Phase space formulation of quantum mechanics and quantum ...

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Phase space formulation of quantum mechanics and quantum cosmology Przemyslaw Malkiewicz,1, ∗ Artur Miroszewski,1, † and Herv´e Bergeron2, ‡ 1

National Centre for Nuclear Research, 00-681 Warszawa, Poland 2 ISMO, UMR 8214 CNRS, Univ Paris-Sud, France (Dated: December 14, 2017)

arXiv:1712.04813v1 [gr-qc] 13 Dec 2017

We develop a phase space formulation of quantum mechanics based on unitary irreducible representations of classical phase spaces. We use a quantum action functional to derive the basic equations. In principle, our formulation is equivalent to the Hilbert space formulation. However, the former allows for consistent truncations to reduced phase spaces in which approximate quantum motions can be derived. We predict that our approach can be very useful in the domain of quantum cosmology and therefore, we use the cosmological phase spaces to establish the basic equations of the new formalism. I.

INTRODUCTION

In this article we propose a certain trajectory approach to quantum mechanics and we develop it with an emphasis on its application to simple quantum cosmological systems. The essence of our approach is to reformulate the Schr¨ odinger equation densely defined on vectors in the Hilbert space in terms of a set of Hamilton’s equations in a phase space. Quantum states are represented by phase space points. The phase space variables encode both classical and nonclassical observables. By classical observables in this context we mean the expectation values of basic operators in a given quantum state. By nonclassical observables we mean all other phase space variables which encode such properties of quantum states as the dispersions of basic observables and so on. Our formalism builds on the notion of coherent states and in particular on the idea of semiclassical framework by J. Klauder [1, 2]. In fact, our formalism is a natural extension of the Klauder framework which is included as a special case. The basic tools behind our formalism are, first, the variational formulation of the quantum dynamics once a quantum Hamiltonian has been provided and, second, coherent states that are constructed with a unitary and irreducible representation of a minimal canonical group in the phase space of the respective classical model. The coherent states are used to incorporate the classical observables into our formalism in a natural (more precisely, covariant) way and, as it is in the original framework, the equations of motion of classical observables, in general, include terms with nonvanishing ~. Nevertheless, our extension of the framework by nonclassical observables brings in some new advantages that are not present in the original framework. In principle, our approach is based on an infinitedimensional phase space. However, its main advantage is that it includes consistent truncations to finitedimensional phase spaces. Specifically, the dimensionality can be as large as to reproduce the exact Schr¨odinger ∗ † ‡

[email protected] [email protected] [email protected]

equation in the form of phase space trajectories in the case of infinitely many dimensions, or as small as to give the roughest quantum corrections to the classical equations of motion in the case of the classical phase space dimensionality. The latter case corresponds, in fact, to the Klauder framework. Our approach can be adjusted to any specific quantum system under consideration and, in particular, used to deal with those quantum dynamics which are too complex to be solved explicitly. Since we are mainly interested in quantum cosmological systems, we will develop our ideas for the case of the classical phase space which is a half-plane rather than a plane. For this particular phase space, a well-suited minimal canonical group is the affine group. Nevertheless, we wish to emphasize that our approach can be easily developed for the case of Weyl-Heisenberg (W-H) and other groups. The great advantage of our approach is that it allows for addressing physical questions which are not possible in the simplest semiclassical framework. For example, as a classical phase space is now extended by nonclassical observables one can investigate in detail the energy transfers between classical and nonclassical observables. It is worth mentioning that that the very existence of such a phenomenon could be mistakenly interpreted in terms of energy transfers from or to hidden spatial dimensions in the universe, which are introduced by the brane-world theories and alike [3, 4]. In fact, the energy carried by a wavefunction naturally spreads into nonclassical observables when a system leaves the classical regime and enters the quantum one. This is in particular expected of cosmological systems when they approach the big-bang singularity. As it was already shown in [5], in this case a quantum repulsive potential may halt the contraction and bring the universe to expansion, that is, prevent the universe form collapsing into the singularity by making it bounce. But then, since the full quantum dynamics is not expected to be simply reverted in time, the dynamics near the bounce is not expected to be fully symmetric in time either. How this exactly happens can be captured within our extended approach by inspecting the evolution of nonclassical observables. Other associated questions could include: How the classical universe emerges from a quantum state? Had been the universe

