Mar 18, 2014 - dimension of a Lie algebra is its dimension as a vector space over F. The ...... [60] Barret OlNeill, Elementos de GeometrÃa Diferencial, Editorial ...
Information theory and semi-quantum MaxEnt C. M. Sarris and A. Plastino March 18, 2014
List of Contents
Preface
2
1. Introductory notions concerning Information Theory
4
2. Statistical mechanics and information theory
6
3. Time dependence and MaxEnt Formalism
9
4. Some notions concerning chaos theory
12
5. A brief digression on Lie algebras
14
6. Semi-quantum systems
16
7. A semi-quantum formalism
19
8. Semi-quantum inherits properties from quantum dynamics
25
9. Geometrical structure of the quantum manifold
33
10. Choosing Lie algebras yields di¤erent invariants
45
11. Consequences of the algebra’s closure
64
12. The SU (2) case
66
13. The SU(1,1) case
71
14. Regular and irregular semi-quantum dynamics
77
15. Semi-quantum dynamics and Lyapunov exponents
85
Epilogue
93
Bibliography
94
1
PREFACE
Semiquantum Hamiltonians represent mixed physical systems for which one part is quantal and the other classical. These parts are coupled in nonlinear fashion. The coupling term contains both quantum and classical degrees of freedom. Physical systems that are appropriately described in this way are are to be found in molecular and solid state physics. Of course, so described is the interaction between a classical measurement’s instrument and the quantum system being measured. Semiquantum systems have a technological counterpart as well, in connection with nano-technology (molecular transistors, quantum dots and SQUiDS, for instance). These special systems can be successfully modelled by quantum billiards, dynamic system in which a particle alternates between motion in a straight line and elastic re‡ections from a boundary. The interest focuses in vibrating quantum billiards, in which a particle of mass m is subjected to elastic interactions con…ned in a potential well V (q), generated by a classical mass M . This book studies in detail the dynamics of time-independent semiquantum systems for which the quantum component is endowed with a Lie symmetry. The pertinent formalism will be developed by appeal to the celebrated Principle of Maximum Entropy (MaxEnt). The focus of attention will be the motion invariants that emerge on account of the physical symmetries. These invariants are of immense help in adequately describing our systems and provide the initial conditions needed for the solution of the concomitant dynamic equations. The invariants’values determine what kind of dynamics prevails. The classical part of our Hamiltonians is of the general form p + V (q); 2m with the nonlinearity arising from the coupling between the classical position ^ of the system to variable q and some of the relevant quantum operators O ^ be discussed. The nonlinear coupling adopts the the appearance q O. Discussing dynamics under the MaxEnt lens has important bene…ts. MaxEnt is a dual approach, for which mean values and their associated Lagrange multipliers have an equal status. This permits studying the time-evolution in terms of each of them. 2
It allows the uncertainty principle to enter the game as an important constraint on the system’s evolution. The range in which the di¤erent invariants can take values further constrains the time-evolution. As a consequence, the selection of adequate initial conditions turns out to be the main theoretical concern, which greatly simpli…es the analysis.
3
Chapter 1 Introductory notions concerning information theory Information theory (IT) treats information as data communication [1], with the primary goal of concocting e¢ cient manners of encoding and transferring data. IT is a branch of applied mathematics and electrical engineering, involving the quanti…cation of information, developed by Claude E. Shannon [2] in order to i) …nd fundamental limits on signal processing operations such as compressing data and ii) …nding ways of reliably storing and communicating data. Since its 1948-inception it has considerably enlarged its scope and found applications in many areas that include statistical inference, natural language processing, cryptography, and networks other than communication networks. A key information-measure (IM) was originally called (by Shannon) entropy, in principle unrelated to thermodynamic entropy. It is usually expressed by the average number of bits needed to store or communicate one symbol in a message and quanti…es the uncertainty involved in predicting the value of a random variable. Thus, a degree of knowledge (or ignorance) is associated to any normalized probability distribution p(i); (i = 1; : : : ; N ), determined by a functional I[fpi g] of the fpi g [3, 2, 4, 5] which is precisely Shannon’s entropy. IT was la axiomatized in 1950 by Kinchin [6], on the basis of four axioms, namely [1], I is a function ONLY of the p(i), I is an absolute maximum for the uniform probability distribution, I is not modi…ed if an N + 1 event of probability zero is added, 4
Composition law. As for the last axiom, consider two sub-systems [ 1 ; fp1 (i)g] and [ 2 ; fp2 (j)g] of a composite system [ ; fp(i; j)g] with p(i; j) = p1 (i) p2 (j). Assume further that the conditional probability distribution (PD) Q(jji) of realizing the event j in system 2 for a …xed i event in system 1. To this PD one associates the information measure I[Q]. Clearly, p(i; j) = p1 (i) Q(jji): Then Kinchin’s fourth axiom states that X I(p) = I(p1 ) + p1 (i) I Q(jji) :
(1.1)
(1.2)
i
An important consequence is that, out of the four Kinchin axioms one …nds that Shannons’s measure S=
N X
p(i) ln [p(i)];
i=1
gives us the only way of complying with Kinchin’s axioms.
