Counter-example where cosmic time keeps its original role in quantum cosmology

Abstract

In the minisuperspace models of quantum cosmology, the absence of time in the Wheeler-DeWitt (constraint) equation, is the main point leading to the generally accepted conclusion that in the quantum cosmology there is no possibility to describe the evolution of the universe procceding in the cosmic time (the time usually used in classical cosmology). We show that in spite of the constraint, under the specific circumstances, the averaging of some of the Heisenberg equations can give nontrivial additional information about explicit time dependence of the expectation values of certain dynamical variables in quantum cosmology. This idea is realized explicitly in a higher dimensional model with a negative cosmological constant and dust as the sources of gravity. When there is an anisotropy in the evolution of the universe, the above phenomenon (i.e. explicit cosmic time dependence of certain expectation values) appears and we find the new quantum effect which consists in "quantum inflationary phase" for some dimensions and simultaneous "quantum deflationary contraction" for the remaining dimensions. The expectation value of the "volume" of the universe remains constant during this quantum "inflation-deflation" process.