2 classical before the bounce? And, what is a good measure of classicality? To finish these introductory remarks, let us notice that our formulation offers an alternative way of looking at quantum mechanics, it is a kind of the hidden-variable theory. We believe that our approach can be a useful tool to those who investigate hidden-variable theories or are simply interested in fundamental principles of quantum mechanics. The outline of the article is as follows. In Sec II we recall the semiclassical framework of Klauder. In Sec III we discuss the variational formulation of quantum dynamics. In Sec IV we develop our formalism. We apply it to two examples in Sec V. In Sec VI we revisit the quantum flat Friedmann model with our approach. We conclude in Sec VII. The appendices deal with some technicalities (self-adjointness and the numerical code) which have been omitted from the main text. II.

COHERENT STATES AND KLAUDER’S FRAMEWORK Coherent states

By coherent states (see e.g. [6]) we mean a continuous mapping from a set of labels, denoted by ~l and equipped with a measure d~l, into unit vectors in Hilbert space, ~l 7→ |~li ∈ H, such that it resolves the identity, Z d~l |~lih~l| = IH .

(1)

(2)

This basic property was used by Klauder [7] in his definition of what he calls an overcomplete family of states (OFS). They provide a bridge between the abstract quantum formalism and the continuous label-space that can be further identified with classical observables. Suppose there exists a unitary irreducible representation on H, U (X ), of a minimal group of canonical transformations in a phase space X of a physical system. Then, the mapping X 3 ξ 7→ |ξi = U (ξ)|ψ0 i ∈ H,

(3)

defines a family of coherent states whose labels describe the classical states of that system [8]. The normalized vector |ψ0 i is called the fiducial vector.

ˆ is the with respect to the normalized |ψi ∈ H, where H quantum Hamiltonian that corresponds to a certain classical Hamiltonian, H. The stationary points of the action (4) are found to satisfy the Schr¨odinger equation, ˆ i∂t |ψi = H|ψi.

The idea of the semiclassical framework based on the coherent states was introduced by Klauder in [1]. Initially, he applied it to the case of the phase space X = R2 and the Weyl-Heisenberg coherent states. The W-H coherent states, |x, pi, are defined as follows |x, pi := D(x, p)|ψ0 i, ˆ

ˆ

1

0

0

D(x0 , p0 ) ◦ D(x, p) = e 2 (xp −px ) D(x + x0 , p + p0 ),

(7)

ˆ Pˆ ) are the position and momentum opand where (Q, erators [9]. The fiducial vector |ψ0 i ∈ H is fixed for all (x, p) ∈ R2 and its choice is almost arbitrary. The only condition that one often imposes is the so called physical centering which relates the classical observables and the expectation values by demanding ˆ pi = x, hq, x|Pˆ |x, pi = p, hx, p|Q|x,

(8)

ˆ and Pˆ are the position and momentum operator, where Q respectively. We find that ihx, p|d|x, pi = pdx,

(9)

where d is the exterior derivative. This result can be guessed (up to the irrelevant total derivative) from the fact that the above one-form must be invariant with respect to the action of the W-H group. The same reasoning applies to all other canonical groups. Now the quantum action functional (4) can be evaluated on the family of coherent states, Z tf ˆ pi S(x, p) = dthx, p|i∂t − H|x, ti tf

Z =

dt (xp ˙ − H s (x, p)) ,

(10)

ti

ˆ pi. Thus, the variation of the where H s = hx, p|H|x, quantum action with respect to the (x, p)-labelled coherent states yields the Hamilton equations for x and p, x˙ =

ti

(6)

where the displacement operator D(x, p) = eixP +ipQ satisfies

Framework

First, let us recall that the quantum dynamics can be obtained via variation of the quantum action, Z tf ˙ = ˆ S(ψ, ψ) dthψ|i∂t − H|ψi, (4)

(5)

∂H s ∂H s , p˙ = − . ∂p ∂x

(11)

On the one hand, the above equation provides an approximation to the exact quantum motion, R 3 t 7→ |x(t), p(t)i ∈ H,

(12)

3 and on the other hand, it establishes a very appealing interpretation of the classical observables and their dynamics within the more fundamental quantum framework. We notice that these equations, in general, differ from the classical equations as the semiclassical Hamiltonian H s may include ~-corrections, Hs = H + O(~).

