7 n := (α/2)4 (n pˆ2 . We calculate with µn (x) :=< Ω0 , En ((−∞, x])Ω0 > etc. of M Ω0 I=1 I n µΨ,Ψ! (x) µΨm ,Ψ!m (x) − µΩ0 (x) µnΩ0m (x) ; < ; < n µnΨ,Ψ! (x) µnΨm ,Ψ!m (x) µΨ,Ψ! (x) µΨ,Ψ! (x) = − n + − µΩ0 (x) µΩ0 (x) µnΩ0 (x) µnΩ0m (x)

= =

µΨ,Ψ! (x)µnΩ0 (x) − µnΨ,Ψ! (x)µΩ0 (x) µΩ0 (x)µnΩ0 (x)

+

µnΨ,Ψ! (x)µnΩ0m (x) − µnΨm ,Ψ!m (x)µnΩ0 (x) µnΩ0 (x)µnΩ0m (x)

< Ψ, [E(x) − En (x)]Ψ" > µΩ0 (x) + µΨ,Ψ! (x) < Ω0 , [En (x) − E(x)]Ω0 > + µΩ0 (x)[µΩ0 (x)+ < Ω0 , [En (x) − E(x)]Ω0 >] [µnΨ−Ψm ,Ψ! (x) + µnΨm ,Ψ! −Ψ!m (x)]µnΩ0m (x) + µnΨm ,Ψ!m (x)[µnΩ0m −Ω0 ,Ω0m (x) + µnΩ0 ,Ω0m −Ω0 (x)]

× [µΩ0 (x) + µnΩ0m −Ω0 ,Ω0m (x) + µnΩ0 ,Ω0m −Ω0 (x)+ < Ω0 , [En (x) − E(x)]Ω0 >] 1 × . (III.5.30) [µΩ0 (x)+ < Ω0 , [En (x) − E(x)]Ω0 >]

Since the finite linear combinations of Fock states are dense, for given , > 0 and unit vectors Ψ, Ψ" , Ω0 we find some m and corresponding unit vectors Ψm , Ψm , Ω0m such that ||Ψ−Ψm ||, ||Ψ" −Ψ"m||, ||Ω0 −Ω0m || < ,. Since En (x), E(x) are projections with uniform operator norm at most unity we find, using the Schwarz inequality e.g. |µΨ−Ψm ,Ψ! (x)| ≤ ||Ψ − Ψm || < , or |µnΨ−Ψm ,Ψ! (x)| ≤ ||Ψ − Ψm || < ,. Next, for given x > 0, , > 0, Ψ, Ψ" , Ω0 there exists n0 = n0 (x, ,, Ψ, Ψ" , Ω0 ) such that ||[En (x) − E(x)]Ψ||, ||[En (x) − E(x)]Ψ" ||, ||[En (x) − E(x)]Ω0 || < , for all n > n0 . It follows e.g. | < Ψ, [En (x) − E(x)]Ψ" > | < ,. The absolute value of (III.5.30) can therefore be estimated by / 0 µnΩ0m (x) + |µnΨm ,Ψ!m (x)| µΩ0 (x) + |µΨ,Ψ! (x)| |(III.5.30)| ≤ , + 2 µΩ0 (x)[µΩ0 (x) − ,] [µΩ0 (x) − ,][µΩ0 (x) − 3,] " $ 2, 1 2 ≤ + µΩ0 (x) − , µΩ0 (x) µΩ0 (x) − 3, 6, ≤ (III.5.31) [µΩ0 (x) − 3,]2 where we have assumed that, given x > 0, Ω0 we have 3, 0 we may choose , such that the last line of (III.5.31) can be made smaller than δ. Hence µΨ,Ψ! (x)/µΩ0 (x) can be approximated by µnΨm ,Ψ!m (x) µnΩ0m (x)

(III.5.32)

arbitrary well. We now wish to calculate (III.5.32) more explicitly. Let ΩnN α := ΩnN ⊗ enα where 7 n on the Hilbert space Fn/ which is the completion of the finite (ΩnN )N is a cyclic system for M linear span of Fock states with excitations at most in the first n degrees of freedom and enα is a / Fock basis of the orthogonal complement (Fn )⊥ . We may identify ΩnN with ΩnN 0 where e0 = 1 is 129

( / 7 n )Ωn the vacuum of (Fn )⊥ . A given vector Ψ can then be written in the form Ψ = N,α ΨnN α (M Nα n for certain measurable functions ΨN α . Up to this point m, n are uncorrelated. Choose n ≥ m. Then the measurable functions n " ΨnmN α , Ψn" mN α , Ω0mN α corresponding to Ψm , Ψm , Ω0m respectively are in fact polynomials and moreover they vanish unless α = 0. Introducing a direct integral representation of F / 7 n based on the Ωn , formula (III.5.32) becomes subordinate to M Nα 'x 0

( dµn (x)[ N ρnN (x)ΨnmN (x) Ψn" mN (x)] 'x ( n n n 2 0 dµ (x)[ N ρN (x)|Ω0mN (x)| ]

(III.5.33)

where ρnN (x) are the Radon–Nikodym derivatives of the spectral measures µnN (B) :=< ΩnN 0 , En (B)ΩnN 0 > with respect to the total measure ( −N ( −α ( −N n 2 2 < ΩnN α , En (B)ΩnN α > 2 µN (B) α( N( µn (B) = N = −N −(N +α) 2 N 2 N,α

(III.5.34)

where we exploited that En (B) does not act on the eα . In order to compute (III.5.33) in the limit x → 0 we now will use a convenient choice of ΩnN . These are simple modifications of the example of n mutually commuting constraints that we discussed in section III.3.1. The Hilbert space F is unitarily equivalent to the Hilbert space L2 (S, dµG ) where S is the set of real valued sequences (pI )∞ measure I=1 and µG is a Gaussian ( α( 2 with white noise covariance, that is µG (exp(iFf )) = exp(− 2 I fI ) where Ff = I fI pI . This follows immediately from µG (eiFf ) :=< Ω, e−

P

(3)

I

(3)

fI [ˆ zI −(ˆ zI )† ]

Ω>

(III.5.35)

and by using the Baker–Campbell–Hausdorff formula. Here, as before, Ω denotes the Fock n vacuum which in the p−representation is given by Ω(p) = 1. We now choose √ the vectors ΩN according to the example in section III.3.1. Introduce the variables yI := pI QI , I = 1, .., n so 7 n = 1 (n y 2 and let hn , l ∈ Ls be the harmonic polynomials of degree s = 0, 1, .. in that M I=1 I s,l 2 the variables yI with respect to the Laplacian in the yI , I = 1, .., n. Here Ls is a finite set of indices which depends on s. We define ΩnN =(s,l) ({pI }) := Ωnsl ({pI }) := cnsl hsl ({yI }) exp(−

n

n

I=1

I=1

1# 2 1 # 2 yI ) exp( yI /QI ) 4 4α

(III.5.36)

where cnls is a normalization constant which is unimportant for what follows. As we showed in sec√ 7 k Ωn are dense in Fn/ . Using that formally dµG (p) = =∞ e−p2I /(2α) dpI / 2πα, tion III.3.1, the M n ls I=1

130

we now compute the corresponding spectral measures ' 7 n ) |Ωn (p)|2 dµG (p) θ(x − M sl n µsl (x) = S ' n (p)|2 dµ (p) |Ω G sl S ' np 1 Pn 2 d − 7 n ) |Ωn (p)|2 I=1 pI θ(x − M √ n e 2α sl Rn 2πα = ' 1 Pn np 2 d − p I=1 I |Ωn (p)|2 √ n e 2α sl Rn 2πα P ' ( n 1 2 n − n 2 n 2 I=1 yI θ(x − 1 n d y e 2 I=1 yI ) |hsl (y)| 2 P = R ' 2 − 12 n n I=1 yI |hn (y)|2 sl Rn d y e P ' (n 2 − 12 n y 1 n 2 n 2 I=1 I θ(x − n d y e I=1 yI ) |hs0 (y)| 2 P = R ' 2 − 12 n n I=1 yI |hn (y)|2 s0 Rn! d y e 1 s+n/2−1 −t = dt θ(x − t) t e Γ(s + n/2) R+ = µns0 (x)

(III.5.37)

where in the second step we have integrated out all but the first n = degrees √ of freedom, in the second we changed variables to yI thereby cancelling the Jacobean nI=1 1/ QI in numerator and denominator, in the third we used that for given s there exists an SO(n) rotation which 7 n and after this the calculation is transforms hs0 into hsl and the rotation invariance of dn y, M exactly the same as in (III.3.19). The total measure therefore becomes as in (III.3.20) ∞

1 # −s 1 µ (x) = 2 µs0 (x) = 2 2Γ(n/2 − 1) n

s=0

!

R+

dt θ(x − t) tn/2−2 e−t/2

and the Radon – Nikodym derivatives were calculated to be ! x dµnsl (x) 2Γ(n/2 − 1) s+n/2−1 −x/2 n = x e [ dt e−t/2 tn/2−2 ]−1 ρsl (x) = dµn (x) Γ(n/2 + s) 0

(III.5.38)

(III.5.39)

which close to x = 0 behave as 2Γ(n/2 − 1) s x [(s + n/2 − 1) − x/2] ≈ 2δs,0 Γ(n/2 + s)

(III.5.40)

Notice that the set of values which the second subindex in hnsl can assume for s = 0 is L0 = {0} so there is only one harmonic polynomial of degree zero. Inserting this into (III.5.33), formula (III.5.32) becomes 'x ( n n ρsl (x)Ψnmsl (x) Ψn" msl (x)] 0 'dµ (x)[ sl ( (III.5.41) x n n n 2 0 dµ (x)[ sl ρsl (x)|Ω0msl (x)| ] which close to zero becomes (the measures µ([0, x]) cancel both in numerator and denominator) ( n ρ (0)Ψn (0) Ψn" Ψnm00 (0) Ψn" sl msl (0) m00 (0) ( sl n msl n = (III.5.42) n 2 |Ω0m00 (0)|2 sl ρsl (0)|Ω0msl (0)|

up to terms which vanish in the limit x → 0 by the intermediate value theorem of Lebesgue integral calculus. (Remember that the Ψnmsl etc. are polynomials and non–vanishing for finitely 131

many s, l only, hence the integrand is actually a smooth function and dµn (x) = dxσ n (x) for a smooth function σ n is absolutely continuous with respect to Lebesgue measure as follows from (III.5.39).) Thus, (III.5.42) approximates (III.5.29) as closely as we want, that is, for all x > 0, δ > 0, Ψ, Ψ" , Ω0 we find m > m0 (x, δ, Ψ, Ψ" , Ω0 ) and n > n0 (x, δ, Ψ, Ψ" , Ω0 ) with n ≥ m such that these two quantities differ at most by δ. Since (III.5.42) depends only on the coefficient Ψnm00 (0) for any m, n it follows that the physical Hilbert space in the approximation given by δ is in one to one correspondence with the one dimensional span of the vector Ωnm;s=0,l=0,α=0, that is with 7 from F / coincides with C. Since this holds for all δ, the physical Hilbert space selected by M C. Since F = F / ⊗ F ⊥ it follows Hphys = F ⊥ . I.e. the physical Hilbert space is – as expected – spanned by the transversal modes.

Notice that this result is independent of the concrete choice of QI , that is, independent of the choice of the trace class operator, which we inserted into the Master Constraint in order to obtain a densely defined Master Constraint Operator.

III.5.2

Linearized Gravity

In this section we will consider the constraints of linearized gravity on Minkowski background. Because of the Minkowski background we can apply the same techniques as for the MaxwellGauss constraint. In the following we will define the Master Constraint Operator for linearized gravity and bring it into a similar form as for Maxwell Theory. For the construction of the physical Hilbert space we can then apply the same line of argumentation as for Maxwell Theory. We will work with the (real) connection formulation of canonical gravity, i.e. the canonical 1 pair of fields (Aja , κβ Eja ) on a three–dimensional manifold Σ, where Aja is an su(2)–connection . and Eja = det(q)eaj is a densitized triad for the spatial metric qab . We use a, b, c, . . . for spatial indices and i, j, k, . . . for su(2)–indizes. The latter ones are raised and lowered with δik and δik , so we do not worry about the position of these indices. κ = 8πGN ewton denotes the gravitational coupling constant and β the (real) Immirzi parameter. Canonical gravity in this formulation is a first class constraint system. The constraints of the full non-linear theory are the Gauss, the diffeomorphism and the scalar constraint: Gj

= ∂a Eja + ,jkl Aka Ela

j Va = Fab Ejb % j & 1 . C = Fab − (1 + β 2 ),jmn Kam Kbn ,jkl Eka Elb κ det(q)

.