5
(1.3)
Chapter 2 Statistical mechanics and information theory It has been argued [7] that the statistical mechanics (SM) of Gibbs is a juxtaposition of subjective, probabilistic ideas on the one hand and objective, mechanical ideas on the other. From the mechanical viewpoint, the vocables “statistical mechanics” suggest that for solving physical problems we ought to acknowledge a degree of uncertainty as to the experimental conditions [1]. Turning this problem around, it also appears that the purely statistical arguments are incapable of yielding any physical insight unless some mechanical information is a priori assumed [7]. This is the conceptual origin of the link SM-IT pioneered by Jaynes in 1957 via his Maximum Entropy Principle (MaxEnt) [3, 4, 8] which allowed for reformulating SM in information terms. Since IT’s central concept is that of information measure (IM) Descartes’scienti…c methodology considers that truth is established via the agreement between two independent instances that can neither suborn nor bribe each other: analysis (purely mental) and experiment [9]. The analytic part invokes mathematical tools and concepts: Mathematics’world , Laboratory. The mathematical realm is called Plato’s Topos Uranus (TP) [1]. Science in general, and physics in particular, may thus be seen as a [TP , “Experiment”] two-way bridge. TP concepts are related to each other in the form of “laws” that adequately describe the relationships obtaining among suitable chosen variables that describe the phenomenon at hand. In many cases these laws are integrated into a comprehensive theory (e.g., classical electromagnetism, based upon Maxwell’s equations) [1, 10, 11, 12, 13, 14, 15]. Jaynes’ MaxEnt ideas describe thermodynamics via the link [IT as a part 6
of TP], [Thermal experiment] [1], or in a more general scenario: [IT] , [Phenomenon at hand]. It is clear that the relation between an information measure and entropy is [IM] , [Entropy S]. One can then assert that an IM is not necessarily an entropy, since the …rst belongs to the Topos Uranus and the later to the laboratory. Of course, in some special cases an association IM , entropy S can be established [1]. Such association is both useful and proper in very many situations [3]. If, in a given scenario, N distinct outcomes (i = 1; : : : ; N ) are possible, three alternatives are to be considered [4]: 1. Zero ignorance: predict with certainty the actual outcome. 2. Maximum ignorance: Nothing can be said in advance. The N outcomes are equally likely. 3. Partial ignorance: we are given the probability distribution fPi g; i = 1; : : : ; N . If our state of knowledge is appropriately represented by a set of, say, M expectation values, then the “best”, least unbiased probability distribution is the one that [4] re‡ects just what we know, without “inventing” unavailable pieces of knowledge [3, 4] and, additionally, maximizes ignorance: the truth, all the truth, nothing but the truth [4]. Such is the MaxEnt rationale. In using MaxEnt, one is not maximizing a physical entropy, but only maximizing ignorance in order to obtain the least biased distribution compatible with the a priori knowledge. Statistical mechanics and thereby thermodynamics can be formulated on an information theory basis if the density operator ^ is obtained by appealing to Jaynes’maximum entropy principle (MaxEnt), that can be stated as follows: Assume that your prior knowledge about the system is given by the values of M expectation values < A1 >; : : : ; < AM >. In such circumstances ^ is uniquely determined by extremalizing I(^) subject to M constraints given, namely, the M conditions < Aj >= T r[^ A^j ], a procedure that entails introducing M Lagrange multipliers i . Additionally, since normalization of ^ 7
is necessary, a normalization Lagrange multiplier should be invoked. The procedure immediately leads one [4] to realizing that I S, the equilibrium Boltzmann’s entropy, if the a priori knowledge < A1 >; : : : ; < AM > refers only to extensive quantities. Of course, I, once determined, a¤ords for complete thermodynamical information for the system of interest [4].