ˆ = x and Pˆ = 1 ∂x . This is equivalent to where Q i ˆ 0 i = 1, hψ0 |Pˆ |ψ0 i = 0. hψ0 |Q|ψ One finds that ˆ 0i ihq, p|d|q, pi = −qdp + hψ0 |D|ψ

(13)

Since in the real world ~ never vanishes, the possibility of modeling the dynamics of the classical observables with nonvanishing ~ could be very useful. Indeed, this possibility becomes particularly important for improving the dynamics of classically singular cosmological models as we show below.

(21)

dq , q

(22)

ˆ = 1 (x∂x + ∂x x) is the dilation operator and where D 2i which confirms the general statement given below Eq. ˆ the quan(9). Now, provided a quantum Hamiltonian H, tum action functional evaluated on the affine coherent states reads Z tf S(q, p) = dt (qp ˙ − H s (q, p)) , (23) ti

Affine group

As we are concerned with gravitational systems, we shall turn to an important example of the phase space that appears in cosmology, namely the half-plane X = R+ × R. The basic observables form a canonical pair, (q, p) ∈ R+ × R,

(14)

where q is the volume of the universe and p is a rate of its expansion. Clearly, the W-H group is not applicable to the present case as one of the canonical variables, q, is confined to the half-line. Instead, we shall employ the affine group [2, 10, 11], Af , that is defined by the multiplication law, p (q 0 , p0 ) ◦ (q, p) = (q 0 q, 0 + p0 ), (15) q and preserves the symplectic structure of the half-plane phase space, p (q 0 , p0 ) ◦ [dq ∧ dp] = d(q 0 q) ∧ d( 0 + p0 ) = dqdp, (16) q 0

0

where (q , p ) is a fixed element of the affine group. There exists a unique (up to sign) unitary irreducible representation of Af , which in H = L2 (R+ , dx) takes the form   eipx x U (q, p)ψ(x) = √ ψ . (17) q q Thus, we define the affine coherent states as |q, pi := U (q, p)|ψ0 i,

(18)

where hx|ψ0 i ∈ L2 (R+ , dx) is the fiducial vector that is subject to the constraint Z dx |ψ0 |2 < 0, (19) x R+ (which follows from the group integrability condition). To tighten the connection between quantum and classical observables we demand ˆ pi = q, hq, p|Pˆ |q, pi = p, hq, p|Q|q,

(20)

ˆ pi. Hence, the variation of the where H s = hq, p|H|q, quantum action with respect to the classical labels (q, p) yields the Hamilton equations, q˙ =

∂H s ∂H s , p˙ = − , ∂p ∂q

(24)

for the stationary trajectories. Free particle dynamics on q > 0

In this article we are going to study a quantum free motion of a particle on the half-line, q > 0. It is a very important example as it formally describes the dynamics of the flat Friedmann universe with a perfect fluid-source [5]. The big-bang singularity is represented by the endpoint, q = 0. The variable q describes the volume and the variable p describes the expansion of the universe (see Sec VI). Let the classical system be defined as follows, H = p2 , ω = dqdp, (q, p) ∈ R+ × R,

(25)

and the quantum Hamiltonian read ˆ = − 4x . H

(26)

We discuss the technical issue of extending the above symmetric operator to a self-adjoint one in Appendix A. For a fiducial vector ψ0 (x) ∈ L2 (R+ , dx), we obtain Z tf S= dt (−qp ˙ − H s (q, p)) , (27) ti

where H s = p2 + ~2

K , q2

(28)

R where K = R+ |ψ00 |2 dx. The respective Hamilton equations (24) include an ~2 -correction that resolves the singularity at q = 0. The particle is repelled away from the singularity by the quantum potential ~2 qK2 and this produces a bounce in its dynamics. See Fig. 1.

4 Its variation with respect to ψ such that ψ, ψ,x , ψ,xx ∈ L2 (R+ , dx) ∩ C ∞ (R+ ) at each t, yields Z tf Z δS = dt (iψ,t δ ψ¯ − iψ¯,t δψ) dx ti tf

Z +

R+

Z

"Z

¯ dx iψδψ

+

(ψ,xx δ ψ¯ + ψ¯,xx δψ) dx

dt ti #tf

R+

Z

tf

+

R+

ti

ti

  ¯ ,x − ψ¯,x δψ ∞ . (30) dt ψδψ 0

Provided that the variations vanish at the endpoints, δψ(ti ) = 0 = δψ(tf ), the stationary points of the quan˙ satisfy the Schr¨odinger equation, tum action S(ψ, ψ) (i∂t + 4) ψ(x, t) = 0.

