Remarks on the issue of time and complex numbers in canonical quantum gravity

TL;DR

This paper explores how factor ordering ambiguities in the Wheeler-DeWitt equation introduce complex terms affecting gauge phases and the emergence of semiclassical time in quantum gravity.

Contribution

It demonstrates that gauge phases cannot be fully eliminated due to factor ordering choices, leading to complex terms in the Wheeler-DeWitt equation and implications for semiclassical time derivation.

Findings

Complex solutions to the Wheeler-DeWitt equation are obtainable.

Factor ordering ambiguity influences gauge phase removal.

Semiclassical time can be derived from complex WDW solutions.

Abstract

We develop the idea that, as a result of the arbitrariness of the factor ordering in Wheeler-DeWitt equation, gauge phases can not, in general, being completely removed from the wave functional in quantum gravity. The latter may be conveniently described by means of a remnant complex term in WDW equation depending of the factor ordering. Taking this equation for granted we can obtain WKB complex solutions and, therefore, we should be able to derive a semiclassical time parameter for the Schroedinger equation corresponding to matter fields in a given classical curved space.