8
Chapter 3 Time dependence and MaxEnt formalism The description of the quantum state of a system is made by means of the statistical or density operator ^ [16, 17, 18] and, the entropy S associated to the state is de…ned as [16, 18] S (^) =
T r (^ ln ^) =
hln ^i :
(3.1)
According to Jayne’s Information Theory (TI) [19, 20], the statistical operator ^ is constructed starting from the knowledge of the expectation values ^ j termed as the constraints of N + 1 operators O D E ^j ; ^ j = T r ^O j = 0; 1; :::; N (3.2) O
where the subindex 0 refers to the normalization condition
T r (^) = 1 (3.3) ^ 0 = I^ must to be included in order to given that the identity operator O ful…ll condition (3.3). As it was established by Alhassid & Levine [18], the constraints must be linearly independent but not necessarily commuting ones. The maximal entropy statistical operator is given by [18] ! N X ^ ^ ; ^ = exp (3.4) 0I j Oj j=1
9
which is expressed in terms of N + 1 Lagrange multipliers 0 ; 1 ; :::; N . 0 is the one associated to the identity operator I^ which must be included into the relevant set in order to ful…ll the normalization condition (3.3). So, 0 is obtained as ( " !#) N X ^ : (3.5) 0 = ln T r exp j Oj j=1
The normalized statistical operator of maximal entropy given by Eq. (3.4) enables one to obtain the entropy S (^) at the maximum as (replacing Eq. (3.4) into Eq. (3.1)) [18] S (^) =
0
+
N X
j
j=1
D
E ^ Oj :
(3.6)
The statistical operator ^; its surprisal, ln ^, as well as any analytical function f of ^ follow the same equation of motion [18] i 1 h^ @f (^) = H(t); f (^) ; @t i~
(3.7)
^ where H(t) is the Hamiltonian of the system which may or may not depend on time explicitly. Alhassid & Levine in ref. [18] prescribed a procedure to specify the statistical operator ^(t) of maximum entropy for any time. Departing from an initial state of maximum entropy at t = t0 , given by Eq. (3.4), they determined that, in order Eq. (3.4) be valid for all time (i.e. in order Eq. (3.4) be an exact solution of Eq. (3.7)), there must exist a set of relevant operators (termed as the constraints) that ful…ll the well-known closure condition [18] h
N i X ^ ^ ^r ; H(t); Oj = i~ grj (t)O
j = 1; :::; N
(3.8)
r=0
“so that the equation of motion of the density operator has thus been converted to a set of coupled equations of motion for the Lagrange parameters. The number of coupled equations equals the number of constraints”[18].
10
@ j X = gjr (t) r ; @t r=0 N
j = 1; :::; N ;
(3.9)
“the boundary conditions of the equation of motion are determined by the requirement that the initial state ^(t0 ) be the state of maximum entropy (3.4) subject to the constraints” [18] ( see Eqs. (3.2)). As a consequence of the fact that the statistical operator obeys Eq. (3.7), the entropy (3.6) is a constant of the motion, i.e. (3.10)
S(t0 ) = S(t)
for any two times t; t0 . In ref. [21], Eq. (3.10) has been used to derive the time evolution of the expectation values of the constraints generated by Eq. (3.8), so as to obtain D E ^j N D E @ O X ^r ; = grj (t) O j = 1; :::; N ; (3.11) @t r=0
Eq. (3.11) is known as the generalized Eherenfest theorem. Finally, one can obtain the mean values of the relevant operators for all times as [18] D
E n o ^ j (t) = T r ^(t)O ^j = O
11
@ @
0 j
;
j = 1; :::; N:
(3.12)
Chapter 4 Some notions concerning chaos theory Chaos theory is a mathematical discipline used in several scienti…c areas including meteorology, physics, engineering, economics, biology, etc. Chaos theory analyzes the behavior of dynamic systems that are highly sensitive to initial conditions (the so-called butter‡y e¤ect). Tiny di¤erences in initial conditions (such as those caused by to rounding errors in numerical computation) yield widely diverging outcomes for such dynamic systems, which makes long-term prediction almost impossible. This happens even though the concomitant systems are deterministic, entailing that their future behavior is fully determined by their initial conditions, with no random factors involved in any way. The deterministic nature of these systems does not make them predictable, though. Such behavior is known as deterministic chaos, or simply chaos. Chaotic behavior can be observed in many natural systems, such as weather. Clues for understanding such behavior are sought by exploring chaotic mathematical models, or using analytical techniques such as recurrence plots and Poincaré maps. Although there is no universally accepted mathematical de…nition of chaos, a frequently employed de…nition states that, for a dynamic system to be typi…ed as chaotic, it should exhibit have the following features it must be sensitive to initial conditions (SIC); it must be topologically mixing; and its periodic orbits must be dense. 12
The demand for sensitive dependence on initial conditions entails that there exists a set of initial conditions of positive measure which do not converge to a cycle of any length. SIC means that each point in such a system is as closely as we may want approximated by other points with signi…cantly di¤erent future trajectories. Consequently, an arbitrarily small disturbance in the current trajectory may lead to signi…cantly di¤erent long-term behavior. It has been shown that the last two properties in the list above actually imply sensitivity to initial conditions. If attention is restricted to intervals, the second property implies the other two. The most signi…cant condition in practice, namely, that of SIC, is redundant in the de…nition, being implied by two (or for intervals, one) purely topological conditions, which are therefore of greater interest to mathematicians. For physicists, in general, SIC is a very important feature for applications. Is is measured in several ways, Lyapunov coe¢ cients being perhaps the most important one.