(31)

We conclude that for each ψ(x) there exists a unique stationary trajectory in the Hilbert space ψ(x, t) such that ψ(x, ti ) = ψ(x). Variation of the reduced quantum action

FIG. 1: Classical and semiclassical dynamics of a free particle on the half-line. As the semiclassical particle approaches the singular state q = 0, it is repelled by the potential, which results in a bounce. We set ~2 K = 2.

Now, suppose we confine the quantum action functional to trajectories in a subspace Γ ⊂ L2 (R+ , dx)∩ C ∞ (R+ ) that is parametrized by real parameters. More precisely, we assume a differentiable map Rn 3 {λi } 7→ ψΓ ∈ Γ. We will consider the reduced action Z tf S(ψΓ , ψ˙ Γ ) = dthψΓ |i∂t + 4|ψΓ i.

(32)

(33)

ti

III.

Its variation yields Z tf Z δS = dt

THEORY OF THE RESTRICTED QUANTUM ACTION

The idea of this work is to extend the phase space description of quantum mechanics due to J. Klauder to a much broader framework that we shall call the trajectory approach to quantum dynamics. For this purpose it is useful to discuss the quantum action formulation of quantum dynamics in somewhat more detail.

ti

Z

tf

+

Z dt

ti

(iψΓ,t δ ψ¯Γ − iψ¯Γ,t δψΓ ) dx

R+

(ψΓ,xx δ ψ¯Γ + ψ¯Γ,xx δψΓ ) dx,

Γ where δψΓ = ∂ψ ∂λi δλi and δλi (ti ) = 0 = δλi (tf ). The stationary points ψΓ (t) satisfy

hδψΓ (t)|i∂t + 4|ψΓ (t)i = 0,

The quantum action is defined on trajectories in the Hilbert space and reads

Z

tf

dthψ|i∂t + 4|ψi. ti

(35)

for any variations δλi (t)’s. In other words, the Schr¨ odinger equation (i∂t + 4) |ψΓ (t)i = 0 holds only in the tangent space to |ψΓ i, namely   ∂|ψΓ i T~λ Γ = span , (36) ∂λi ~λ

Variation of the quantum action

˙ = S(ψ, ψ)

(34)

R+

(29)

and, in general, T~λ Γ 6= Γ. Given an orthonormal basis, e1 , e2 , . . . , in the tangent space T~λ Γ, the equation of motion (35) reads X i∂t |ψΓ i = hei | − 4|ψΓ i · |ei i. (37) i

5 Suppose we gradually enlarge the subspace Γ and its tangent space T~λ Γ by increasing the number of real parameters λi . Then, the orthonormal basis is enlarged accordingly en+1 , en+2 , . . . . Thus, for a fixed |ψΓ i, its time derivative i∂t |ψΓ i becomes progressively a better and better approximation to the exact −4|ψΓ i as the series lim

n→∞

n X hei |4|ψΓ i · |ei i = 4|ψΓ i,

(38)

i=1

converges by the virtue of Parseval’s identity. Notice that the convergence is defined for each point separately.

How to confine quantum motion?

The Klauder semiclassical framework is based on a fixed family of coherent states. Each element of a given family satisfies the constraints, ˆ pi = q, hq, p|Pˆ |q, pi = p, hq, p|Q|q,

(39)

which tighten the relation between classical observables and their quantum counterparts. We may view the families of coherent states as sections of a certain fiber bundle [12]. Namely, the total space is the Hilbert space (or, its dense subspace), the base space is the space of all possible expectation values of the basic operators, and the fibers are made of state vectors that give equal expectation values, ˆ π : H 3 |ψi 7→ (hψ|Q|ψi, hψ|Pˆ |ψi) ∈ R+ × R.

(40)

The “coherent” sections are defined by fixing a fiducial vector |ψ0 i in the fibre (1, 0) and then by transporting it to all the other fibers via the unitary group action, |ψ0 i 7→ U (q, p)|ψ0 i.