(III.5.43)

In the last line Kaj is given by β Kaj = Aja − Γja , where Γja is the spin connection of the triad eaj . F = 2(dA + A ∧ A) is the field strenght and the curvature of the connection A. We assume that the fields are asymptotically flat and adopt the boundary conditions from [62], that is Aja (and Kaj ) falls off as r −2 and Eja ∼ flat Eja + O(1/r) at infinity, where flat Eja is a densitized triad for a flat metric. Linearized Gravity in the connection formulation was previously considered in [63], however there complex connection variables C Aja = Γja + iKaj were used. Since we have β Kaj = Aja − Γja ,

132

we can express the complex connection in terms of the real variables as C

Aja =

i j i Aa + (1 − )Γja β β

.

(III.5.44)

Moreover [63] gave the ADM-energy functional in terms of the complex connection as ! . % & 1 H= d3 x det(q) C Aab C Aba − C Aaa C Abb (III.5.45) κ Σ

as the surface term one has to add to where C Aab = C Aja ejb . This energy functional can be derived ' make the integrated Hamiltonian constraint C(N ) = Σ d3 xN C (where N is the lapse function) differentiable. The differentiability is achieved with respect to the fields Aja , Eja fulfilling the asymptotic flat boundary conditions. Rewriting the energy functional (III.5.45) with help of (III.5.44) into the real connection variables gives ! . 5 6 1 H= 2 d3 x det(q) (1+β 2 )[Γab Γba − Γaa Γbb ] + [Aab Aba − Aaa Abb ] − 2[Γab Aba − Γaa Abb ] β κ Σ (III.5.46) where we introduced Aab = C Aja ejb and Γab = Γja ejb . We come now to the linearization of the theory around Minkowski initial data Aja = 0 and Eja = δja . We will apply the following theorem, proved in a more general setting in [64], which deals with the linearization of a finite dimensional first class constrained systems: Theorem III.5.5. Let (M, Ω) be a finite dimensional symplectic manifold with coisotropic constraints Cα and Hamiltonian H. Suppose that m0 ∈ M is a point on the constraint surface at which the Hamiltonian vectorfield of H vanishes. Consider the tangent space V := Tm0 (M ) and introduce the linearized constraints, quadratic Hamiltonian, and linearized symplectic structure respectively by Cαlin : V → C ∞ (V ); H lin : V → C ∞ (V );

∂Cα (m0 )xa ∂ma 1 ∂2H x &→ H lin := H(m0 ) + (m0 )xa xb 2 ∂ma ∂mb a b a b ab Ωab lin := {x , x }lin := {m , m }(m0 ) = Ω (m0 )

x &→ Cαlin :=

.

(III.5.47)

Here xa are understood as coordinates of a vector fields x in the linearized phase space V , given by the relation xa = x·ma where ma are coordinates on the phase space M (seen as C ∞ -functions on M ). The statement is: The Cαlin are coisotropic for the symplectic vector space (V, Ωlin ), even Abelian, and H lin is invariant under their Hamiltonian flow. Moreover, the linearization of the reduced theory coincides with the reduction of the linearized theory. We will use the prescription of the theorem for our infinte dimensional theory and check afterwards whether the statement holds also for this case. Thus we take m0 = (Aja = 0, Eja = δja ) 133

and consider the tangent space at this point, coordinatized by lin Aja and lin Hja , so that we can write Aja = lin Aja and Eja = δja + lin Hja . We insert the latter into (III.5.43) and keep only terms of linear order in (lin Aja , lin Hja ). The result is = ∂b lin Hba + ,abc lin Acb Glin a Valin lin C

= ∂a Abb − ∂b Aab lin

lin

= ,abc ∂a Abc lin

(III.5.48) (III.5.49) (III.5.50)

where we introduced lin Aab := lin Aja δjb and lin H ab = lin Hja δbj and all indices are pulled with respect to the flat background metric δab . (Hence we will not worry about index positions.) The induced symplectic structure is {lin Hab (x), lin Acd (y)}lin = βκδac δdb δ(x, y). To compute the linearized Hamiltonian we notize that (III.5.46) is at least quadratic in Γab and Aab . So we need Γab in terms of (lin Aja , lin Hja ) to linear order, which is lin

Γab =

1 bcd lin , (∂a Hcd + ∂d (δac lin Hee − lin Hac − lin Hca )) . 2

(III.5.51)

This can be computed by using the condition 0 = ∂a Eja + ,jkl Γka Ela for the spin connection Γka . . We can then replace Γ, A, det(q) by lin Γ, lin A, 1 respectively in order to arrive at the quadratic Hamiltonian " ! 1 lin 3 d x (1+β 2 )[lin Γab lin Γba − lin Γaa lin Γbb ] + [lin Aab lin Aba − lin Aaa lin Abb ] H = 2 β κ Σ $ lin b lin a lin a lin b −2[ Γa Ab − Γa Ab ] . (III.5.52) Now it is merely a computational effort to check, that the linearized constraints (III.5.48) are indeed Abelian and that the Hamiltonian (III.5.52) is invariant (on the constraint hypersurface) with respect to the Hamiltonian flow of the linearized constraints. Summarizing, we use for the linearized gravitational field canonical variables (lin Aab , lin H ab ) (and drop in the following the superfix ‘lin’), subject to the Abelian constraints (III.5.48). The boundary conditions for the linearized fields can be read off from the boundary conditions for the non-linear theory to be Aab ∼ r −2 and H ab ∼ r −1 . As in the Maxwell case we will use Fock space methods to quantize this system, so we have to introduce a pair of complex conjugated fields whose smeared forms will be finally promoted into annihilation and creation operators. Our choice is ) * 1 −1/2 1/2 zab = √ W · (Aab + ,acd ∂c Hdb ) − iβW · Hab and z¯ab , (III.5.53) 2βlp √ where we introduced the abbreviation W = −∆. The non-vanishing Poisson brackets are i!{¯ zab (x), zcd (y)} = δ(x, y)δac δab . With this choice the Hamiltonian can be written as a manifestly positive function, as will be shown later.

134

Now we plug Hab Aab

ilp √ W −1/2 · (zab − z¯ab ) =: lp W −1/2 · pab 2 βlp 1/2 ilp = √ W · (zab + z¯ab ) − √ W −1/2 · ,acd ∂c (zdb − z¯db ) 2 2 1/2 −1/2 =: βlp W · xab − lp W · ,acd ∂c pdb =

(III.5.54)

into the constraints (III.5.48) arriving at

˜ a := Ga − W −2 · ∂a C − W −2 · ∂b ,abc Vc G % & = βlp W 1/2 · ,acb − W −2 · ,dbc ∂a ∂d + W −2 · ,adb ∂d ∂c · xbc + lp W −1/2 · ∂b δac pbc $ % & % 1/2 −1/2 Va = βlp W · ∂a δbc − ∂c δab · xbc + lp W · ,bdc ∂a ∂d − ,bda ∂c ∂d · pbc % & C = βlp W 1/2 · ,abc ∂a xbc + lp W −1/2 · − W 2 δbc − ∂b ∂c · pbc (III.5.55)

where we introduced a more convenient equivalent set of constraints. For the definition of the (classical) Master Constraint we use the Hilbert spaces h = L2 (R3 , d3 x) ' and h3 , with inner product on h3 being by < fa" , fa >h3 := d3 xδab f¯a" fb . Later on we will ' 3 given 9 " ac bd also need h with < fab , fab >h9 := d xδ δ f¯" ab fcd. (Here we deviate from the usual mathematical notation and write the indices in the inner product in order to keep track of the ordering of the indices. For h3 this is not necessary, therefore the indices are sometimes omitted.) Now we can define the Master Constraint as % & ˜ K1 · G ˜ >h3 + < V, K2 · V >h3 + < C, K3 · C >h M = 12 < G, (III.5.56)

where K1 , K2 are positive definite operators on h3 and K3 is a positive definite operator on h. These operators have to be chosen in such a way that the Master constraint is well defined (on the classical phase space) and can be promoted into a densely defined operator on the quantum configuration space. In the following we will rewrite the Master constraint using the fields xab and pab , smeared with some basis of h9 . To begin with the construction of this basis we introduce an orthonormal basis bI of h consisting of real valued smooth functions of rapid decrease. From (3) this we define a basis for the longitudinal modes in h3 by bIa := W −1 · ∂a bI and complete these (1) (2) to a full orthonormal basis of h3 by choosing smooth, transversal functions bIa , bIa of rapid decrease. Finally we equip h9 with the following basis (the index i assumes values i = 1, 2): % & (i) (1i) bIbc := ,acb + W −2 · ,adb ∂d ∂c · bIa left long. right transv. (3)

(3)

bIbc := −W −1 · ∂b bIc = −W −2 · ∂b ∂c bI

left and right long.

(4i) (i) bIbc := W −1 · ∂c bIb (6) (3) bIbc := −2−1/2 W −1· (∂a δbc − ∂c δab )bIa = −2−1/2 W −2· (∆δbc (7) (3) bIbc := 2−1/2 W −2 · ,bdc ∂a ∂d bIa = −2−1/2 W −1 · ,bdc ∂d bI

left transv. right long. − ∂c ∂b )bI

symm. transv., trace part antisymm. transv. (III.5.57) (8)

(9)

We complete this system to an orthonormal basis of h9 by choosing a basis bIbc , bIbc of the symmetric transverse traceless modes34 . Furthermore we introduce another basis of the left 34

The completeness of this basis is verified in appendix III.7.2.

135

longitudinal right transversal and left transversal right longitudinal modes: (1! i)

(i)

(1i)

bIbc := −W −1 · ∂b δac bIa = W −1 · ,cda ∂d bIba (4! i)

(i)

left long. right transv. (4i)

bIbc := −W −2 · ,bda ∂c ∂d bIa = −W −1 · ,bda ∂d bIac

left transv. right long. . (III.5.58)

Since the operator δ : tc &→ ,cda ∂d ta squares to W 2 = −∆ on the subspace of the transversal modes in h3 , one can also write (1i)

bIbc

(4i)

bIbc

(1! i)

= W −1 · ,cda ∂d bIba

(4! i)

= −W −1 · ,bda ∂d bIac

(III.5.59)

. (2)

(1)

If one chooses the basis of the transversal modes in h3 such that bIc = W −1 ,cda ∂d bIa one obtains (1! 2)

(1! 1)

(11)

(12)

bIbc = bIbc

bIbc = bIbc

(4! 2)

bIbc = −bIbc

(4! 1)

(41)

bIbc = −bIbc

(42)

(III.5.60)

.

Now we come to the calculation of the first term in the Master Constraint (III.5.56) # ˜ K1 · G ˜ >h3 = ˜ a , b(i) >h3 < b(i) , K1 · b(j) >h3 < b(j) , G ˜ a >h3 (III.5.61) < G, h3 h3 (i)

=

i=1,2

+lp < W −1/2 · ∂b δac pbc , bIa >h3 % & (i) βlp < xbc , W 1/2 · ,acb + W −2 · ,adb ∂d ∂c · bIa >h9 (i)

=

−lp < pbc , W −1/2 · ∂b δac bIa >h9 # " % & (j) lp β < xbc , ,acb + W −2 · ,adb ∂d ∂c · bJa >h9 J;j=1,2

−1

=

(j) · ∂b δac bJa

$

(j)

(III.5.62)

(i)

− < pbc , W >h9 < W 1/2 bJa , bIa >h3 # % & (1j) (1! j) (j) (i) lp β < xbc , bJbc >h9 + < pbc , bJbc >h9 < W 1/2 bJa , bIa >h3 J;j=1,2

(1)

(2)

where in the second line we used that bIa , bIa are transversal and used an integration by parts. (i) In the second last line we expanded the transversal vectors W 1/2 · bIa , i = 1, 2 in terms of the (j) basis bJa , j = 1, 2. In a similar way ˜ a , b(3) >h3 = lp < pbc , W 1/2 · b(3) >h9 h9 < W 1/2 · bJa , bIa >h3 J

136

.