13
Chapter 5 A brief digression on Lie algebras Lie algebras are algebraic structures which were introduced to study the concept of in…nitesimal transformation. The term Lie algebra (after Sophus Lie) was introduced in the 1930s. In older literature, the name “in…nitesimal group” is employed instead. A Lie algebra is a vector space g over some …eld F , like that of the real numbers, together with a binary operation [ ; ] : g g ! g called the Lie bracket, which veri…es the following postulates: Bilinearity: [ax + by; z] = a[x; z] + b[y; z]; [z; ax + by] = a[z; x] + b[z; y] for all scalars a, b in F and all elements x; y; z 2 g. Alternating on g : [x; x] = 0 for all x 2 g. The Jacobi identity: [x; [y; z]] + [z; [x; y]] + [y; [z; x]] = 0 x; y; zing.
for all
Note that the bilinearity and alternating properties imply anticommutativity, i.e., [x; y] = [y; x], for all elements x; y 2 g. If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of SU(n) is written as su(n). A collection of elements of a Lie algebra g are said to be generators of the Lie algebra if the smallest subalgebra of g containing them is g itself. The dimension of a Lie algebra is its dimension as a vector space over F . The cardinality of a minimal generating set is always less than or equal to its dimension. 14
Let us have N given generators and consider a special operator H. We speak of a partial Lie algebra under commutation with H if the commutator of H with any of the generators can be expressed as a linear superposition of these generators .
15
Chapter 6 Semi-quantum systems There exist many situations in which a semiquantum description of a system has been attempted [22, 23, 24]. M. A. Porter [22] made an exhaustive compilation of physical systems for which this kind of description is relevant. One may highlight vibrating quantum billiards as a useful abstraction of the ensuing semiquantum dynamics [25]. A billiard is a dynamic system in which a particle alternates between motion in a straight line and elastic re‡ections from a boundary. As the particle hits the boundary, it is re‡ected from it without losing speed. Billiard systems are Hamiltonian idealizations of the actual game of billiards. However, the region contained by the boundary can have variegated shapes, not only rectangular, and can even be multidimensional. Billiards may also be studied for non-Euclidean geometries. The study of billiards which are kept out of a region, rather than being kept within a region, is known as outer billiard analysis. The motion of the particle in the billiard is a straight line, with constant energy, between re‡ections with the boundary (a geodesic if the Riemannian metric of the billiard table is not ‡at). All re‡ections are elastic: the angle of incidence before the collision is equal to the angle of re‡ection after the collision. The sequence of re‡ections is described by the billiard map that completely characterizes the motion of the particle. Billiards capture all the complexity of Hamiltonian systems, from integrability to chaotic motion, without the di¢ culties of integrating the equations of motion to determine the concomitant Poincaré map. It is known that a billiard system with an elliptic table is integrable. The quantum version of the billiards is readily studied in several ways. The classical Hamiltonian for the billiards is now replaced by the stationary-state of an appropriate Schrödinger equation. Quantum billiards provide a good scenario 16
for the analysis of quantum chaos. One considers quantum billiards with time-varying surfaces, which yield a signi…cant example of quantum chaos that does not require the high quantum-number limits. Porter [25] analyzed vibrating quantum billiards within the framework of Riemannian geometry and derived a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations imply that these conditions are also su¢ cient. Porter proved this theorem for one degree-of-freedom boundary vibrations and brie‡y discussed a generalization to billiards with two or more degrees-of-vibrations. It was encountered that the requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and that they emerge from orthogonality relations