FIG. 2: We illustrate the quantum dynamics that takes place in the fiber bundle. The fibers consist of state vectors with the same expectation values, q and p, of the ˆ and Pˆ . In the Klauder framework, basic operators, Q given a fiducial vector |ψ0 i, it is the UIR of the affine group U (q, p) that fixes a section in the bundle to which the quantum motion is confined. In our approach, we introduce extra parameters λi ’s to parametrize the fiducial space. As a result, the quantum motion takes place both along the sections given by U (q, p)|ψ0 i and along the fibers as the extra parameters can vary.

2. Such a framework would keep the connection between quantum states and classical observables while adding more dimensions to the phase space, which would describe purely quantum features of quantum states. The number of the extra features would be controlled by the dimensionality of the fiducial space. Moreover, as we showed above, one expects that as the fiducial space is enlarged, the accuracy of this description is increased and it converges to a fully quantum mechanics expressed in terms of trajectories in a phase space of infinite dimension.

(41)

In other words, the orbits of the group define the “coherent” sections. There are as many families of coherent states as fiducial vectors, |ψ0 i, and the particular choice of the fiducial vector fixes purely quantum characteristics of the coherent states such as dispersions of the basic observables. They are nonclassical parameters that are completely fixed by the fiducial vector and are not allowed to evolve as they normally would do. Thus, the dynamical contribution from nonclassical observables is completely neglected in the Klauder framework and the only dynamical observables are the expectation values (q, p) whose dynamics can be very rough. A way to improve this framework is to consider a fiducial space rather than a fiducial vector. It translates into confining the quantum motion to families of families of coherent states instead of a single family of coherent states. Such a framework allows the quantum motion to take place along the fibers of fixed expectation values of the basic observables. This idea is presented in Fig.

IV.

EXTENSION OF KLAUDER’S FRAMEWORK Quantum action

We will now extend the Klauder semiclassical framework based on the affine coherent states. Instead of fixing a fiducial vector |ψ0 i, we shall consider a fiducial space, |ψ0 (λj )i, which contains the vectors labelled by λj , j = 1, . . . . Hence, we obtain a family of families of affine coherent states, |q, pi~λ = U (q, p)|ψ0 (λj )i,

(42)

which are labelled by λj , j = 1, . . . . The quantum action restricted to those families of affine coherent states reads  Z tf  q˙ K S= dt −q p˙ + D − Gi λ˙ i − (p2 + 2 ) , (43) q q ti

6 where

Dynamics

1 Gi [λj ] = hψ0 (λj )| ∂λi |ψ0 (λj )i, i 1 1 D[λj ] = hψ0 (λj )|x ∂x + ∂x x|ψ0 (λj )i, 2i 2i K[λj ] = hψ0 (λj )| − 4x |ψ0 (λj )i,

(44) (45) (46)

2

where we assume that ψ0 (λj ) ∈ L (R+ , dx) ∩ L2 (R+ , dx/x) and that the conditions of normalization and for the expectation values for the basic observables hold, hψ0 (λj )|ψ0 (λj )i = 1, hψ0 (λj )|Pˆ |ψ0 (λj )i = 0, ˆ 0 (λj )i = 1. hψ0 (λj )|Q|ψ (47) We assume the fiducial space to be linear and consist of the fiducial vectors of the form: X |ψ0 (λj )i = λj |ej i, λj ∈ C, (48) such that hej |ei i = Nji . Then, 1 ¯j , Nji λ i ¯ j λi , D[λj ] = Dji λ ¯ j λi , K[λj ] = Kji λ

Gi [λj ] =

We introduce γj = λj eidj ln q and find  ¯ j λi  dj δji λ ¯ idγj d¯ γj = idλj dλj − dqd . q

ω = dqdp + idγj d¯ γj , i(dj −di ) ln q ¯ Kji λj λi = e Kji γ¯j γi =: kij γ¯j γi , i(d −d ) ln q j i ¯ j λi = e Qji λ Qji γ¯j γi =: qij γ¯j γi , i(d −d ) ln q ¯ j λi = e j i Pji λ Pji γ¯j γi =: pij γ¯j γi .