(III.5.63)

Thus, utilizing #

(i! )

I,J;i,j=1,2,3

(i)

(i)

(j)

(j ! )

(j)

< W 1/2 bI ! a , bIa >h3 < bIa , K1 · bJa >h3 < bJa , W 1/2 bJ ! a >h3 (i! )

(j ! )

=< bI ! a , W 1/2 K1 W 1/2 · bJ ! a >h3

(III.5.64)

we get ˜ a >h3 ˜ a , K1 · G h9 + < pbc , bIbc >h9

% & (i) (j) (1j) (2j) < bIa , W 1/2 K1 W 1/2 · bJa >h3 β < bJbc , xbc >h9 + < bJbc , pbc >h9 # (3) (3) (3) (3) +lp2 < pbc , bIbc >h9 < bIa , W 1/2 K1 W 1/2 · bJa >h3 < bJbc , pbc >h9 I,J

(III.5.65)

where we assume, that K1 (and later also K2 ) commutes with the projector on the transversal modes. We will see that this is always possible to achieve. ˜ a we get for the Va contribution In an analogous way as for G # % & (4i) (4! i) < Va , K2 · Va >h3 = lp2 β < xbc , bIbc >h9 + < pbc , bIbc >h9 I,J;i,j=1,2

% & (i) (j) (4j) (4! j) < bIa , W 3/2 K2 W 3/2 · bJa >h3 β < bJbc , xbc >h9 + < bJbc , pbc >h9 #% & (6) (7) +2lp2 β < xbc , bIbc >h9 + < pbc , bIbc >h9 I,J

% & (3) (3) (6) (7) < bIa , W 3/2 K2 W 3/2 · bJa >h3 β < bJbc , xbc >h9 + < bJbc , pbc >h9 . (III.5.66)

To obtain the C contribution we need % & < C, bI >h = lp < β W 1/2 · ,abc ∂a xbc + W −1/2 · − W 2 δbc − ∂b ∂c · pbc , bI >

= −βlp < xbc , W 1/2 · ,abc ∂a bI >h9 +lp < pbc , W −1/2 · (−W 2 δbc − ∂b ∂c ) · bI >h9 % & (7) (6) = −21/2 lp β < xbc , W 3/2 bIbc >h9 + < pbc , W 3/2 bIbc >h9 #% & (7) (6) = −21/2 lp β < xbc , bJbc >h9 + < pbc , bJbc >h9 < W 3/2 bJ , bI > (III.5.67) J

arriving at

< C, K3 C >h =

# I,J

= 2lp2

< C, bI >h< bI , K2 bJ >h< bJ , C >h

#% & (7) (6) β < xbc , bIbc >h9 + < pbc , bIbc >h9 < bI , W 3/2 K3 W 3/2 · bJ >h I,J

%

(7)

(6)

β < bIbc , xbc >h9 + < bIbc , pbc >h9

137

&

.

(III.5.68)

The expression of the constraints in terms of the fields zab and z ab is quite complicated, therefore we will perform a canonical transformation which will simplify the constraints. We will describe the canonical transformation in terms of the coordinates (α)

(α)

(α)

We define new coordinates x˜I (1i)

x ˜I

(α)

:=< bIbc , xbc >h9

xI

(α)

and p˜I

and pI

(1i)

p˜I

(3)

(8i)

(8i)

(1! i)

(1! i)

xI

(3)

p˜I = −xI 1 1 (4i) (4i) (4! i) p˜I = √ ( pI − xI ) 2 β 1 1 (6) (6) (7) p˜I = √ ( pI − xI ) β 2 1 1 (7) (7) (6) p˜I = √ ( pI − xI ) 2 β (8i)

= xI

Here, for instance xI

1 1 (1i) (1! i) = √ ( pI − xI ) 2 β

(3)

x ˜ I = pI 1 (4i) (4i) (4! i) x ˜I = √ (βxI + pI ) 2 1 (6) (6) (7) x ˜I = √ (βxI + pI ) 2 1 (7) (7) (6) x ˜I = √ (βxI + pI ) 2 x ˜I

(III.5.69)

.

by

1 (1i) (1! i) = √ (βxI + pI ) 2

(3)

(α)

:=< bIbc , pbc >h9

p˜I

(8i)

= pI

.

(1! i)

(1j)

(III.5.70)

stands for

(1! i)

:=< bIbc , xbc >h9 =

#

(1j)

< bJbc , xbc >h9 < bIbc , bJbc >h9

(III.5.71)

,

J;j=1,2

and the sum over J, j reduces to just one term if one uses a basis with the property (III.5.60). In this case the canonical transformation restricts to finite dimensional subspaces, indexed by I. (α) This gives new complex fields z˜I , defined by (α)

z˜I

1 (α) (α) = √ (˜ x − ipI ) 2 I

1 (α) (α) (α) z¯˜I = √ (˜ x + ipI ) . 2 I

(III.5.72)

The Master Constraint is then lp2 3 M= 4

#

I,J;i,j=1,2

(ij)

(1i) (1j)

Q1IJ x ˜I x ˜J

+2

#

(33)

I,J

+2

# I,J

(ij)

(3) (3)

Q1IJ x ˜I x ˜J +

(ij)

#

(ij)

(4i) (4j)

Q2IJ x ˜I x ˜J

I,J;i,j=1,2 (33) Q2IJ

(6) (6) x ˜I x ˜J

+2

# I,J

(7) (7)

Q3IJ x ˜I x ˜J

4

(III.5.73)

where Q1IJ , Q2IJ and Q3IJ are the matrix elements of Q1 = W 1/2 K1 W 1/2 , Q2 = W 3/2 K2 W 3/2 and Q3 = W 3/2 K2 W 3/2 respectively. We now come back to our assertion that the ADM–energy functional can be written as a manifestly positive function in terms of the fields zab and z¯ab . We will calculate the energy functional in a specific gauge and then generalize to the whole phase space. To this end we note (α) that the constraint hypersurface is given by the vanishing of all the x˜I , I ∈ I where α runs through all the values 1i, 3, 4i, 6, 7, that is all modes except the symmetric transverse traceless 138

(α)

ones. An natural gauge condition is to require the vanishing of p˜I , I ∈ I where the index α runs through the same values as above. Thus in this gauge the fields Aab and Hab are symmetric transversal traceless covariant tensors of second rank. (Hence this gauge is called STT gauge.) In the STT gauge formula (III.5.51) simplifies to " Γ"ab = −,acd ∂c Hdb

(III.5.74)

where the prime indicates that the fields are in the STT gauge. Using that −,acd ∂c squares to −∆ on transversal fields the energy functional (III.5.52) can be written as " $ ! 1 lin 3 2 " " " " " " H = d x β Γab Γab + (Aab − Γab )(Aab − Γab ) β2κ Σ " $ ! 1 3 2 " " " " " " d x β Hab (−∆)Hab + (Aa b + ,acd ∂c Hdb )(Aa b + ,aef ∂e Hf b ) = β2κ Σ ! √ " ) . (III.5.75) = 2 d3 x z " ab (! −∆ · zab Σ

Since the energy functional is a gauge-invariant function (on the constraint hypersurface) we can rewrite it in terms of the fields which are not necessary in the STT gauge by introducing the projector P (8) onto the symmetric transversal traceless modes (see (III.7.76)): ! √ lin H = 2 d3 x z ab (! −∆ · P (8) · z)ab + terms vanishing on the constraint surface. (III.5.76) Σ

The canonical transformation (III.5.70) leaves the STT modes unaffected, hence one can express the ADM-energy functional equally well in the z˜ab , z˜ab fields by simply replacing the old with the new fields. We conclude that the energy functional is a manifestly positive function. We will now quantize linearized gravity and examine the conditions on the operators Ki , i = 1, 2, 3. As in the Maxwell case we quantize the theory by introducing the kinematical algebra generated from " (α) (α) zˆ ˜ = < b , z˜bc >h9 (III.5.77) I

Ibc

(α) (zˆ˜I )† .

and the corresponding adjoint operators This algebra is represented on the Fock space F generated from the cyclic vacuum vector Ω. The commutation relations are the usual ones 5

(α) (γ) 6 z˜ˆI , zˆ ˜J = 0

5

(α) (γ) 6 (zˆ ˜I )† , (zˆ˜J )† = 0

The Master Constraint Operator is given by 7 = M

5

(α) (γ) 6 z˜ˆI , (zˆ˜J )† = δIJ δαγ

.

(III.5.78)

lp2 3 ( ( ( (ij) ˆ(1i) ˆ(1j) (33) ˆ(3) ˆ(3) (ij) ˆ(4i) ˆ(4j) ˜I x ˜J + 2 I,J Q1IJ x ˜I x ˜J + I,J;i,j=1,2 Q2IJ x ˜I x ˜J I,J;i,j=1,2 Q1IJ x 4 4 ( ( (33) ˆ(6) ˆ(6) ˆ˜(7) x ˆ(7) +2 I,J Q2IJ x ˜I x ˜J + 2 I,J Q3IJ x (III.5.79) I ˜J (ij)

(ij)

where Q1IJ , Q2IJ and Q3IJ are the matrix elements of Q1 = W 1/2 K1 W 1/2 , Q2 = W 3/2 K2 W 3/2 and Q3 = W 3/2 K2 W 3/2 respectively. 7 is a densely defined operator A calculation completely analogous to (III.5.15) reveals that M 3 on F if and only if Q1 , Q2 and Q3 are trace class operators on h respectively h. 139

One possible choice for these operators is Qj

= P ⊥ e−h P ⊥ + P / e−h P /

for

j = 1, 2

−h

Q3 = e

(III.5.80)

where h is the three dimensional harmonic oscillator operator and P ⊥ , P / are the projectors onto the transversal and longitudianl modes in h3 respectively. K1 = P ⊥ W −1/2 e−h W −1/2 P ⊥ + P / W −1/2 e−h W −1/2 P / and analogously K2 commute with the projector P ⊥ . The following calculation shows that the classical Master Constraint is well defined with this ˜ a , K1 · G ˜ a > since the other terms can be choice of the Qi ’s. We consider only the part < G treated in a similar way. As in the Maxwell case we will expand this term into coherent states ψz (III.5.17) for the three dimensional harmonic oscilator h. In the following calculation we will see ˜ a and (P / · G) ˜ a for a = 1, 2, 3 as scalar functions. To emphasise this, we the components (P ⊥ · G) will place a dot above the a: a. ˙ We will also replace the label z ∈ C3 in ψz by za˙ ∈ C3 , a˙ = 1, 2, 3. We can then write: # # ˜ a , K1 · G ˜ a >h3 = ˜ a˙ , e−h P (γ) W −1/2 G ˜ a˙ >h h < P (γ) W −1/2 G a ˙ π3

˜ a˙ >h × < ψza˙ , e−h ψza!˙ >h< ψza!˙ , P (γ) W −1/2 G ! ! # d3 za˙ d3 z¯a˙ d3 za"˙ d3 z¯a"˙ ˜ a˙ , ψz >h < W −2 ∂b ∂b P (γ) W −1/2 G a ˙ 3 3 3 3 π π C C

a=1,2,3 ˙ γ=⊥,/

=

#

˜ a˙ >h × < ψza˙ , e−h ψza!˙ >h< ψza!˙ , W −2 ∂c ∂c P (γ) W −1/2 G # ! d3 za˙ d3 z¯a˙! d3 z " d3 z¯" 1 1 3 a˙ a˙ ˜ a˙ , W − 2 ∂b ψz >h < W − 2 ∂b P (γ) W − 2 G a ˙ 3 3 π π C3 C3

a=1,2,3 ˙ γ=⊥,/

≤

#

3

1

1

˜ a˙ >h × < ψza˙ , e−h ψza!˙ >h< W − 2 ∂c ψza!˙ , W − 2 ∂c P (γ) W − 2 G ! 3 3 ! 3 " 3 " # d za˙ d z¯a˙ d za˙ d z¯a˙ −1/2 (γ) −1/2 ˜ 2 ||W ∂b P W Ga˙ ||h3 3 3 3 π π3 C C

a=1,2,3 ˙ γ=⊥,/

×||W −3/2 ∂b ψza˙ ||h3 ||W −3/2 ∂b ψza!˙ ||h3 | < ψza˙ , e−h ψza!˙ >h |

(III.5.81)

The first factor in the last line can be simplified to ˜ a˙ ||23 = ||(P (γ) G) ˜ a˙ ||2 ≤ ||G ˜ a˙ ||2 ||W −1/2 ∂b P (γ) W −1/2 G h h h

(III.5.82)

since partial derivatives commute with the projectors P (γ) . The remaining integrals in (III.5.81) are the same as in the sixth line of (III.5.20), so we can use the estimates (III.5.20–III.5.24), showing that the Master constraint is a well defined phase space function. If we assume, as is the case for the Qi ’s in (III.5.80), that it is possible to choose bI and (1) (2) (1) (2) (3) bIa , bIa such that bI , bIa , bIa and bIa = W −1 · ∂a bI are eigenstates of Q1 , Q2 and Q3 , we can 140

write the Master Constraint Operator more compactly as 7 = M

lp2 # (α) (α) 2 ˆ˜ ) qI (x I 2

,

(III.5.83)

I,α

where the index α assumes values α = 1i, 3, 4i, 6, 7 with i = 1, 2 and the qIα are positive eigenvalues (multiplied by two for α = 3, 6, 7) of the Qj ’s. Thus this Master Constraint Operator has the same structure as the Master Constraint Operator for the Maxwell theory (III.5.14). Going through the same procedure as in this case, see section III.5.1.2, we see that the physical Hilbert ˜ by applying just space is unitarily equivalent to the Fock space generated from the vacuum Ω the creation operators generating symmetric–transversal–traceless modes. This result coincides with a Reduced Phase Space Quantization in a STT gauge.