satis…ed by solutions of such equation. A second theorem was also demonstrated in that paper that yields a general form for the coupled ordinary di¤erential equations that describe quantum billiards with one degree-of-vibration boundaries. Such set of equations may be used to illustrate KAM theory and also constitutes a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nano-structures. Indeed, many semiquantum Hamiltonians are found in the literature [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. In ref. [26], L. E. Ballentine de…nes a semi-quantum system as: “one composed by a quantum part coupled to a classical part. The essential structure of all these models is a classical part acting directly on the quantum part, with the quantum part reacting back on the classical part through the expectation value of some observable”[26]. He points out that “we refer to a system as semiquantum if one part is treated classically and the other part quantum mechanically”[26]. Thus, a semi-quantum system may be represented by a Hamiltonian of the form [23, 37, 35] ^ =H ^ q + Hcl + Hint ; H
17
(6.1)
^ q and Hcl stand for the quantum and classical parts of the system, where H respectively, and Hint is an interaction term in which a quantum and a classical degree of freedom are coupled. Such Hint term causes the system (6.1) to be non-linear. We are particularly interested in semi-quantum systems in which the quantum degrees of freedom are given by a set of, say, N relevant operators that closes a partial Lie algebra under commutation with the Hamiltonian (6.1). The classical degrees of freedom are the 2n canonical conjugate variables qi and pi . Accordingly, Eq. (6.1) may be re-written in the fashion [35] ^ = H
n XX j
^j + aj (qi ; pi )O
i=1
n XX k
Fk (qi ; pi );
(6.2)
i=1
^ q and Hint terms, the classical variables where the …rst term includes the H (qi ; pi ) are contained in the coe¢ cients aj (qi ; pi ), and the second term is a purely classical one. With the help of the Maximum Entropy Principle (MaxEnt) one is always able to develop a semiquantum formalism to deal with semi-quantum nonlinear Hamiltonians of the type (6.2), in which a set of relevant operators is involved to ful…ll the closure condition (3.8). This formalism was developed in refs. [23, 37, 35] and, in the following, we are going to outline it.
18
Chapter 7 A semi-quantum formalism Consider a system that possesses, both, quantum and classic degrees of freedom, with a coupling amongst them, that we call semiquantum [22, 23, 26, 38, 39, 40]. The associated Hamiltonian is of the general form (6.2) [35]. As the classical degrees of freedom act like as if they were simple parameters, the Hamiltonian (6.2) may be put into the form ! n n XX XX ^ = ^j + ^ H aj (qi ; pi )O Fk (qi ; pi ) I; (7.1) j
i=1
k
i=1
with I^ the identity operator. Now, we can generalize the prescription given by Alhassid & Levine in ref. [18] for the semi-quantum case. The idea is to …nd the su¢ cient number of required constraints of the kind (3.2), such that the statistical operator of maximum entropy (3.4) be valid for any time t. We want to stress the fact that, as the identity operator I^ does commute with any other operator of the set, the classical term of Eq. (7.1) “disappears”in the quantum commutation operation (3.8) and then we still retain the the maximum entropy statistical operators’ form given by Eq. (3.4) even in the semi-quantum case. Consequently, introducing the surprisal expression ln ^(t) =
N X
^
r (t)Or
(7.2)
r=0
into the equation of motion (3.7) (and, taking into account that the time dependence of ^(t) is contained only in the Lagrange parameters: @[ln@t^(t)] =
19
N P
r=0
@ r ^ Or ), @t
we obviously obtain the same Alhassid & Levine’s expression i~
N X @ j=0
j
@t
^j = O
N X
j
j=0
h
i ^ ^ H(t); Oj ;
(7.3)
^ where H(t) is the non-linear semi-quantum Hamiltonian given by Eq. (7.1), which may or may not depend upon time explicitly. Eq. (7.2) is an exact solution of Eq. (3.7) if the set of constraints ful…ll the semi-quantum closure condition [35] h
N X n i X ^ ^ ^r ; H(t); Oj = i~ grj (qi ; pi )O
(7.4)
r=0 i=1