(51)

where L reads    ¯ j λi  ¯ j λi  1 Kji λ Dji λ 2 ¯ ˙ − Nji λj λi − p + . q˙ p + q i q2 (53) From the above action one derives the Hamiltonian formalism ¯ j λi Kji λ H = p2 + , (54) q2   ¯ j λi 1 Dji λ ¯j ) + dλi d(− Nji λ (55) ω = dqd p + q i

(56)

Note that the action (52) yields the symplectic structure for both the classical and nonclassical observables. We follow the Dirac procedure [13] and define the total Hamiltonian ¯ j λi + c2 Qji λ ¯ j λi + c3 Pji λ ¯ j λi , (57) HT = H + c1 Nji λ where ci ∈ R are to be determined with the use of the consistency conditions, ¯ j λi ) = {Nji λ ¯ j λi , HT } = 0, ∂t (Nji λ (58)

(61)

Thus, we may turn to a canonically equivalent formalism in which

(50)

ti

¯ j λi ) = {Qji λ ¯ j λi , HT } = 0, ∂t (Qji λ ¯ j λi ) = {Pji λ ¯ j λi , HT } = 0. ∂t (Pji λ

¯ j λi Kji λ ¯ j λi + c2 Qji λ ¯ j λi + c3 Pji λ ¯ j λj , + c1 δji λ q2 (59)   ¯ dj δji λj λi ¯j . ω = dqd p + + idλj dλ (60) q

HT = p2 +

(49)

where Nji , Dji and Kji are hermitian. The quantum action reads now (after removing total time derivatives) Z tf S= Ldt, (52)

with the quadratic constraints ¯ j λi = 1, Qji λ ¯ j λi = 1, Pji λ ¯ j λi = 0. Nji λ

Since Nji is the identity operator and Dji is a hermitian operator, they can be simultaneously diagonalized. Suppose that they are diagonal, i.e. Nji = δji , Dji = dj δji . Then,

(62) (63) (64) (65)

and HT = p2 +

kji γ¯j γi + c1 δji γ¯j γi + c2 qji γ¯j γi + c3 pji γ¯j γi . q2 (66)

Let us define [M N ]ji γ¯j γi := {Mji γ¯j γi , Nji γ¯j γi } 1 γj γi . = (Mjk Nki − Njk Mki )¯ i Now, the consistency relations yield c1 = arbitrary (phase shif t generator), c2 = − c3 = −

∂p (2p ∂qji

+

1 γj γi q 2 [pk]ji )¯

[pq]ji γ¯j γi (2p

∂qji ∂q

+

1 γj γi q 2 [qk]ji )¯

[qp]ji γ¯j γi

(67)

(68)

,

(69)

.

(70)

It follows that the normalization condition is a first-class constraint that generates a pure gauge transformation (an overall phase-shift) and thus, the coefficient c1 is arbitrary. On the other hand, the physical centering conditions are second-class and the vaules of the coefficients c2 and c3 are determined. The equations of motion take the form q˙ = 2p, (71) kji γ¯j γi kji,q γ¯j γi p˙ = 2 − − c2 qji,q γ¯j γi − c3 pji,q γ¯j γi , q3 q2 (72) kji γi γ˙ j = −i 2 − ic1 δji γi − ic2 qji γi − ic3 pji γi . (73) q

7 A basis for the fiducial space

In what follows we propose a set of orthonormal vectors |ei i, i = 0, 1, 2 . . . that diagonalize the dilation operator Dij . Observe the following unitary transformation: L2 (R+ , dx) 3 ψ(x) 7→ φ(y) = ey/2 ψ(ey ) ∈ L2 (R, dy) (74) It transforms the dilation, position and momentum operator as follows ˆ = x 1 ∂x + 1 ∂x x 7→ 1 ∂y , D 2i 2i i ˆ Q = x 7→ ey , 1 1 Pˆ = ∂x 7→ e−y/2 ∂y e−y/2 i i

(75)

|ψ0 i = λ1 |e1 i + λ2 |e2 i + λ3 |e3 i.

(76)

In this case neither the absolute values |λi | nor the respective phases are preserved during the evolution. In Fig. 3 we compare the dynamics of the classical observables and of the extra parameters between the two- and three-parameter cases.

(77)

Notice that the dilation operator is the momentum operator on y ∈ R. Let us take the harmonic oscillator eigenvectors: y2

1 e− 2 √ Hn (y) hy|ψn i = √ 2n n! 4 π

(78)

where Hn are the Hermite polynomials. If we restrict the considerations to the even eigenvectors, i.e. |en i = |ψ2n i,

(79)

we obtain 1 Nij = hei |ej i = δij , Dij = hei | ∂y |ej i = 0. i

(80)

In this case the dynamical analysis becomes very simple. Indeed, the equations of motion (71) become q˙ = 2p, ¯ j λi Kji λ p˙ = 2 , q3 Kji λi γ˙ j = −i 2 − ic1 δji λi − ic2 Qji λi − ic3 Pji λi , q

V.