III.6

Summary and Discussion

We have seen that the Master Constraint Programme can deal with a broad selection of examples. In particular it can cope with systems, where a priori one would expect difficulties, as systems with structure functions, systems where the constraints generate a non–amenable gauge group, such that zero is not in the spectrum of the Master Constraint Operator or systems where one would expect severe ultra–violet divergencies. Let us discuss the results of the examples in detail. For the systems in section III.3 we recover the results one would expect. The first example III.3.1 which has an Abelian constraint algebra, shows that the Direct Integral Decomposition suppresses spurios solutions, i.e. wavefunctions, which are solutions to the Master Constraint operator but not to the individual constraints pˆj . That the spurius solutions do not appear in the physical Hilbert space, is connected to the fact that the Master Constraint Programme leads automatically to a self–adjoint representation of the Dirac observables. Example III.3.2 is a second class system and we showed that one can apply the Master Constraint Programme also to such systems. This was already emphasized by Klauder in his Affine Quantization Programme for gravity [20]. Of course, because of the second class property of the system one has now to allow for a quantum correction proportional to ! to the Master Constraint Operator. Nevertheless one finds for the physical Hilbert space and for the representation of the Dirac observables the results one would expect beforehand. In the next example III.3.3 the constraints generated a semi–simple compact Lie group. Here we have chosen the Master Constraint Operator such that it coincides with the Casimir operator of the group. This Casimir operator has a discrete spectrum and hence the physical Hilbert space is simply given by its null eigenspace. In other words we had just to look for the subspace of the kinematical Hilbert space, where the gauge group is represented trivially. In example III.3.4 we had to deal with structure functions. Such systems are notoriously difficult to deal with in other approaches to Dirac quantization, such as group averaging [17].35 The reason is that because of the structure functions the constraints do not generate a proper Lie group. In our example we succeeded in constructing a physical Hilbert space. A further complication arose because the Master Constraint Operator displayed a mixed spectrum near zero. We 35

A suggestion how to deal with structure functions employing BRST methods has been made in [65]. However, so far the method is only formal and it has not been shown that it produces a non–negative physical inner product.

141

therefore got a contribution from the pure point spectrum and a contribution from the continuous spectrum to the physical Hilbert space. The first contribution was unitarily equivalent to an L2 –Hilbert space in the Dirac observable t. The latter contribution was a one–dimensional space, corresponding to the delta-function δ(t) at the origin of the configuration space as a solution to the Master Constraint. The next example III.3.5 was also an example where the Master Constraint Operator had a mixed spectrum and served as an illustration of a technical point encountered in the previous example. Our example was in some respect special, because it was possible to quantize the individual constraints as self–adjoint operators. Generically, for instance for the constraints in general relativity, this is not possible for the following reason [66] : Let {Cj , Ck } = fjk l Cl with non–trivial, real valued structure functions fjk l . Suppose that the Cˆj are self–adjoint then in quantum theory we expect a relation of the form [Cˆj , Cˆk ] = i!(fˆjk l Cˆl + Cˆl (fˆjk l )† )/2 where the symmetric ordering is forced on us due to the antisymmetry of the commutator. Since fjk l is real valued fˆjk l −(fˆjk l )† should vanish as ! → 0. Now suppose that Ψ is a generalized solution36 of all constraints, Cˆj Ψ = 0 for all j. Applying the commutator l )† ]Ψ for all j, k. Now as ! → 0, the expression i![C ˆl , (fˆl )† ] becomes the we find 0 = i![Cˆl , (fˆjk jk Poisson bracket !2 {Cl , fjk l } to lowest order in !. Unless this classical quantity vanishes, the l )† ]Ψ = 0 which have solution Ψ not only satisfies Cˆj Ψ = 0 but the additional constraints [Cˆl , (fˆjk no classical counterpart. Iterating like this it could happen, in the worst case, that Ψ satisfies an infinite tower of new constraints, leaving us with the only solution Ψ = 0. In general the solution space will be too small as to capture the physics of the classical reduced phase space, displaying a quantum anomaly. In our example we had that {Cl , fjk l } = 0, l = 1, 2, 3 which is why we succeeded in using self–adjoint constraint operators there. However also if the individual constraints cannot be quantized as self–adjoint operators this does not represent an obstacle for the Master Constraint Programme. We just need to quantize the Master Constraint as a self–adjoint operator and this can (formally) be done for example as 7 = ( Cˆk Cˆ † also if the individual constraint operators are not self–adjoint. M k k In section III.4 we have examples with constraints generating a semi–simple non–compact Lie group, namely SO(2, 1) and its double cover SL(2, R). The key property of these groups concerning the Master Constraint Programme is that these are non–amenable groups. This causes that the spectrum of the Master Constraint Operator is supported on a genuine subset of the positive real line not containing zero. Our proposal to subtract the lowest point of the spectrum from the Master Constraint, which can be considered as a quantum correction37 because it is proportional to !2 worked and produced an acceptable physical Hilbert space. In order to construct the physical Hilbert space we made heavily use of the representation theory of the constraint algebra and of the observable algebra. In particular we employed the theory of dual pairs explained in appendix III.7.1. We did not have to perform the Direct Integral Decomposition explicitly. Instead we used that the representation of the observable algebra on the physical Hilbert space is determined by the decomposition of the representation More precisely we should look for solutions in a space of distributions Φ∗ dual to a dense and invariant (under ˆ † f ) = 0 for all f ∈ Φ and all j. the constraints) subspace Φ ⊂ H in the form Ψ(C j 37 The fact that the correction is quadratic in ! rather than linear in contrast to the normal ordering correction of the harmonic osciallor can be traced back to the fact that harmonic oscillator Hamiltonian can be considered the Master Constraint for the second class pair of constraints p = q = 0 while the sl(2, R) constraints are first class. 36

142

of the constraint algebra. Note that the decomposition of a Lie group representation in its irreducible components is an example for a Direct Integral Decomposition. We compared our results with the results of a group averaging procedure in [18] and of (Refined) Algebraic Quantization [55].38 For both examples in section III.4 these methods resulted in a representation of the observable algebra with superselection sectors, which differs from the results obtained in this work. However, as was observed in [18], the appearance of superselection sectors in the Refined Algebraic Quantization scheme may depend on the choice of a dense subspace in the kinematical Hilbert space. This is an auxilary structure which one has to choose in the Refined Algebraic Quantization but which is not needed in the Master Constraint Programme. A further difference is that the physical wavefunctions obtained in the Refined Algebraic Quantization satisfy exactly the constraints whereas we had to add a quantum correction to the Master Constraint. It would be interesting to know whether this is related to the appearance respectively absence of superselection sectors in the representation of the observable algebra. The constraints impose certain restrictions on this representation and it may be that one has to modify these restrictions by a quantum correction in order to obtain a representation without superselection sectors. A further study of this issue would be valuable. In section III.5 we applied the Master Constraint Programme to free field theories, namely Maxwell theory and linearized gravity. The basic lesson that we have learnt from these examples is that, surprisingly, the Master Constraint Programme can cope with the UV singularities that one expects from squaring operator valued distributions. Take the Maxwell case for simplicity. The infinite number of classical Gauss' constraints G(x) = ∂a E a (x) can classically be encoded in the single Master contraint M" = 12 d3 xG(x)2 . This is classically well defined because G(x)2 decays at infinity as r −6 and, moreover, it has well defined distributional second derivatives with 7 " = ∞ is respect to the phase space differentiable structure. However, quantum mechanically, M hopelessly divergent. This cannot be even repaired by subtracting an infinite constant because ' 7 " , in contrast to the electromagnetic field energy H = 1 d3 xδab δab (E a (x)E b (x)+B a (x)B b (x)) M 2 has a different structure in terms of creation and annihilation operators. This is best seen by ' 3 √ 3 −ik·x using the √ Fourier transform of the annihilation operators z ˆ (k) = d x/ 2π e zˆa (x) where a √ √ za (x) = ( 4 −∆Aa (x) − i( 4 −∆)−1 E a (x))/ 2α and α is the Feinstrukturkonstante. Then we get ! ka kb ˆ = ! ) [ˆ za (k)ˆ zb (k)† + zˆa (k)† zˆb (k)] (III.6.1) H d3 k ||k|| (δab − ||k||2 ! " α 7 d3 k ||k|| ka kb [ˆ za (k)ˆ zb (k)† + zˆa (k)† zˆb (k) − zˆa (k)ˆ zb (−k) − zˆa (k)† zˆb (−k)† ] M = 2 7 " : is neither ˆ : is densely defined and positive while : M Thus, the normal ordered expression : H densely defined nor (formally) positive. We used the flexibility in the definition of the Master Constraint in order to circumvent this problem. What we did is to introduce a positive integral kernel K(x, x" ) (that is, a 7 = positive on the one particle Hilbert space h = L2 (R3 , d3 x)) and to define M ' 3 ' 3operator " " " d x d x K(x, x )G(x)G(x ) which still classically encodes all the constraints. The idea is that the kernel serves to smoothen the operator valued distributions into well behaved opera7 to be densely defined is that tors. We found that the necessary and sufficient condition for M 38

Refined Algebraic Quantization is really meant as a refinement of the Algebraic Quantization scheme and group averaging is a method in order to implement Refined Algebraic Quantization.

143

√ 3/2 √ 3/2 Q = −∆ K −∆ be trace class. For any such choice of K the physical Hilbert space selected by the Master Constraint from the full Fock space of all modes was the correct one, the Fock space for the transversal modes. Thus, the physical Hilbert space constructed is independent of the concrete choice of K which matches nicely with the fact that the classical constraint surface defined by M = 0 is independent of the choice of K. The application of the Master Constraint Programme to the examples was successful but in many of the examples also technically involved. For the application to Loop Quantum Gravity it might be helpful to develop approximation schemes which implement the main ideas of the Master Constraint Progamme.

144

III.7

Appendix

III.7.1

Review of the Representation Theory of SL(2, R) and its various Covering Groups

III.7.1.1

sl(2, R) Representations

In this section we will review unitary representations of sl(2, R), see [67, 68]. In the defining two–dimensional representation the sl(2, R)–algebra is spanned by " $ " $ " $ −1 −1 0 1 −1 1 0 0 1 h= n1 = n2 = (III.7.1) −1 0 1 0 0 −1 2i 2i 2i with commutation relations 6 5 h, n1 = in2

5

6 n2 , h = in1

5

6 n1 , n2 = −ih .

(III.7.2)

We introduce raising and lowering operators n± as complex linear combinations n± = n1 ± in2 of n1 and n2 , which fulfill the algebra 5 6 5 + −6 h, n± = ±n± n , n = −2h . (III.7.3) The Casimir operator, which commutes with all sl(2, R)-algebra operators, is C = −h2 + 12 (n+ n− + n− n+ ) = −h2 + n21 + n22

.

(III.7.4)

We are interested in unitary irreducible representations of sl(2, R), i.e. representations where h, n1 and n2 act by self–adjoint operators on a Hilbert space, which does not have non-trivial subspaces, that are left invariant by the sl(2, R)–operators. According to Schur’s Lemma the Casimir operator acts on an irreducible space as a multiple of the identity operator C = c Id. Since n1 and n2 are self–adjoint operators, the raising operator n+ is the adjoint of the lowering operator n− and vice versa. (For notational convenience we often do not discriminate between elements of the algebra and the operators representing them.) In general the sl(2, R)-representations do not exponentiate to a representation of the group ˜ R). Since h is the generator of the compact SL(2, R) but to the universal covering group SL(2, ˜ subgroup of SL(2, R) it will have discrete spectrum (and therefore normalizable eigenvectors) in a unitary representation. If the sl(2, R)-representation exponentiates to an SL(2, R) representation, h has spectrum in { 12 n, n ∈ Z}. If in this group representation the center ±Id acts trivially, it is also an SO(2, 1) representation, since SO(2, 1) is isomorphic to the quotient group SL(2, R)/{±Id}. In this case h has spectrum in Z. Now, assume that |h > is an eigenvector of h with eigenvalue h. Using the commutation relations (III.7.3) one can see, that n± |h > is either zero or an eigenvector of h with eigenvalue h ± 1. By repeated application of n+ or n− to |h > one therefore obtains a set of eigenvectors {|h + n >} and corresponding eigenvalues {h + n}, where n is an integer. This set of eigenvalues may or may not be bounded from above or below. Similarly, one can deduce from the commutation relations that n+ n− |h > and n− n+ |h > are both eigenvectors of h with eigenvalue h (or zero). Apriori these eigenvectors do not have to be a multiple of |h >, since it may be, that h has degenerate spectrum. But this is excluded by the relations n− n+ = h2 + h + C , (III.7.5) n+ n− = h2 − h + C 145

obtained by using (III.7.3,III.7.4). (Remember, that h and C act as multiples of the identity on |h >.) From this one can conclude that the set {|h + n >} is invariant under the sl(2, R)-algebra (modulo multiples) and hence can be taken as a complete basis of the representation space. One can use the relations (III.7.5) to set constraints on possible eigenvalues of h and C. Consider the scalar products < h ± 1|h ± 1 > = < h|(n± )† n± |h >=< h|n∓ n± |h >

= < h|h2 ± h + C|h >= (h2 ± h + c) < h|h > .