the grj (qi ; pi ) are the coe¢ cients of a N N matrix G(qi ; pi ), whose nature is semi-quantum, given that the aj (qi ; pi ) terms in Eq. (7.1) may contain the classical degrees of freedom qi ; pi . In the semi-quantum closure condition (7.4) we detect the …rst di¤erence between the full quantum case of ref. [18] and the semi-quantum scenario. However, as it was done in ref. [18], we still can replace Eq. (7.4) into Eq. (7.3) and, considering that the relevant operators generated by Eq. (7.4) are linearly independent, we obtain [35] @ r XX = grj (qi ; pi ) j ; @t j=0 i=0 N
n
r = 1; :::; N ;
(7.5)
Thus, for the semi-quantum case, the equation of motion for the density operator (3.7) also has been converted to a set of coupled equations (7.5). There exist a second di¤erence with respect to the full quantum case of ref. [18]: while the set of Eqs. (3.9) is a set of coupled linear equations, the set of Eqs. (7.5) is a set of non-linear coupled equations due to the fact that coe¢ cients grj (qi ; pi ) may contain the classical degrees of freedom qi ; pi : Now, for the mean values of the quantum observable we obtain D E ^j n N X D E @ O X ^ r (t); = grj (qi ; pi ) O j = 1; :::; N ; (7.6) @t r=0 i=1 20
The normalization condition (3.3) enables us to obtain the 0 Lagrange parameter in similar fashion as one does it in the full quantum case (see Eq. (3.5)). 0 de…nes the di¤erentiable manifold 0
=
0
[ 1 (t); :::;
(7.7)
N (t)] ;
where the 0 s obey the non-linear equations of motion (7.5) Accordingly, the dynamic evolution of the system itself gives rise to this “hypersurface” 0 : Thus, for the semi-quantum case we have obtained a density operator ^(t) of maximum entropy of the kind (3.4), paying the price of having obtained a set of non-linear coupled equations of motion for the Lagrange parameters. However, as we were able to close the algebra (see Eq. (7.4)) we still can obtain obtain the mean values of the quantum degrees of freedom in the fashion D E n o ^ j (t) = T r ^(t)O ^j ; O j = 1; :::; N (7.8) and this means we have integrated the quantum degrees of freedom of the non-linear semi-quantum system given by Eq. (7.1). In fact, Eq. (7.8) is a primitive of the equation of motion (7.6). Thus, if we take the time derivative of Eq. (7.8), we obtain D E ^j d O dt
=
nh io d ^j = T r T r ^(t)O dt
i d h ^j ; ^(t)O dt
j = 1; :::; N
(7.9)
As in the Schrödinger representation the quantum operators do not depend on time explicitly, all the time dependence is contained in the MaxEnt density operator ^(t) through the time dependence of the Lagrange parameters ^ @O (Accordingly, @tj = 0). Then, from Eq. (7.9), we obtain (see also Eq. (3.7)) D E ^j d O dt
= Tr
d^(t) ^ Oj dt
= Tr
i 1 h^ ^j ; H(t); ^(t) O i~
j = 1; :::; N
(7.10) Let us now take into account the invariance of the trace under commutation
21
D E ^j d O
1 h^ ^ i Oj ; H(t) ^(t) = i~ h i 1 ^ j ; H(t) ^ ^(t) O = = Tr i~ h i 1 ^ ^ = Tr ^(t) H(t); Oj ; i~ = Tr
dt
j = 1; :::; N:
(7.11)
so that, …nally, taking into account the semi-quantum closure condition (7.4), Eq. (7.11) may be cast as D
^j d O dt
E
=
Tr
(
^(t)
N X n X
=
r=0 i=1 N X n X
=
r=0 i=1
N X n X
^r grj (qi ; pi )O
r=0 i=1
)
=
n o ^r = grj (qi ; pi )T r ^(t)O D E ^ r (t); grj (qi ; pi ) O
j = 1; :::; N
(7.12)
which is the generalized Ehrenfest theorem given by Eq. (7.6). In short, if we are able to close a semi Lie algebra under commutation with the semi-quantum non-linear Hamiltonian (7.1), then we are also able to integrate the equations of motion of the quantum degrees of freedom, even ^ j as they do. Eqs. though they exhibit a non-linearity of the kind aj (qi ; pi )O (7.8) also may be obtained as D
E ^ j (t) = O
@ @
0
;
(7.13)
j = 1; :::; N:
j
Eqs. (7.13) tells us that nD the set by the E conformed D E o mean values of the quan^ ^ tum degrees of freedom O1 (t); :::; ON (t) de…nes the tangent space of the 0 manifold at each instant t of the evolution, and at each “point” [ 1 (t); :::; N (t)] of the manifold T
0
= span
D
E ^ j (t) = O
@ @ 22
0 j
;
j = 1; :::; N
:
(7.14)
The MaxEnt density operator may be used to calculate the mean value of the Hamiltonian (7.1) n n D E D E XX XX ^ = T r ^H ^ = ^j + H aj (qi ; pi ) O Fk (qi ; pi ): j
i=1
k
(7.15)
i=1
The entropy at the maximum acquires the form S (^) =
