Quantum Geometrodynamics of the Bianchi-Ix Model in Extended Phase Space

Abstract

A way of constructing mathematically correct quantum geometrodynamics of a closed universe is presented. The resulting theory appears to be gauge-noninvariant and thus consistent with the observation conditions of a closed universe, by that being considerably distinguished from the traditional Wheeler -DeWitt one. For the Bianchi-IX cosmological model it is shown that a normalizable wave function of the Universe depends on time, allows the standard probability interpretation and satisfies a gauge-noninvariant dynamical Schrödinger equation. The Wheeler -DeWitt quantum geometrodynamics is represented by a singular, BRST-invariant solution to the Schrödinger equation having no property of normalizability.

The role of BRST charge as a generator of gauge transformations in quantization of gauge theories and Gravity

“…In the approach to Hamiltonian dynamics proposed in [6,7] H is the Hamiltonian in extended phase space. Thanks to the differential form of gauge condition in (2.2), the Hamiltonian (2.5) can be obtained by the usual rule H = πṄ + pȧ +Pθ +˙θ P − L which is applicable to unconstrained systems.…”

The Exact Cosmological Solution to the Dynamical Equations for the Bianchi-Ix Model

“…However, in quantum cosmology appropriate mathematical operations are just formal and the question arises, whether these operations are mathematically and physically justified. This question has been explored in our paper 1 . Our main result consisted in the demonstration that in a closed universe without asymptotical states BRST-invariance is not equivalent to gauge invariance.…”

“…In the previous paper 1 we studied quantum geometrodynamics (QGD) of the Bianchi-IX model in the framework of extended phase space (EPS) approach elaborated by Batalin, Fradkin and Vilkovisky (BFV) 2−6 . It is generally accepted that in the BFV approach one singles out a BRST-invariant sector which is supposed to coincide with a gauge-invariant one.…”

Dirac prescription from BRST symmetry in FRW space-time

“…In [13] the BV approach has been adopted to develop a proper BRST invariant Lagrangian for a Bianchi IX model in the presence of a differential gauge condition. Thanks to the presence of time derivatives of the lapse function in the total Lagrangian, the Hamiltonian in extended phase space is free from constraints and it can be defined simply by a Legendre transformation.…”

“…In this work, starting from the Hamiltonian formulation developed in [13], we outline how to find out the proper BRST charge. In a theory as GR, one deals with Lagrangian containing second order derivatives, which can be avoided by perfoming some partial integrations and discarding boundary contributions.…”