(III.7.6)

Since the norm of a vector has to be positive one obtains the inequalities h2 ± h + c ≥ 0

(III.7.7)

for the spectrum of h and the value of the Casimir C = c Id. To summarize what we have said so far, we can specify a unitary irreducible representation with the help of the spectrum of h and the eigenvalue of the Casimir c. The spectrum of h is non-degenerate and may be unbounded or bounded from below or from above. Together with c the spectrum has to fulfill the inequalities (III.7.7). In this way one can find the following irreducible representations of sl(2, R) (For an explicit description, how one can find the allowed representation parameters, see [67, 68]): 6 (a) The principal series P (t, ,) where t ∈ { 12 + ix, x ∈ R ∧ x ≥ 0} and , = h(mod 1) ∈ (− 12 , 12 . The spectrum of h is unbounded and given by {, + n, n ∈ Z}. The Casimir eigenvalue is c = t (1 − t) ≥ 14 . (For t = 12 , , = 12 the representation P (t, ,) is reducible into D(1) and D ∗ (1) see below.)

(b) The complementary series Pc (t, ,) where

1 2

< t < 1 and |,| < 1 − t.

The spectrum of h is unbounded and given by {, + n, n ∈ Z}. The Casimir eigenvalue is 0 < c = t (1 − t) < 14 (c) The positive discrete series D(k) where k > 0. Here, the spectrum of h is bounded from below by 12 k and we have spec(h) = { 12 k + n, n ∈ N}. The value of the Casimir is c = 12 k − 14 k2 ≤ 14 . (d) The negative discrete series D∗ (k) where k > 0. In this case the spectrum of h is bounded from above by − 12 k and we have spec(h) = {− 12 k − n, n ∈ N}. The value of the Casimir is c = 12 k − 14 k2 ≤ 14 . (e) The trivial representation. As mentioned above representations with an integral h-spectrum can be exponentiated to representations of the group SO(2, 1), if the spectrum includes half integers one obtains representations of SL(2, R) and for spec(h) ∈ { 14 n, n ∈ Z} representations of the double cover of SL(2, R) (the metaplectic group). Finally we want to show, how one can uniquely determine the action of the sl(2, R)-algebra in a representation from the principal series. (The other cases are analogous, but we need this case

146

in section III.4.2.5.) To this end we assume that the vectors ||h >> are normalized eigenvectors of h with eigenvalue h. Applying n± gives a multiple of ||h ± 1 >>: n± ||h >>= A± (h)||h ± 1 >>

(III.7.8)

.

Using relation (III.7.5) one obtains for the coefficients A± (h) n∓ n± ||h >>= A∓ (h ± 1)A± (h)|h >>= (h2 ± h + c)||h >>

.

(III.7.9)

A+ (h) =>= > = A− (h + 1) .

(III.7.10)

Furthermore

The solution to these equations is A+ = c+ (h) (h + t)

A− (h) = c− (h) (h − t)

(III.7.11)

where |c± | = 1 and c+ (h)c− (h+1) = 1. Solutions with different c± are related by a phase change for the states ||h >>. III.7.1.2

Oscillator Representations

Here we will summarize some facts about oscillator representations, following [51, 69]. #R) the double cover of The oscillator representation is a unitary representation of SL(2, SL(2, R) (the so-called metaplectic group) and is also known under the names Weil representation, Segal–Shale–Weil representation or harmonic representation. The associated representation ω of the Lie algebra sl(2, R) on L2 (R) is given by h = 12 (a† a + 12 )

n1 = 14 (a† a† + aa) n+ = n1 + in2 = 12 a† a†

n2 = −i 14 (a† a† − aa)

n− = n1 − in2 = 12 aa

(III.7.12)

where we introduced annihilation and creation operators a=

√1 (x 2

+ ip) and

a† =

√1 (x 2

− ip) with

d p = −i dx

.

(III.7.13)

The operator h is (half of) the harmonic oscillator Hamiltonian and represents the infinitesimal generator of the two-fold covering group of SO(2). It has discrete spectrum spec(h) = { 14 + 12 n, n ∈ N} and its eigenstates are the Fock states |n >= (n!)−1/2 (a† )n |0 >; a|0 >= 0, which form an orthonormal basis of L2 (R). As can be easily seen, the representation (III.7.12) leaves the spaces of even and odd number Fock states invariant, therefore the representation is reducible into two subspaces. These subspaces are irreducibel since one can reach each (un-)even Fock state |n > by applying powers of n+ or n− to an arbitrary (un-)even Fock state |n" >. Since we have an h-spectrum which is bounded from below by 14 for the even number states and 34 for the uneven number states, the corresponding representations are D(1/2) and D(3/2) from the positive discrete series (of the metaplectic group). The Casimir of the oscillator representation is a constant: C(ω) = −h2 + 12 (n+ n− + n− n+ ) = 147

3 16

.

(III.7.14)

This confirms the finding ω " D(1/2) ⊕ D(3/2), since we have C(D(1/2)) =

11 4 2 (2

− 12 ) =

3 16

=

13 4 2 (2

− 32 ) = C(D(3/2))

.

(III.7.15)

In chapter III.4.2 we use an sl(2, R)-basis {h−ui = 2h, h+ui = 2n1 , d = 2n2 }. If one would rewrite the (ui )-representation (III.4.50) in terms of {h, n1 , n2 } it would differ from the representation (III.7.12) by minus signs in n1 and n2 . Nevertheless the (ui )-representation is unitarily equivalent to the representation (III.7.12), where the unitary map is given by the Fourier transform F. This can be easily seen by using the transformation properties of annihilation and creation operators under Fourier transformation: FaF−1 = ia and Fa† F−1 = −ia† . The contragredient oscillator representation ω ∗ on L2 (R) is given by h∗ = − 12 (a† a + 12 )

n∗1 = − 14 (a† a† + aa)

n∗2 = −i 14 (a† a† − aa)

n+∗ = n∗1 + in∗2 = − 12 aa

n−∗ = n∗1 − in∗2 = − 12 a† a†

(III.7.16)

Here, h∗ has strictly negativ spectrum spec(h∗ ) = {− 14 − 12 n, n ∈ N}. An analogous discussion to the one above reveals that the contragredient oscillator representation is the direct sum of the representations D∗ (1/2) and D∗ (3/2) from the negative discrete series (of the metaplectic group). 3 The Casimir evaluates to the same constant as above C(ω ∗ ) = 16 = C(D∗ (1/2) = C(D∗ (3/2)). III.7.1.3

Tensor Products of Oscillator Representations

Now one can consider the tensor product of p oscillator and q contragredient oscillator representations. The tensor product (⊗p ω) ⊗ (⊗q ω ∗ ) will be abbreviated by ω (p,q) . The representation space is (⊗p L2 (R)) ⊗ (⊗q L2 (R)) wich can be identified with L2 (Rp+q ). The tensor product representation (of the sl(2, R)-algebra) is given by h(p,q) =

1 2

p # (a†j aj + 12 ) − j=1

(p,q)

n1

=

1 4

(p,q)

= − 4i

(a†j a†j + aj aj ) −

p+q # j=1

p+q #

(a†j aj + 12 )

j=p+1

p # j=1

n2

1 2

1 4

p+q #

(a†j a†j + aj aj )

j=p+1

(a†j a†j − aj aj )

(III.7.17)

where aj and a†j denote annihilation and creation operators for the j-th coordinate in Rp+q : aj =

√1 (xj 2

+ ipj ) and a† =

√1 (xj 2

− ipj ) with

pj = −i∂j

.

(III.7.18)

On L2 (Rp+q ) we also have a natural action of the generalized orthogonal group O(p, q) given by g ·f (5x) = f (g−1 (5x)), where g ∈ O(p, q) and 5x ∈ Rp+q . This action commutes with the sl(2, R)

148

action, which can rapidly be seen, if we calculate the following sl(2, R) basis: +

e

= 2h

(p,q)

+

(p,q) 2n1

p p+q # # j 2 = (x ) − (xj )2 = gij xi xj j=1

−

e

= 2h

(p,q)

−

(p,q)

d = −2n2

(p,q) 2n1

=

j=1

=

1 2

p+q #

j=p+1

p #

(pj ) − 2

p+q #

(pj )2 = gij pi pj

j=p+1

(xj pj + pj xj ) = 12 gji (xj pi + pi xj )

.

(III.7.19)

j=1

Here gij is inverse to the metric gij = diag(+1, . . . , +1, −1, . . . , −1) (with p positive and q negative entries) so that gji := gik gkj = δji , where in the last formula and in the right hand sides of (III.7.19) we summed over repeated indices. Since O(p, q) leaves by definition the metric gij invariant, the sl(2, R)–operators (III.7.19) (and all their linear combinations) are left invariant by the O(p, q)–action: ρ(g−1 )sˆ ρ(g) = s, where s is an element from the sl(2, R)-algebra representation and ρ(g) denotes the action of g ∈ O(p, q) on states in L2 (Rp+q ). The action of O(p, q) induces a unitary representation ρ of O(p, q) on L2 (Rp+q ) (defined by ρ(g)f (5x) = f (g−1 (5x)) for f (5x) ∈ L2 (Rp+q )). The derived representation of the Lie algebra so(p, q) is given by: Ajk = xj pk − xk pj ,

j, k = 1, . . . , p

(III.7.20)

j, k = p + 1, . . . , p + q

(III.7.21)

Cjk = x pk + x pj ,

j = 1, . . . p, k = p + 1, . . . , p + q

Bjk = x pk − x pj , j j

k k

.

(III.7.22)

The operators Ajk and Bjk span the Lie algebra so(p) × so(q) of the maximal compact group O(p) × O(q). (From this one can conclude that Ajk and Bjk have discrete spectra.) #R)) there is a remarkable theorem, which For the representations ρ(O(p, q) and ω (p,q) (SL(2, we will cite from [69]: ˜ R)) generate each other commutants in The groups of operators ρ(O(p, q)) and ω (p,q) (SL(2, the sense of von Neumann algebras. Thus there is a Direct Integral Decomposition ! p+q L2 (R ) " σs ⊗ τs ds (III.7.23) #R), and σs and τs are irreducible where ds is a Borel measure on the unitary dual of SL(2, #R), respectively. Moreover σs and τs determine each other representations of O(p, q) and SL(2, almost everywhere with respect to ds. This means, that if we are interested in the decomposition of the ρ(O(p, q))-representation, #R), which may be an easier task. (We used this in example we can equally well decompose SL(2, III.4.1.) Furthermore, this theorem is very helpful if one of the two group algebras represents the constraints (say so(p, q)) and the other coincides with the algebra of observables (as is the case in examples III.4.1 and III.4.2). The constraints would then impose that the physical Hilbert space has to carry the trivial representation of so(p, q). Now if the trivial representation is included in the decomposition (III.7.23) we can adopt as a physical Hilbert space the isotypical 149

component of the trivial representation (i.e. the direct sum of all trivial representations which appear in the decomposition of L2 (Rp+q ) with respect to the group O(p, q))). The above cited theorem ensures that this space carries a unitary irreducible39 representation of the observable algebra. The scalar product on this Hilbert space is determined by this representation. The same holds if we have the sl(2, R) algebra as constraints and so(p, q) as the algebra of observables. To determine the representation of the observable algebra on the physical Hilbert space the following relation between the (quadratic) Casimirs of the two algebras involved is administrable (see [51]): # # # (p,q) (p,q) 2 2 2 4(−(h(p,q) )2 + (n1 )2 + (n2 )2 ) = − A2jk − Bjk + Cjk + 1 − ( p+q 2 − 1) j= . (A†+ )k+ (A†− )k− (B+ ) (B− ) |0, 0, 0, 0 > " " k+ k− k+ k−

(III.7.34)

where |0, 0, 0, 0 > is the state which is annihilated by all four annihilation operators and " , k " ∈ N. These states are eigenstates of H , O k+ , k− , k+ − 12 and O34 with eigenvalues − " " k := eigenval(H− ) = k+ + k− − k+ − k−

j := eigenval(O12 ) = k+ − k−

" " j " := eigenval(O34 ) = −k+ + k−

(III.7.35)

.