T r f^(t) ln ^(t)g =
0
+
N X j=1
j (t)
D
E ^ j (t) O
(7.16)
and is a constant of the motion also for the semi-quantum dynamics. The equations of motion for the mean values of the quantum degrees of freedom may be obtained in a similar way as it in the full quantum case [18, 21, 35], that is to say, if we take into account that the mean values of the relevant operator must ful…ll [41] D E ^j i d O 1 h^ ^j ; = H(t); O (7.17) dt i~ and replacing Eq. (7.4) into (7.17), we …nd [35] D E ^j N X n D E d O X ^ r ; j = 1; :::; N: = grj (qi ; pi ) O (7.18) dt r=1 i=1 Eq. (7.18) shows that the time evolution of the mean values of the quantum degrees of freedom will be a¤ected by the classical ones. On the other hand, the set of coupled equations is non-linear. Concerning the system’s classical degrees of freedom, the energy is taken to coincide with the quantum expectation value of the semi-quantum Hamiltonian [23, 37, 24, 31, 32] given by Eq. (7.15) and the temporal evolution of the classical variables are [37, 35] D E ^ @ H dqi = ; i = 1; :::; n (7.19) dt @pi D E ^ @ H dpi = ; i = 1; :::; n (7.20) dt @qi 23
Although Eqs. (7.19) and (7.20) generate the dynamic evolution of the classical variables, its nature is semiquantum as well, because it involves the mean values of the quantum variables. Accordingly, the interplay between classical an quantal variables via i) the matrix G(qi ; pi ) and D E acquires D E special D relevance E ^ ^ ^ ii) the energy H = H qi ; pi ; Oj , since they are the keys to visualize the entanglement between the quantum and classical degrees of freedom of a semi-quantum system of the type (7.1). Summing up, we see that the semi-quantum dynamics of the system (7.1) develops in a semi-quantum phase space D E D E ^ ^ N (t); q1 ; :::; qn ; p1 ; :::; pn g; VSq = spanf O1 (t); :::; O
whose dimension is N + 2n. This is so because the N quantum variables are linearly independent and so are too the 2n classical ones. Accordingly, in this semi-quantum phase space, the quantum D E meanDvalues E span the quantum ^ ^ manifold of the system QM = spanf O1 (t); :::; ON (t)g, whose dimension is N while the classical variables span the classical manifold of the system CM = spanfq1 ; :::; qn ; p1 ; :::; pn g whose dimension is 2n; so that one has VSq = QM CM [35]. The classical manifold is a symplectic one, while the quantum manifold has its own geometrical structure, that we will explore below.
24
Chapter 8 Semi-quantumness inherits properties from quantum dynamics There exists many features in semi-quantum dynamics that are direct consequences of the semi-quantum closure condition (7.4) concerning semi-quantum systems of the type (7.1). The particular form of this Hamiltonians ! n n XX XX ^ = ^j + ^ H aj (qi ; pi )O Fk (qi ; pi ) I; (8.1) j
i=1
k
i=1
enables us to treat them in the same fashion as we deal with purely quantum systems concerning the treatment the quantum degrees of freedom’s dynamics. In this sense, we can say that semi-quantum dynamics (SQD) shares many characteristics with quantum dynamics (QD) and inherits from the latter it many features which are applied straightforwardly. The key issue is that the semi-quantum system (8.1) must ful…ll the semi-quantum closure condition (7.4). For the sake of clarity, we are going to enumerate most of full quantum dynamics’characteristics, properties, and tools that are also valid in the semiquantum scenario, as we will be seen in the following Chapters. As the statistical operator ^(t) and its surprisal ln ^(t) =
N P
^
r (t)Or
r=0
obey the Liouville equation (3.7), it is possible to derive general dynamic invariants that hold for any positive integer n [42, 43] 25
I
(n)
=
*
N X
^
r (t)Or
r=0
!n +
*
N X
^
r (t)Or
r=0
+!n
;
(8.2)
^ r ’s belong to the set of relevant operators generated by the where the O closure condition (3.8), and the r (t)’s are their associated Lagrange multipliers. For the particular case n = 2, the former invariant (8.2) adopts the form
I
(2)
=
N X N X
i (t) j (t)Kij (t)
(8.3)
i=1 j=1
in terms of the so-called correlation coe¢ cients [44] Kij (t) =
1 2
h
^i; O ^j O
i
(t) +
D
E D E ^ j (t): ^ i (t) O O
(8.4)