¯ j λi [QK]ji λ =− 2 ¯ j λi . q [QP ]ji λ

QUANTUM DYNAMICS OF THE FRIEDMANN UNIVERSE

Let us see how one can apply the formalism developed herein to a quantum cosmological model, the quantum radiation-filled flat Friedmann universe with a bounce. For more details on the framework we refer to [5]. The metric of the classical model reads: ds2 = −N 2 dt2 + q 2 (d~x)2 ,

(87)

where N is a nonvanishing and otherwise arbitrary lapse function. The Hamiltonian constraint reads  C = N q −1 −p2 + pT ,

(82)

where T and pT are canonical variables that describe the radiation and

(83)

(84)

NUMERICAL EXAMPLES

In what follows we consider two simple examples. In the first example, we set the fiducial space to be twodimensional, |ψ0 i = λ1 |e1 i + λ2 |e2 i,

VI.

(86)

(81)

where c1 is arbitrary and ¯ j λi [P K]ji λ c2 = − 2 ¯ j λi , c3 q [P Q]ji λ

in time while the respective phases are dynamical. The classical observables q and p undergo a simple bounce as in the case of Eq. (28) to which solutions are presented in Fig. 1. This result is not surprising as there is, in fact, no extra degrees of freedom. The counting of the extra degrees of freedom gives: 4 (two complex parameters) 2 (two second-class constraints from the physical centering) -2 (a first-class constraint and the respective gauge transformation from the normalization condition) = 0. In the second example, we set the fiducial space to be three-dimensional,

(85)

where the vectors |ei i are defined below Eq. (78) and λi ∈ C. We find that the absolute values |λi | are constant

q = a, p = a2 H,

(88)

(89)

are canonical variables that describe the geometry, the scale factor a and the Hubble rate H times the scale factor squared, respectively. We solve the Hamiltonian constraint with respect to pT , set the lapse function N = q and employ the variable T as the internal clock. Then, the reduced phase space is given just by the canonical pair (q, p) ∈ R+ × R and the physical Hamiltonian reads H = p2 .

(90)

The above Hamiltonian can be promoted to the quantum Hamiltonian of Eq. (26). Then, we can use our approach to determine the quantum dynamics of the Friedmann universe in terms of a trajectory. In Fig. 4 we will plot the dynamics of the classical variables a and H and their

8

FIG. 3: We compare the cases of two and three extra complex parameters, λ1 , λ2 and λ1 , λ2 , λ3 , respectively. The two top plots show the dynamics of the extra parameters. For the two-parameter case, the extra parameters only rotate in the complex plane. For the three-parameter case, the extra parameters exhibit very rich dynamics with both rotation and contraction/expansion. The bottom plot shows the dynamics of the classical observables q and p and despite the fact that the initial conditions for these observables are the same, the two-parameter (dashed) trajectory gives a bounce at smaller values of q than the three-parameter As the initial condition we q (solid) one. q

9 set λ1 (0) = 10 , λ2 (0) = − and p(0) = −2.

1 10 ,

λ3 (0) = 0, q(0) = 10

FIG. 4: The top plot shows the bouncing evolution of the isotropic geometry in the half-plane (a, H). The two bottom plots show the evolution of the dispersions σa and σH of the scale factor and the Hubble rate, respectively. We see the first evidence that the dynamics is not symmetric in time around the bounce. As the initial condition we set the initial data from the three-parameter case of Sec V.

dispersions. Note the following relations, q ˆ 2 |q, pi − hq, p|Q|q, ˆ pi2 , σq = hq, p|Q q σp = hq, p|Pˆ 2 |q, pi − hq, p|Pˆ |q, pi2 , s p2 σa = σq , σH = 4 6 σq2 + q −4 σp2 . q

(91) (92) (93)

In this section we just present one example of the extended phase space formulation of a quantum cosmological model. In our future work [14] we investigate the quantum Friedmann model much more thoroughly.