The common eigenspaces V (j, j " ) of the operators O12 and O34 are left invariant by the sl(2, R)algebra (III.7.32), since there only appear combinations of A+ A− , B+ B− , their adjoints and number operators, which leave the difference between particels in the plus polarization and particels in the minus polarization invariant. Moreover the kinematical Hilbert space is a direct sum of all the (Hilbert) subspaces V (j, j " ) (since these V (j, j " ) constitute the spectral decomposition of the self adjoint operators O12 and O34 ): # L2 (R4 ) = V (j, j " ) . (III.7.36) j,j ! ∈Z

The scalar product on V (j, j " ) is simply gained by restriction of the L2 -scalar product to V (j, j " ). The space V (j, j " ) still carries an sl(2, R)–representation, which can be written as a tensor product, where the two factor representations are (A)

H− = A†+ A+ + A†− A− + 1 (A)

H+ = −(A+ A− + A†+ A†− ) D (A) = i(A†+ A†− − A+ A− )

(B)

H−

(B)

H+

† † = −(B+ B+ + B− B− + 1) † † = −(B+ B− + B+ B− )

† † D(B) = i(B+ B− − B+ B− ) .

(III.7.37)

" , k " >, k − k = j ∧ −k " + k " = j " }, which is also an Each V (j, j " ) has a basis {|k+ , k− , k+ + − − + − (A) (B) (A) eigenbasis for H− and H− . Therefore it is easy to check that H− has on V (j, j " ) a lowest (A)

eigenvalue given by (|j| + 1), more generally the spectrum of H− 152

is non-degenerate and given

(A)

(B)

by spec(H− ) = {|j| + 1 + 2n, n ∈ N}. Similarly, H− has a highest eigenvalue −(|j| + 1) (B) on V (j, j " ) and the spectrum is spec(H− ) = {−(|j| + 1 + 2n), n ∈ N}. From this one can deduce that the representation given on V (j, j " ) is isomorphic to D(|j| + 1) ⊗ D(|j " | + 1), i.e. a tensor product of a positive discrete series representation and a representation fom the negative discrete series. 7 Now, what we want achieve is a spectral composition of the Master Constraint Operator M " on each of the subspaces V (j, j ). (Clearly, the Master Constraint Operator leaves these spaces invariant.) The Master Constraint Operator is the sum of a multiple of the sl(2, R)-Casimir 2 . The latter two operators commute, so we can diagonalize them simultanously. This and 2H− problem was solved in [59]. There, another realization of the representation D(|j|+1)⊗D(|j " |+1) was used, hence to use the results of [59], we have to construct a (unitary) map, which intertwines between our realization and the realization in [59]. To this end, we will depict the realization used in [59], at first for representations from the positive and negative discrete series. The Hilbert spaces for these realizations are function spaces on the open unit disc in C. For the positive discrete series D(l), l ∈ N − 0 the Hilbert space, which we will denote by Hl , consists of holomorphic functions and for the negativ discrete series D ∗ (l) , l ∈ N0 the Hilbert space (H∗l ) is composed of anti-holomorphic functions. The scalar product is in both cases ! l−1 < f, h >l = f (z)h(z)(1 − |z|2 )l−2 dx dy (III.7.38) π D where D is the unit disc and dx dy is the Lebesgue measure on C. (For l = 1 one has to take the limit l → 1 of the above expression.) An ortho–normal basis is given by 1

fn(l) := (µl (n))− 2 z n

(n ∈ N) with µl (n) =

Γ(n + 1)Γ(l) Γ(l + n)

(III.7.39)

for the positiv discrete series; for the negative series an ortho–normal basis is 1

fn(∗l) := (µl (n))− 2 z n

(n ∈ N) .

(III.7.40)

In this realizations the sl(2, R)-algebra acts as follows: for the positive discrete series D(l) d dz

(l)

= l + 2z

(l)

= −lz − (z + z −1 )z

H− H+

D (l)

d dz d = ilz + i(z − z −1 )z dz

(III.7.41)

and for the negative discrete series D∗ (l) d dz

(∗l)

= −l − 2z

(∗l)

= −lz + (z + z −1 )z

H− H+

D(∗l)

d dz d = −ilz + i(z − z −1 )z dz 153

.

(III.7.42)

(l)

(∗l)

(l)

(∗l)

The aforementioned bases {fn } and {fn } are eigen-bases for H− resp. H− with eigenvalues {l + 2n} resp. {−l − 2n} (where always n ∈ N). The representation space of the tensor product D(l) ⊗ D∗ (l" ) is the tensor product Hl ⊗ H∗l! , (l) (∗l! ) which has as an ortho–normal basis {fn ⊗ fn! , n, n" ∈ N}. The tensor product representation is obtained by adding the corresponding sl(2, R)-representatives from (III.7.41) and (III.7.42). " , k " >, k −k = j ∧ −k " +k " = j " } Now, considering the properties of the bases {|k+ , k− , k+ + − − + − (|j|+1)

(∗(|j ! |+1))

for V (j, j " ) and {fn ⊗ f n! } for H|j|+1 ⊗ H∗(|j ! |+1) it is very suggestive to construct a unitary map between these two Hilbert spaces by simply matching the bases: U:

H|j|+1 ⊗ H∗(|j ! |+1) → V (j, j " ) (∗(|j ! |+1))

fn(|j|+1) ⊗ fn!

!

" " &→ (−1)n |k+ , k− , k+ , k− >

2n = k+ + k− − |j| , "

2n =

" k+

+

" k−

"

− |j | ,

where

j = k+ − k− ,

" " j = −k+ + k−

(III.7.43)

.

One can check that this map intertwines the sl(2, R)-representations. (For this to be the case ! the factor (−1)n in (III.7.43) is needed.) Since this map maps an ortho–normal basis to an ortho–normal basis it is an (invertible) isometry and can be continued to the whole Hilbert space (which justifies the notation in (III.7.43)). We will later use this map to adopt the results of [59] to our situation. In the following we will sketch how the spectral decomposition of the Casimir operator in the D(l) ⊗ D∗ (l" )-representation is achieved in [59]. The Casimir operator is C =

(l) 1 4 (−(H−

(∗l! )

(l)

(∗l! ) 2

+ H− )2 + (H+ + H+

!

) + (D(l) + D(∗l ) )2 )

= −(1 − z1 z 2 )2 ∂z1 ∂z 2 + l" (1 − z1 z 2 )z1 ∂z1 + l(1 − z1 z 2 )z 2 ∂z 2 − 14 (l − l" )2 + 12 (l + l" ) − l l" z1 z 2 . (III.7.44)

!

(l)

(∗l! )

ll = (H +H This operator commutes with all sl(2, R)-generators and in particular with H− − − ), ! ll i.e. it leaves the eigenspaces of H− invariant. To take advantage of this fact one introduces new coordinates z = z1 z 2 , w = z1 and rewrites functions in Hl ⊗ H∗l! as a Laurent series in w (where 1

!

the coefficients are functions of z). Since functions of the form f (z)w 2 (k−l+l ) are eigenfunctions ll! with eigenvalue k one has effectively achieved the spectral decomposition of H ll! . (The of H− − number 12 (k − l + l" ) is always a whole number, since k is (un)even iff (l − l" ) is (un)even.) The linear span of all these functions (with fixed k) completed with respect to the subspace-topology coming from Hl ⊗ H∗l! ) is a Hilbert space, abbreviated by H(k, l, l" ). Since the power of w is fixed, this Hilbert space is a space of functions of z. The scalar product in this Hilbert space is characterized by the fact that {(µl (n + 12 (k − l + l" ))µl! (n)z n

| max(0, 12 (−k + l − l" ) ≤ n < ∞}

(III.7.45)

is an ortho–normal basis. ll! -eigenspaces One can restrict the Casimir (III.7.44) to this Hilbert space (since it leaves the H− invariant) obtaining " d2 d Ck = (1 − z) − z(1 − z) 2 − 12 (k − l + l" + 2 − (k + l + 3l" + 2)z) + dz dz $ " " " −1 1 " 1 . (III.7.46) 2 l (k + l + l ) + 4 (l + l )(2 − l − l )(1 − z) 154

Likewise the Master Constraint Operator restricts to H(k, l, l" ) and can be written as Mk = 4Ck + 2k2

(III.7.47)

.

These operators are ordinary second order differenential operators and their spectral decomposition is effected in [59] by using (modifications of) the Rellich-Titchmarsh-Kodaira-theory. We will not explain this procedure but merely cite the results. The eigenvalue equation for the Master Constraint (Mk − λ)f = 0 on H(k, l = |j| + 1, l" = " |j | + 1) has two linearly independent solutions (since it is a second order differential operator), a near z = 0 regular solution being 1

!

fk,j,j ! (z, t) = (1 − z)1−t− 2 (|j|+|j |+2) F (1 − t + 12 (−|j| + |j " |), 1 − t + 12 k, 1 + 12 (k − |j| + |j " |); z)

(III.7.48)

for k − |j| + |j " | ≥ 0 and 1

1

!

!

fk,j,j ! (z, t) = (1 − z)1−t− 2 (|j|+|j |+2) z 2 (−k+|j|−|j |) ×

×F (1 − t − 12 k, 1 − t + 12 (|j| − |j " |), 1 + 12 (−k + |j| −| j " |); z) (III.7.49) √ for k − |j| + |j " | ≤ 0, where t = 12 (1 + 1 − λ + 2k2 ), Re(t) ≥ 12 and F (a, b, c; z) is the hypergeometric function. For λ(k, |j|, |j " |) in the spectrum of the Master Constraint Operator these solutions are generalized eigenvectors of the Master Constraint Operator. The spectrum has a continuous part and a discrete part. There is a discrete part only if k > 0 for |j| − |j " | ≥ 2 or k < 0 for |j| −| j " | ≤ 2.: λdiscr = 4t(1 − t) + 2k2 ≥ 2k2 − k2 + 2|k|

with

t = 1, 2, . . . , 12 min(|k|, ||j| − |j " ||) for even k

with t = 32 , 52 . . . , 12 min(|k|, ||j| − |j " ||) for odd k and

λcont = 1 +

x2

+

2k2

>0

5

x ∈ 0, ∞

&

.

(III.7.50) (III.7.51)

The spectral resolution of a function f (z) in H(k, l = |j|+ 1, l" = |j|+ 1) is for k − |j|+ |j " | ≥ 0 ! 1 +i∞ # 2 f (z) = A(λdiscr ) + (2t − 1)µ(j, j " , k, t) < f, fk,j,j ! (·, t) > fk,j,j ! (z, t) dt 1 2

λdiscr

µ(j, j " , k, t) =

1 sin πt cos πt × iπ 2 Γ(|j| + 1)Γ(|j " | + 1)Γ2 ( 12 (k − |j| + |j " | + 2))

× |Γ(t + 12 k)Γ(t − 12 (2 − |j| − |j " |))|2 |Γ(t − 12 (|j| − |j " |))|2 and for k − |j| + |j " | ≤ 0 f (z) =

#

λdiscr

µ(j, j " , k, t) =

B(λdiscr ) +

!

