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Classical and Quantum Gravity

Conditional probabilities and entropy in (minisuperspace) quantum cosmology

Henry E. Kandrup

Published Jun 1, 1988• 15 citations
Publisher

Abstract

This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q 0 , . . ., q N , one is instructed to pick one of the q A , say q 0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q 1 , . . ., q N ) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i , lambda i ; a) in some O a -representation as conditional probabilities p i for measuring the eigenvalues lambda i of some operator O a , one is led to consider an observable dependent entropy S 0 (a)=- Sigma i p i log p i . And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' f w and then use that f w to define a 'Wigner function entropy' S w .

DOI
10.1088/0264-9381/5/6/011

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This paper exploits an analogy with a time-independent Schrodinger equation to propose a concrete interpretation of a 'wavefunction of the universe' psi which solves the Wheeler-DeWitt equation for a minisuperspace quantum cosmology, and then uses this interpretation to define information theoretic notions of entropy. Given a wavefunction psi depending, in a configuration space representation, on some set of variables q 0 , . . ., q N , one is instructed to pick one of the q A , say q 0 identical to a, as a 'time' and then view psi (Q; a) as an unnormalised conditional probability amplitude for finding the values Q identical to (q 1 , . . ., q N ) at 'time' a. By identifying the 'natural' measure dQ on an a=constant hypersurface in minisuperspace, rho (Q, Q'; a)dQ varies as psi *(Q; a) psi (Q'; a)dQ is interpreted as representing a density matrix in the Q-representation; and, by viewing the Q as abstract operators for fixed a, rho is interpreted as an abstract density matrix. This interpretation admits no a priori notion of unitary evolution with respect to a 'translation' da, unitarity being viewed instead as a derived concept arising only in a semiclassical limit. The object then is to use rho to define reasonable measures of 'coarse-grained entropy' in accord with physical intuition which could serve as a diagnostic for an 'arrow of time'. Thus, e.g., by interpreting the diagonal components rho ( lambda i , lambda i ; a) in some O a -representation as conditional probabilities p i for measuring the eigenvalues lambda i of some operator O a , one is led to consider an observable dependent entropy S 0 (a)=- Sigma i p i log p i . And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' f w and then use that f w to define a 'Wigner function entropy' S w .
The quasi-Heisenberg picture of minisuperspace model is considered. The The quasi-Heisenberg picture of minisuperspace model is considered. The suggested scheme consists in quantizing of the equation of motion and interprets all observables including the Universe scale factor as the time-dependent (quasi-Heisenbeg) operators acting in the space of solutions of the Wheeler--DeWitt equation. The Klein-Gordon normalization of the wave function and corresponding to it quantization rules for the equation of motion allow a time-evolution of the mean values of operators even under constraint H=0 on the physical states of Universe. Besides, the constraint H=0 appears as the relation connecting initial values of the quasi-Heisenbeg operators at t=0t=0t=0. A stage of the inflation is considered numerically in the framework of the Wigner--Weyl phase-space formalism. For an inflationary model of the ``chaotic inflation'' type it is found that a dispersion of the Universe scale factor grows during inflation, and thus, does not vanish at the inflation end. It was found also, that the ``by hand'' introduced dependence of the cosmological constant from the scale factor in the model with a massless scalar field leads to the decrease of dispersion of the Universe scale factor. The measurement and interpretation problems arising in the framework of our approach are considered, as well.
In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in nonrelativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the SSS matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit, and are closely related to an intersection number operator. The definitions of class operators given here handle the key case in which the underlying classical system has multiple crossings of the boundaries of the regions of interest. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the well-defined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology.
Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the SSS-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.
The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.
The unsatisfactory status of the interpretation of the wave function of the Universe in canonical quantum gravity is reviewed. The ``naive interpretation'' obtained by straightforwardly applying the standard interpretive rules to the canonical quantization of general relativity is manifestly unacceptable; the ``WKB interpretation'' has only a limited domain of applicability; and the ``conditional probability interpretation'' requires one to pick out a ``preferred time variable'' (or preferred class of such variables) from among the dynamical variables. Evidence against the possibility of using a dynamical variable to play the role of ``time'' in the conditional probability interpretation is provided by the fact (proven here) that in ordinary Schr\"odinger quantum mechanics for a system with a Hamiltonian bounded from below, no dynamical variable can correlate monotonically with the Schr\"odinger time parameter t, and thus the role of t in the interpretation of Schr\"odinger quantum mechanics cannot be replaced by that of a dynamical variable. We also argue that the interpretive problems of quantum gravity are not alleviated by the incorporation of observers into the theory. Faced with these difficulties, we seek a formulation of canonical quantum gravity in which an appropriate nondynamical time parameter is present. By analogy with a parametrized form of ordinary Schr\"odinger quantum mechanics, we make a proposal for such a formulation. A specific proposal considered in detail yields a theory which corresponds at the classical level to general relativity with an arbitrary, unspecified cosmological constant. In minisuperspace models, this proposal yields a quantum theory with satisfactory interpretive properties, although it is unlikely that this theory will admit sufficiently many observables for general spacetimes. Nevertheless, we feel that the approach suggested here is worthy of further investigation.
Standard quantum mechanics describes the evolution of physical systems through unitary Schr¨odinger dynamics, while decoherence and wavefunction collapse are introduced through additional postulates or environmental interactions. Although highly successful operationally, the theory does not provide a fundamental dynamical mechanism explaining the emergence of classical localization from microscopic evolution. n this work we propose an entropy-weighted causal dynamics framework, referred to as Quantum Causal Entropy (QCE), in which each microscopic causal transition contributes both a phase increment and an entropy cost. Starting from a local causal update rule, Ψ → eiθℓ−∆SℓΨ, we derive a modified evolution equation in which the standard Hamiltonian generator is supplemented by an entropy-production operator. The resulting entropy-corrected Schr¨odinger equation takes the form iℏ dΨ dt =H − iℏλSˆΨ, where Sˆ represents the entropy rate associated with microscopic causal processes and λ is an entropy coupling constant. We show that the standard unitary Schr¨odinger evolution is recovered in the limit of vanishing entropy coupling, while finite entropy production leads naturally to decoherence, interference suppression, and effective measurement localization without invoking an explicit collapse postulate. A corresponding density-matrix evolution equation is derived, providing a thermodynamic description of coherence decay. The framework therefore provides a dynamical link between quantum evolution, information entropy, and emergent classicality, and suggests experimentally testable corrections to coherence behavior in controlled quantum systems