9 VII.

CONCLUSION

ACKNOWLEDGMENTS

A. M. was supported by Narodowe Centrum Nauki with Decision No. DEC-2012/06/A/ST2/00395. In this article we present a phase space formulation of quantum mechanics. We start from the semiclassical framework introduced by J. Klauder many years ago and we extend it by inclusion of nonclassical observables that are equipped with a symplectic form. The obtained infinite-dimensional phase space trajectories are, in principle, equivalent to the exact solutions of the Schr¨odinger equation, though it is the possibility for consistent truncations to finite phase spaces that makes our approach attractive. We show that the respective Hamilton equations are not too complicated and can be successfully used for numerically integrating the dynamics. Our trajectory approach is a tool that opens many new possibilities in the studies of quantum cosmological systems. In the present article we test our approach with two simple examples. We postpone a detailed study of cosmological systems to our next papers. We believe that our approach can be helpful in establishing a definition of the classicality of cosmological systems. Given such a definition, we may be able to explain the classicality of the universe and probe the effects of the lack of classicality in the past of the universe on the present-day cosmological observables. We may learn if the universe can move back and forth between the classical and quantum phases. Finally, we could verify whether the universe could had been classical before the bounce. We investigate these and other issues in the forthcoming paper [14]. As a final remark, let us make a brief comparison of our approach to the Bohm-de Broglie (BdB) approach used in quantum cosmology [15]. According to the BdB formulation, a given solution to the Schr¨ odinger equation, the so-called pilot-wave, is a source of an extra quantum term in the classical equations of motion which determine a complete set (i.e., for arbitrary initial data) of quantum trajectories in the classical phase space. Furthermore, it is the technology rather than physics that disables to predict which particular trajectory a physical system is going to follow. Interestingly, it is claimed that the technology-induced uncertainties generically evolve into the wavefunction-induced uncertainties and thus, the two become indistinguishable. Whereas in our approach a given solution to the Schr¨ odinger equation is represented by a unique trajectory in an infinite dimensional phase space that includes both “classical” and “nonclassical” observables and a system follows a predictable trajectory. However, according to the Copenhagen interpretation, measuring devices are incompatible with the observables used in this description and necessarily lead to uncertainties. Whether our approach can stimulate new ways of thinking about the measurement is an open issue.

APPENDIX A

The symmetric operator of Eq. (26),   ∂2 ∂2 − 2 , D − 2 = Cc∞ (R+ ), ∂x ∂x

(94)

defined on smooth functions with a compact support is symmetric. There exist infinitely many self-adjoint extensions of the operator, which can be obtained by extending the domain to C ∞ (R+ ) ∩ L2 (R+ , dx) with the boundary condition, ψ(0) = µψ 0 (0),

(95)

where µ ∈ [0, ∞] labels the extensions [16]. In the article, we impose the Dirichlet condition, ψ(0) = 0 (or, µ = 0), though this particular choice has no essential consequences for the obtained framework, and other choices of µ may be easily included. To ensure that we are consistent with this choice throughout the article we must demand that any fiducial vector satisfies, ψ0 (0) = 0.

(96)

Indeed, one may verify that the eigenvectors of the dilation operator introduced in Eq. (78) satisfy the above condition. APPENDIX B

We relate the expectation values of the basic observˆ and Pˆ in coherent states |q, pi to the phase space ables Q observables q and p by demanding (20), which in the case of a many-parameter fiducial vector yields two last conditions of Eq. (56). Hence, in order for q and p to correspond to the aforementioned expectation values one needs to impose the awkward constraints (56) on the fiducial vector labels λi ∈ Cn . This problem can be overcome rather easily after one notices that in the absence of the constraints, ¯ j λi , ˆ pi = qhψ0 |Q|ψ ˆ 0 i = qQji λ hq, p|Q|q, ¯ j λi . hq, p|Pˆ |q, pi = p + hψ0 |Pˆ |ψ0 i = p + Pji λ

(97) (98)

Hence, one may ignore the constraints (56) and treat q and p as auxiliary parameters. The genuinely classical observables can be then defined as follows, ¯ j λi , ps = p + Pji λ ¯ j λi . q s = qQji λ

(99)

¯ j λi and Pji λ ¯ j λi are constant along The quantities Qji λ the motion. In the studied examples we therefore first

10 solve the dynamics for q and p and next plot the evolution of q s and ps . In terms of the geometric viewpoint that we develop at the end of Sec. III, the employment of fiducial vectors which do not satisfy the constraints

(56) is equivalent to fixing the respective family of coherent states via a state in a fiber different than (1, 0), which is clearly an admissible procedure provided that one recalculates the expectation values as shown above.

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