Applications of a new proposal for solving the ‘‘problem of time’’ to some simple quantum cosmological models

Abstract

We apply a recent proposal for defining states and observables in quantum gravity to some simple models. First, we consider the toy model of a Klein-Gordon particle in an external potential in Minkowski spacetime and compare our proposal to the theory obtained by deparametrizing with respect to a choice of time slicing prior to quantization. We show explicitly that the dynamics defined by the deparametrization approach depends upon the choice of time slicing. On the other hand, our proposal automatically yields a well defined dynamics -manifestly independent of the choice of time slicing at intermediate times -but there is a "memory effect": After the potential is turned off, the dynamics no longer returns to the standard, free particle dynamics. Next, we apply our proposal to the closed Robertson-Walker quantum cosmology with a homogeneous massless scalar field. We choose our time variable to be the size of the universe, so the only dynamical variable is the scalar field. It is shown that the resulting theory has the expected semi-classical behavior up to the classical turning point from expansion to contraction, i.e., given a classical solution which expands for much longer than the Planck time, there is a quantum state whose dynamical evolution closely approximates this classical solution during the expanding phase. However, 1 when the "time" takes a value larger than the classical maximum, the scalar field becomes "frozen" at the value it had when it entered the classically forbidden region. The Taub model, with and without a homogeneous scalar field, also is studied, and similar behavior is found. In an Appendix, we derive the form of the Wheeler-DeWitt equation on minisuperspace for the Bianchi models by performing a proper quantum reduction of the momentum constraints; this equation differs from the usual form of the Wheeler-DeWitt equation, which is obtained by solving the momentum constraints classically, prior to quantization.

Arrow of time in a recollapsing quantum universe

“…Although it is far from clear what kind of "norm" one has to use for a wave function in quantum cosmology 1 , we regard it as appropriate to proceed in accordance with standard quantum mechanics and to exclude exponentially increasing solutions in "classically forbidden" regions. (Note 1 An interesting proposal to use the Klein-Gordon norm in this context has recently been made in [20]. In its present formulation, however, this proposal has only been applied to describe expanding universes.…”

A path-integral approach to the problem of time

“…The first condition is a replacement of unitary evolution operators. The second one is necessary because previous attempts to obtain evolution through a turning point, based on deparameterization or the proposal of [20], have led to a "freezing" of evolution after a turning point [21,22] which does not agree with classical evolution. No freezing happens in our formalism, as seen in equation (26).…”

First-quantized theory of expanding universe from field quantization in mini-superspace

“…Another good point is that we can treat the dynamics of the single-universe state, while in typical third-quantization models [19][20][21][22][23][24], we always suffer from creation of infinite number of universes [25][26][27]. However, we have to introduce a mass scale of the quantum field, which is a disadvantage of the theory.…”