1 +i∞ 2 1 2

(III.7.52)

(2t − 1)µ(j, j " , k, t) < f, fk,j,j ! (·, t) > fk,j,j ! (z, t) dt

1 sin πt cos πt × iπ 2 Γ(|j| + 1)Γ(|j " | + 1)Γ2 ( 12 (−k + |j| −| j " | + 2))

× |Γ(t − 12 k)Γ(t − 12 (2 − |j| −| j " |))|2 |Γ(t + 12 (|j| − |j " |))|2 155

(III.7.53)

where in the following we do not need A(λdiscr ) and B(λdiscr ) in explicit form. This gives the following resolution of a function f (z1 , z 2 ) in H|j|+1 ⊗ H∗(|j ! |+1)) : #! f (z1 , z 2 ) = discr. part + k

1 +i∞ 2

(2t − 1)µ(j, j " , k, t) < f, fk,j,j ! (·, ·, t) > fk,j,j ! (z1 , z 2 , t) dt

1 2

(III.7.54)

where

1

fk,j,j ! (z1 , z 2 , t) := fk,j,j ! (z1 z 2 , t) z12

(k−|j|+|j !|)

,

(III.7.55)

and the sum is over all whole numbers k with the same parity as (j − j " ). Now we can use the map U in (III.7.43) to transfer these results to the subspaces V (j, j " ) of the kinematical Hilbert space L2 (R4 ). To this end we rewrite (III.7.55) into a power series in z1 and z 2 using the definition of the hypergeometric function Γ(c) # Γ(a + n)Γ(b + n) n F (a, b, c ; z) = z (III.7.56) Γ(a)Γ(b) Γ(c + n)Γ(1 + n) n=0

and

(1 − z)1−d = For k − |j| + |j " | ≥ 0 we obtain

#

n=0

#

f (t; k, j, j " ) = U (fk,j,j ! (z1 , z 2 )) =

m=0

where

Γ(d + k − 1) zk Γ(d − 1)Γ(k + 1)

" " am |k+ (m), k− (m), k+ (m), k− (m) >

k+ = m + 12 (k + j + |j " |) and

(III.7.57)

.

k− = m + 12 (k − j + |j " |)

" k+ = m + 12 (|j " | − j " )

" k− = m + 12 (|j " | + j " )

(III.7.58)

(III.7.59)

1 1 1 am = (−1)m (µ(|j|+1) (m + (k − |j| + |j " |))) 2 (µ(|j ! |+1) (m)) 2 × 2 Γ(1 + 12 (k − |j| + |j " |)) × × Γ(1 − t + 12 (−|j| + |j " |))Γ(1 − t + 12 k) m # Γ(1 − t + 12 (−|j| + |j " |) + l)Γ(1 − t + 12 k + l) Γ(t + 12 (|j| + |j " |) + (m − l)) × Γ(1 + 12 (k − |j| + |j " |) + l)Γ(1 + l) Γ(m − l + 1)Γ(t + 12 (|j| + |j " |)) l=0

.

(III.7.60)

For k − |j| + |j " | ≤ 0 the coefficient am in (III.7.58) is obtained from (III.7.60) by replacing k 1

!

with −k, switching |j| and |j " | and multiplying with (−1) 2 (−k+|j|−|j |) . We could use the vectors f (t; k, j, j " ) to construct the spectral decomposition of L2 (R4 ). However, we want to achieve a spectral measure, which is independent of k, j and j " . For this purpose we normalize the solutions (III.7.58) to "

|t, k, j, j >=

"

µ(j, j " , k, t) i sin πt cos πt 156

$1 2

f (t; k, j, j " ) .

(III.7.61)

Now we can decompose a vector |f >∈ L2 (R4 ) as follows |f > = discrete part + 1 # ! 2 +i∞ i (1 − 2t) sin πt cos πt < f |t, k, j, j " > |t, k, j, j " > dt k,j,j !

1 2

(III.7.62)

!

where the sum is over all whole numbers k, j, j " with (−1)k = (−1)j−j . From this it follows, that L2 (R4 ) decomposes into a direct sum (for the discrete part) and direct integral of Hilbert spaces H(t), where in each H(t) an ortho–normal basis is given by the vectors |t, k, j, j " >. As explained in section III.4.2 our physical Hilbert space H"" consists of vectors with t = 12 , k = 0 and |j| = |j " |. In this case these vectors are given by # " " |j, j " > = |t = 12 , k = 0, j, j " > = bm |k+ (m), k− (m), k+ (m), k− (m) > m

bm = (−1)m

Γ(m + 1) # (Γ( 12 + l))2 Γ(|j| + 12 + (m − l)) Γ(|j| + 1 + m) (Γ(1 + l))2 Γ(m − l + 1)

(III.7.63)

l=0

" (m), k " (m) given by (III.7.59). with k+ (m), k− (m), k+ − Now we want to check our results by calculating the action of the Master Constraint Operator and of the observables on the states (III.7.63). The Master Constraint Operator rewritten in terms of annihilation and creation operators is % 7 = 2 2N (A+ )N (A− ) + N (A+ ) + N (A− ) + 2N (B+ )N (B− ) + M & † † N (B+ ) + N (B− ) + 2 + 2A†+ A†− B+ B− + 2A+ A− B+ B− + % &2 N (A+ ) + N (A− ) − N (B+ ) − N (B− ) (III.7.64)

where N (i) stands for the number operator for quanta of type i. The eigenvalue equation (M − λ)|j, j " >= 0 for the states (III.7.63) can be written as an equation for the coefficients bm : 0 = (8(m + |j| + 1)m + 4|j| + 4 − λ) bm + 4m(m + |j|) bm−1 + 4(m + 1)(m + |j| + 1) bm+1

(III.7.65)

(The coeffecient b−1 is defined to be zero.) One can check, that the coefficients (III.7.63) fulfill this equation for λ = 1: For this purpose one introduces ˜bm = (−1)m Γ(|j| + 1 + m) bm = Γ(m + 1)

m # (Γ( 12 + l))2 Γ(|j| + 12 + (m − l)) (Γ(1 + l))2 Γ(m − l + 1)

(III.7.66)

l=0

and realizes that the ˜bm ’s are the coefficients in the power expansion of the function Γ(|j| + 1/2) f0,j,j ! (z, t = 12 ) (with |j| = |j " |) from (III.7.48). This function fullfills the differential equation M0 · f = f where Mk=0 is the differential operator from (III.7.47). One can rewrite the differential equation for f into a equation for the coefficients ˜bm in a power expansion for f . If one replaces ˜bm with bm according to the first part of equation (III.7.66) one will get equation (III.7.65). Therefore the coefficients bm fulfill this equation. 157

The observables can be written as Q1 = Q2 = Q3 = P1 = P2 = P3 = Q+ = Q− = P+ = P− =

† † † † i 2 (A+ B+ − A− B− − A+ B+ + A− B− ) † † † † −1 2 (A+ B+ + A− B− + A+ B+ + A− B− ) 1 2 (N (A+ ) − N (A− ) + N (B+ ) − N (B− )) † † † † i 2 (A+ B− − A+ B− − A− B+ + A+ B+ ) † † † † −1 2 (A+ B− + A+ B− + A− B+ + A+ B+ ) 1 2 (N (A+ ) − N (A− ) − N (B+ ) + N (B− )) † −i √1 (Q1 + iQ2 ) = √ (A†+ B+ + A− B− ) 2 2 † +i √1 (Q1 − iQ2 ) = √ (A†− B− + A+ B+ ) 2 2 † −i √1 (P1 + iP2 ) = √ (A†+ B− + A− B+ ) 2 2 † +i √1 (P1 − iP2 ) = √ (A†− B+ + A+ B− ) 2 2

(III.7.67)

(III.7.68)

.

In section III.4.2 we concluded that on the physical Hilbert space H"" the observable algebra is generated by operators of the form Θ(Q3 )Qi Θ(Q3 ) and Θ(P3 )Pi Θ(P3 ). Therefore we will just depict the action of Q± on states with zero P3 -eigenvalue and of P± on states with zero Q3 -eigenvalue. One can determine from this the action of the observable algebra on H"" . To begin with, we consider the action of Q+ on states |j, −j >, j ≥ 0, i.e. states with P3 -eigenvalue zero and nonnegative Q3 -eigenvalue: # |j, −j > = bm (j)|m + j, m, m + j, m > m=0

m

bm (j) = (−1)m

Γ(m + 1) # (Γ( 12 + l))2 Γ(|j| + 12 + (m − l)) Γ(|j| + 1 + m) (Γ(1 + l))2 Γ(m − l + 1)

. (III.7.69)

l=0

On these states Q+ acts as # −i Q+ |j, −j > = √ ((m + 1)bm+1 (j) + (m + j + 1)bm (j)) |m + j + 1, m, m + j + 1, m > 2 m=0 −i √ (j + 1 ) 2 2

=

(∗)

|j + 1, −(j + 1) >

(III.7.70)

.

For j < 0 we have Q+ |j, −j >

= =

(∗)

−i √ 2

#

(m bm−1 (j) + (m + |j|)bm (j)) |m, m + |j| − 1, m, m + |j| − 1 >

m=0

√i (j 2

+ 12 ) |j + 1, −(j + 1) >

(III.7.71)

.

For the equalities marked with a star (∗) we have to check the relations (m + 1)bm+1 (j) + (m + j + 1)bm (j) = (j + 12 )bm (j + 1) m bm−1 (|j|) + (m + |j|)bm (j) = (|j| − 158

1 2 )bm (|j|

for j ≥ 0

− 1)

for j < 0

. (III.7.72)

The last equation is verified by using the ˜bm (j)’s defined in (III.7.66), which are the coefficients of f|j|(z) := Γ(|j| + 1/2) f0,j,±j (z, t = 12 ) = Γ(|j| + 1/2)(1 − z)−|j|−1/2 F (1/2, 1/2, 1; z) (III.7.73) in a power expansion in z. Then, rewriting of the identity (1 − z)f|j| (z) = (|j| − 1/2) f|j|−1 (z) into an equation for the ˜bm (j) and furthermore for the bm (j) results in the last equation of (III.7.72). For the first equation one starts with the differential equation M0 · f|j|(z) = f|j| (z) and replaces there (1 − z)−1 f|j|(z) with (|j| + 1/2)−1 f|j|+1(z). This then translates into the first equation of (III.7.72) for the coefficients bm (j). The relations (III.7.72) will also ensure the following equalities: Q− |j, −j > =

P+ |j, j > = P− |j, j > =

√i (j 2 −i √ (j 2 √i (j 2

− 12 )|j − 1, −(j − 1) > + 12 )|j + 1, j + 1 > − 12 )|j − 1, j − 1 >

(III.7.74)

.

These formulas differ by phase factors from the formulas in III.4.2.5. One can adjust these phase factors to one by choosing a new basis |j, ,j >" = (−i)j |j, ,j >. Therefore the results of this section and section III.4.2 are consistent.

III.7.2

Completeness of the basis

Here we will verify that (III.5.57) is indeed a basis of the Hilbert space of square integrable covariant tensors of second rank h9 = (L2 (R3 ))9 . We will begin by establishing, that the modes listed there are complete. To this end we introduce the projector p onto the transversal modes in h3 = (L2 (R3 ))3 (p · v)a := pba · vb := (δab + W −2 · ∂a ∂b ) · vb (III.7.75)

and define the following projectors on h9 (P (1) · T )ab = (δac − pca ) · pdb · Tcd

2 left long. right transv. modes

(P

(3)

(P

(4)

2 left transv. right long. modes

· T )ab = · T )ab =

(P (6) · T )ab = (P

(7)

(P

(8)

· T )ab = · T )ab =

(δac − pca ) · (δbd − pdb ) · Tcd pca · (δbd − pdb ) · Tcd cd 1 2 pab · p · Tcd d c d 1 c 2 (pa · pb − pb · pa ) · Tcd d c d cd 1 c 2 (pa · pb + pb · pa − pab p )

1 left and right long. mode 1 symm. transv. trace part mode 1 antisymm. transv. mode · Tcd

2 symm. transv. tracefree modes

(III.7.76)

Using the projector property p · p = p, it is easy to see that the projectors P (α) are orthogonal to each other and satisfy P (α) · P (β) = δαβ P (α) . For instance (P (6) · P (8) · T )ab = =

ef 1 c d c d cd 1 2 pab · p · 2 (pe · pf + pf · pe − pef p ) · cd cd ef cd 1 4 pab (p + p − p δef p ) = 0

Tcd (III.7.77)

where we used pef δef = 2. Furthermore the sum of all the projectors in (III.7.76) is the identity on h9 . 159

(α)

Next we have to show that the bI in (III.5.57) are a complete basis of the subspace of α-modes in h9 . We begin with the left longitudinal and right transversal modes, rewriting them as follows & (i) % (1i) (i) (III.7.78) bIbc := ,acb + W −2 · ,adb ∂d ∂c · bIa = −W −2 · ∂b ,cda ∂d bIa .

