A probabilistic interpretation of the wave function of the Universe is proposed in which ``probability'' and ``unitarity'' are inherently approximate concepts. Their validity is limited by the accuracy of the semiclassical approximation for the entire Universe. This approach (which is a logical extension of DeWitt's) defines a positive-semidefinite probability distribution and satisfies the correspondence principle with classical cosmology and with quantum mechanics. In particular, one recovers the standard probabilistic interpretation of the wave function for a small subsystem of the Universe.
A probabilistic interpretation of the wave function of the Universe is proposed in which probability'' and unitarity'' are inherently approximate concepts. Their validity is limited by the accuracy of the semiclassical approximation for the entire Universe. This approach (which is a logical extension of DeWitt's) defines a positive-semidefinite probability distribution and satisfies the correspondence principle with classical cosmology and with quantum mechanics. In particular, one recovers the standard probabilistic interpretation of the wave function for a small subsystem of the Universe.
A non-probabilistic interpretation of quantum mechanics asserts that we get a prediction only when a wave function has a peak. Taking this interpretation seriously, we discuss how to find a peak in the wave function of the universe, by using some minisuperspace models with homogeneous degrees of freedom and also a model with cosmological perturbations. Then we show how to recover our classical picture of the universe from the quantum theory, and comment on the physical meaning of the backreaction equation.
For a FRW-spacetime coupled to an arbitrary real scalar field, we endow the solution space of the associated Wheeler-DeWitt equation with a Hilbert-space structure, construct the observables, and introduce the physical wave functions of the universe that admit a genuine probabilistic interpretation. We also discuss a proposal for the formulation of the dynamics. The approach to quantum cosmology outlined in this article is based on the results obtained within the theory of pseudo-Hermitian operators. 1
We consider the problem of extracting physical predictions from the wave function of the universe in quantum cosmological models. We state the features of quantum cosmology an interpretational scheme should confront. We discuss the Everett interpretation, and extensions of it, and their application to quantum cosmology. We review the steps that are normally taken in the process of extracting predictions from solutions to the Wheeler-DeWitt equation for quantum cosmological models. Some difficulties and their possible resolution are discussed. We conclude that the usual wave function-based approach admits at best a rather heuristic interpretation, although it may in the future be justified by appeal to the decoherent histories approach.
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 .
Quantum cosmology is the quantum theory of the entire universe. Although strange at first sight, it is appropriate because (1) our world appears to be fundamentally quantum, (2) the classical description of gravity breaks down at singularities it would give at the beginning of the universe, and (3) our universe has many properties that cannot be explained by a classical description of them. A quantum state of the universe should obey certain constraints, such as the Wheeler-DeWitt equations. One approach to interpreting such a state is {\it Sensible Quantum Mechanics} (SQM), in which nothing is probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued {\it awareness operators}. These may be defined even when the quantum state itself is not normalizable, though there still seem to be problems of divergences when one has an infinite amount of inflation, producing infinitely large spatial volumes with presumably infinite measures of perceptions. Ratios of the measures for sets of perceptions (if finite) can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so there is nothing indeterministic in a specific SQM. One can do a Bayesian analysis to test between different specific SQMs. One can also make statistical predictions of what one might perceive within a specific SQM by invoking the {\it Conditional Aesthemic Principle}: among the set of all conscious perceptions, our perceptions are likely to be typical.
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.