Changing the Hilbert space structure as a consequence of gauge transformations in "extended phase space" version of quantum geometrodynamics

TL;DR

This paper explores how gauge transformations in quantum geometrodynamics affect the Hilbert space structure and the form of the Schrödinger equation, highlighting continuous and discrete changes due to different classes of gauge transformations.

Contribution

It analyzes the impact of various gauge transformations on the Hilbert space structure and Schrödinger equation within the path integral approach to quantum geometrodynamics.

Findings

Gauge transformations can cause smooth or discrete changes in the Schrödinger equation.

Different classes of gauge transformations affect the Hilbert space structure in distinct ways.

Time-dependent gauges allow for continuous evolution of solutions.

Abstract

In the earlier works on quantum geometrodynamics in extended phase space it has been argued that a wave function of the Universe should satisfy a Schrodinger equation. Its form, as well as a measure in Schrodinger scalar product, depends on a gauge condition (a chosen reference frame). It is known that the geometry of an appropriate Hilbert space is determined by introducing the scalar product, so the Hilbert space structure turns out to be in a large degree depending on a chosen gauge condition. In the present work we analyse this issue from the viewpoint of the path integral approach. We consider how the gauge condition changes as a result of gauge transformations. In this respect, three kinds of gauge transformations can be singled out: Firstly, there are residual gauge transformations, which do not change the gauge condition. The second kind is the transformations whose parameters can be related by homotopy. Then the change of gauge condition could be described by smoothly changing function. In particular, in this context time dependent gauges could be discussed. We also suggest that this kind of gauge transformations leads to a smooth changing of solutions to the Schrodinger equation. The third kind of the transformations includes those whose parameters belong to different homotopy classes. They are of the most interest from the viewpoint of changing the Hilbert space structure. In this case the gauge condition and the very form of the Schrodinger equation would change in discrete steps when we pass from a spacetime region with one gauge condition to another region with another gauge condition. In conclusion we discuss the relation between quantum gravity and fundamental problems of ordinary quantum mechanics.