The invariant (8.3) is called the centered second order invariant [42, 43]. It is possible to demonstrate that Eq. (8.3) represents a positive definite quadratic form, this means that the dynamic invariant is always positive and that its value may be …xed by the initial conditions imposed on the system through the equations of motion (3.9) and (3.11) [45]. In order to prove the positive de…niteness of the quadratic form (8.3), it is necessary and su¢ cient to demonstrate that the correlation matrix K(t) (whose elements are the correlation coe¢ cients de…ned by Eq.(8.4)) is the matrix associated to an inner product [45]. Because of the positive de…niteness of the quadratic form (8.3), D Eit is ^ 1 (t); possible to de…ne an inner product in the space V = spanf O D E ^ N (t)g generated by the mean values of the relevant operators :::; O [45]: : V V ! R=
26
D
^i O
E D
h i 1 ^ ^ T r ^(t) Oi ; Oj = 2 + D E 1 ^ ^ ^j O ^i Kij (t) = Oi Oj + O 2 (t) ^j O
E
^ i T r ^(t)O ^j T r ^(t)O D
^i O
E
(t)
D
^j O
E
= (t)
@2 0 (8.5) : @ i@ j
(the can …ndDa detailed demonstration of the equality D interested reader E D E E 2 1 ^iO ^j + O ^j O ^i ^i ^j O O O = @@ i @ 0 j in ref. [46]). 2 (t)
(t)
(t)
As a consequence of the closure condition and of the existence of the metric (8.5), it is possible to recover the generalized uncertainty principle [47, 48, 49] as the summation over the principle minors of order 2 belonging to the correlation matrix. In fact, for any pair of noncommuting operators belonging to the relevant set and generated by the closure condition (3.8), the uncertainty relation always holds [44]
^j O
2
^k O
2
E 1D^ ^ ^k O ^j Oj Ok + O 2
D
^j O
ED E ^k O
Dh
2
^i; O ^j O
iE2
:
(8.6) The left hand side of Eq. (8.6) is a principal minor of order 2 be2 ^j = longing to the correlation matrix K(t) (the metric), where O D E D E2 ^ j . Keeping in mind that through Eq. (3.8) we are able to ^2 O O j …nd a …nite set of non-commuting operators, we can de…ne the following expression, which is obtained as the summation over the principal minors of order 2 belonging to the metric matrix K(t) [47, 48, 49]
IH =
N X N X j=1 k=1 j 0; N
(8.10)
where i are the quadratic form’s positive eigenvalues (or equivalently, the correlation matrix’s positive eigenvalues). Eq. (8.10) also may be expressed as ~ 2 (t) 1 I (2) 1
+
~ 2 (t) 2 I (2)
+ ::: +
2
~ 2 (t) N I (2)
= 1:
(8.11)
N
Eq. (8.11) de…nes an “hypersurface”, a N 1 ellipsoid surface because the quadratic form’s eigenvalues i are all positive. Accordingly, the invariant I (2) represents the invariance of the squared norm ku(t)k2 . The evolution of the system is restrained to take place on the ellipsoid’s surface. In fact, as the system evolves, the Lagrange multipliers and the mean values of the relevant set evolve according to the equations 28
of motion (3.9) and (3.11), but the squared norm remains always the same. There exists a particular circumstance in which the generalized uncertainty principle (8.7) remains as a constant of the motion: when the matrix G(t) (whose coe¢ cients grj (t) are de…ned through the closure condition (3.8)) is an antisymmetric one [47]. With the help of Eqs. (3.8) and (3.11) we obtain the following equations of motion D
^2 d O j dt
D E ^ ij d L dt
D
E
= D
N X r=0
E
=
2
N X r=0
D E ^ ir grj (t) L E
D
E ^ gri (t) Lri ; N X r=0
D
(8.12)
D E ^ jr ; gri (t) L
(8.13)
ED E ^ ij = 1 O ^iO ^j + O ^j O ^i ^i O ^ j . Now, if we take time with L O 2 derivative in the left hand side of Eq. (8.7), with the help of Eqs. (3.11), (8.12), and (8.13), we see that the generalized uncertainty principle (8.7) becomes an invariant of the motion if the matrix G(t) is antisymmetric (the reader can …nd the proof in ref. [47]). Due to the algebra’s closure D Econdition, D one E derives s complete set of non^ ^ commuting operators f O1 (t); :::; ON (t)g, termed the relevant set. The full quantum Hamiltonians for which the closure condition (3.8) holds and whose dynamics is governed by an antisymmetric matrix G(t), also exhibits other invariants. We are going to enumerate these invariants and, since the invariance proof is similar for all of them, we will prove it only for the …rst: 1) I=
N X N D X
^ i2 O
i=1 j=1 i