This equation can be proved by using that every transversal vector t1a can be written as the curl operator applied to another transversal vector t1a = ,abc ∂b t2c . Additionally one utilizes that ,abc ∂b · ,cde ∂d = −∆ δae on the subspace of transversal vectors. Now we have for any left longitudinal tensor Tab Tab = (δac − pca ) · Tcb = W −1 ∂a · (−W −1 · ∂c Tcb ) =: W −1 · ∂a fb

(III.7.79)

so it can be written as a gradient of a vector W −1 · fb . If Tab is additionally right transversal this vector has to be transversal. Therefore {bJcd = W −1 · ∂a tdJ }j∈J is a(n) (ortho–normal) basis for the left longitudinal right transversal modes if {tdJ }J∈I is a(n) (ortho–normal) basis for the transversal modes in h3 . Now if the latter is the case then {t"aJ := W −1 · ,abd ∂b tdJ }J∈J is also an ortho–normal basis for the transversal vector modes, since the operator ta &→ W −1 · ,abd ∂b ta is unitary on this subspace (on the transversal modes it squares to the identity and on the whole ! space h3 to the projector p). So we have proved, that b(1i) and b(1 i) is a basis for the left longitudinal right transversal modes. In a similar way one can deal with α = 3, 4i, 4" i. For the completeness proof of (6)

(3)

bIbc = −2−1/2 W −1 · (∂a δbc − ∂c δab )bIa

=

(3) bIa =W −1 ·∂a bI

(3)

2−1/2 (δbc + W −2 · ∂c ∂b )bI

(III.7.80)

we write for a tensor Tab ∈ P (6) (h9 ): Tab = (P (6) · T )ab = 12 pab pcd Tcd = 12 (δab + W −2 · ∂a ∂b ) · Tcc

(III.7.81)

where in the second equation we used that Tab is right and left transversal. Tcc is a square integrable function, so it can be expanded into the basis {bI }I∈J of h. We conclude, that (6) {bIbc }I∈J is a complete basis of the symmetric and transversal modes with tracepart. (7) To verify the completeness of the antisymmetric and transversal basis bIbc = −2−1/2 W −1 · ,bdc ∂d bI , we use that every antisymmetric tensor Tab can be expressed as Tab = ,abc tc

where tc = 12 ,cde Tde

.

(III.7.82)

The transversality condition ∂a Tab = 0 translates into the vanishing of the curl ,abc ∂a tc of tc , so that tc has to be longitudinal, i.e. can be written as the gradient of a function W −1 · f . If Tab is square integrable, so is tc and hence f , so f can be expanded into a basis bI of h. We conclude, (7) that {bIbc }I∈J is a complete basis of the antisymmetric and transversal modes. This finishes our verification, that the basis (III.5.57) is complete.

160

Discussion and Outlook In this thesis we were concerned with the problem to extract gauge invariant information out of a gauge system. This problem is the main obstacle for the construction of a canonical quantum theory of gravity. In Part II we developed and discussed the concepts of partial and complete observables in order to construct Dirac observables, i.e. gauge invariant functions, on the classical level. In Part III we considered the Master Constraint Programme, which is a technique for the construction of a physical Hilbert space, i.e. a Hilbert space of gauge invariant states. Our approach in Part II allowed us to unify several ideas into one framework, that is the Bubble Time Formalism, the construction of Dirac observables via gauge invariant extensions and the concept of partial and complete observables, which existed before only for systems with one constraint. In this way we could also write down and prove the Poisson algebra of complete observables and generalize and clarify the role of the physical Hamiltonians, which can be seen as generators of dynamics. Note however that our approach is more flexible than the Bubble Time Formalism or the gauge invariant extensions. We can also deal with clock observables, which do not provide a perfect gauge fixing, as the formalism of partially invariant partial observables shows. Moreover in the case of general relativity we managed to formulate quantities, which are invariants in the covariant formalism, i.e. with respect to the space–time diffeomorphism group, as Dirac observables in the canonical formalism. We hope that this will facilitate a comparison, based on observable quantitites, between canonical and covariant approaches to quantum gravity. An important new technical point in Part II is the use of a weakly Abelian constraint algebra. Such constraint algebras are much easier to obtain and to handle than strongly Abelian constraint algebras, because the latter will typically involve square roots of not necessarily positive functions. However a weakly Abelian constraint algebra is sufficient in order to use the advantages provided by the Abelianess. For instance we could give a system of partial differential equations for complete observables and even write down a formal solution in terms of a power series for complete observables. Furthermore we could reduce the number of constraints one has to deal with in the construction of certain complete observables for general relativity from infinity to one. We hope that this will help to develop approximation methods for the construction of Dirac observables, because one could use techniques which already exist for the case of one constraint. In Part III we applied the Master Constraint Programme to a wide variety of examples, including constraint algebras with structure functions, constraint algebras which generate a non–compact Lie group and free field theories. For this we had to use the Direct Integral Decomposition technique, which to our knowledge was not done before in the context of constraint quantization. This required a careful spectral analysis of the Master Constraint Operators. In the examples III.4, where the constraints generate covering groups of SO(2, 1) we could employ the representation theory of these groups and in particular the theory of dual pairs [69]. One can generalize the technique outlined there to other examples with a dual pair structure, for instance [56]. In the free field examples III.5 we had to find a densely defined Master Constraint Operator and then to deal with the difficulty, that the Master Constraint Operator involved excitations in infinitely many degrees of freedom. The Master Constraint Programme offers the advantage that one can replace a very complicated constraint algebra, possibly with structure functions, by just one constraint. This 162

allowed us to handle also an example, namely III.3.4, with structure functions, which in other approaches, as for instance Refined Algebraic Quantization, is very troublesome. The examples in Part III have shown, that the Master Constraint Programme is a promising method, so that hopefully it can also be applied to Loop Quantum Gravity. There one has to deal with infintely many constraints with structure functions. However the Master Constraint summerizes all these constraints into one constraint, which then has to be analysed. This will be nevertheless very complicated and it may happen that one has to develop some approximation scheme. In the end the question is, wether the physical Hilbert space supports enough semi– classical states and whether these lead to the correct classical limit. The construction of the physical Hilbert space is in principle possible without explicit knowledge of Dirac observables. However if these are available they facilitate the construction very much. If one has succeeded with the constuction, quantum Dirac observables are given by the self–adjoint operators on this Hilbert space. But in order to check the classical limit one needs operators on this Hilbert space which can be interpreted in classical terms. These are usually obtained by quantizing the classical Dirac observables. Classical Dirac observables can be provided by using the methods of Part II. One advantage of the concept of complete observables developed there is, that complete observables have an immediate physical interpretation. Moreover in the case of general relativity we succeeded in expressing space–time diffeomorphism invariant quantities as Dirac observables in the canonical formalism. The question also here is, whether it will be possible to obtain explicit and exact expressions for the complete observables. One could start by considering approximations. One kind of approximation can be obtained by using the power series (II.5.41) for complete observables. Since it is a power series in (τk − Tk ) the convergence should be good at least in a small region around the phase space points x with τk = Tk (x). If one omits higher powers of (τk − Tk ) the resulting phase space functions will be in general not gauge invariant anymore. However one can try to quantize these on the kinematical Hilbert state and apply these to semi–classical states peaked around phase space points lying on the constraint hypersurface and satisfying Tk (x) = τk . Note that because these states should be peaked around Tk = τk they are necessarily gauge variant, i.e. the quantum constraint equations are not satisfied. One has to carefully balance and control the errors which result from the gauge variance of the state and from omitting the higher order terms in the power series. The better the peakedness property at Tk = τk the smaller is the error from omitting higher order terms but the larger is the error resulting from the gauge variance of the state. Furthermore one has to check how the result is influenced by the fact that one is working with the kinematical inner product and not with the physical one. Of course, if an explicit expression, for instance via the power series, for a complete observable is known, one can try to quantize it on the kinematical Hilbert space. If this is possible it will also have a representation on the physical Hilbert space, as defined in the Master Constraint Programme. An interesting question is to characterize complete observables directly on the quantum level. For systems with one constraint it is suggested in the first two references in [11] to consider the conditions (i) (ii)

ˆ =0 [Fˆ[f ;T ] (τ ), C] F[f ;T ] (τ → Tˆ) = fˆ .

The second condition is a quantum analogue of the boundary condition (II.4.2) for complete observables. The symbols τ → Tˆ mean that one has to replace the classical parameter τ with the 163

operator Tˆ. The resulting operator (if it can be defined) does not commute with the constraint anymore. Obviously these conditions make only sense on the kinematical Hilbert space. For systems with more constraints it is in general necessary to consider also weak observables, one would therefore only require that the analogues of these conditions for systems with more constraints hold on the subspace of wave functions which satisfy the quantum constraint equations. An immediate problem is then that in general these are not elements of the kinematical Hilbert space. Another question is, whether these conditions define the complete quantum observable uniquely. Already on the classical level this is only given, if the equation T = τ defines a good gauge in the sense of section II.8. On the quantum level there arises additional the problem that the second condition suffers from factor ordering ambiguities [70]. For the characterization of quantum complete observables on the physical Hilbert space ˆ τ of the physical Hamiltonians, defined in section II.9. These one could use quantizations H j physical Hamiltonians are also complete observables, so the following is a consistency check. Note also, that the physical Hamiltonians depend on the choice of momenta conjugated to the clock variables. This choice will determine the set of remaining canonical coordinates {Qk , Pk }p−n k=1 . For these it is possible to generate τ –transformations by applying the physical Hamiltonians. A quantum analogue would be the Heisenberg equation of motion i!

∂ ˆ ˆ jτ ] F (τ ) = [Fˆ[Xk ;Ti ] (τ ), H ∂τj [Xk ;Ti ]

where Xk = Qk or Xk = Pk . These equations would implement a dynamical evolution of the observables on the physical Hilbert space. But before we can obtain such a quantum dynamics for the case of general relativity we have to solve many deep but very interesting problems.

164

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Danksagung Diese Arbeit w¨ are ohne die Hilfe von so vielen Leuten nicht m¨ oglich gewesen und zu meiner Schande werde ich wohl nicht alle aufz¨ ahlen k¨ onnen. Als Erstes m¨ ochte ich mich bei meinem Betreuer Thomas Thiemann bedanken, vor allem f¨ ur seinen Optimismus, die unz¨ ahligen Hilfestellungen und die oft stundenlangen Diskussionen u ¨ber Physik und den Rest der Welt. Dann m¨ ochte ich mich bei den Mitarbeitern der Institutionen bedanken, die diese Arbeit erm¨ oglicht haben, n¨ amlich das Max–Planck–Instituts f¨ ur Gravitationsphysik in Golm, das Perimeter Institute for Theoretical Physics in Waterloo und die Universit¨ at Potsdam. Auch verdanke ich dem Max–Planck–Institut und dem Perimeter Institute Reisemittel, die mir erlaubten meine Arbeit auch an anderen Orten vorzustellen und zu diskutieren. Dank geht daher auch an Arundhati Dasgupta und Viqar Husain an der University of New Brunswick, Hanno Sahlmann an der PennState University, Jorma Louko in Nottingham, Max Niedermaier in Tours und Renate Loll an der Universiteit Utrecht f¨ ur viel Gastfreundlichkeit und f¨ ur viele Diskussionen. Das Perimeter Institute hat die Ankunft in Kanada sehr erleichtert, besonders durch die Organisation von Karen Wiatowski und die gute Seele des Instituts, Rita Schwander. Durch das Promotionsstipendium der Studienstiftung des deutschen Volkes war es mir m¨ oglich, problemlos den Aufenthaltsort nach Kanada und wieder zur¨ uck nach Deutschland zu wechseln. Daf¨ ur vielen Dank! Verschiedenen Personen an der Universit¨ at Postdam danke ich f¨ ur die Unterst¨ utzung seit dem Beginn meines Studiums, besonders Prof. Klein, Prof. Wilkens und Prof. Gerhard–Multhaupt. F¨ ur die Bereitschaft diese Arbeit zu begutachten, m¨ ochte ich mich bei Prof. Wilkens, Thomas Thiemann, Martin Bojowald und Renate Loll bedanken. F¨ ur viele Diskussionen, Unternehmungen und Fahrdienste danke ich Dorothea Bahns, Benjamin Bahr, Niklas Beisert, Yves Bodenthin, Martin Bojowald, Johannes Br¨ odel, Johannes Brunnemann, Florian Conrady, Olaf Dreyer, Thomas Fischbacher, Stefan Fredenhagen, Kristina Giesel, Bruno Hartmann, Christian Hillmann, Thomas Klose, Tomasz Konopka, Antje Leisner, Fotini Markopoulou, Markus P¨ ossel, Thomas Quella, Hanno Sahlmann, Aureliano Skirzewski, Thomas Thiemann und Oliver Winkler. F¨ ur das Korrekturlesen und viele hilfreiche Verbesserungsvorschl¨ age geht großer Dank an Benjamin Bahr, Johannes Brunnemann, Christian Hillmann und Thomas Thiemann. Ganz besonders m¨ ochte ich mich bei meinem Bruder Stephan bedanken, den ich allzu oft als meinen Sekret¨ ar und pers¨ onlichen System–Administrator missbrauchen musste. Herzlichsten Dank an Antje, Yves und Aron, f¨ ur die großartige Doktoranden–WG und f¨ ur vieles vieles mehr.

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