Conditional probabilities and entropy in (minisuperspace) quantum cosmology

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 q0, ., qN, one is instructed to pick one of the qA, say q0 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 (q1, ., qN) 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 Oa-representation as conditional probabilities pi for measuring the eigenvalues lambda i of some operator Oa, one is led to consider an observable dependent entropy S0(a)=- Sigma ipi log pi. And similarly, at least when the a=constant hypersurfaces are flat, one can map psi into a 'Wigner function' fw and then use that fw to define a 'Wigner function entropy' Sw.