Gauge invariance of the wave functional in mixed momentum/coordinate representations

TL;DR

This paper investigates which operator representations in gauge theories ensure gauge-invariant wave functionals after quantum constraints are applied, revealing that mixed momentum/coordinate representations are necessary for gauge invariance.

Contribution

It demonstrates that pure coordinate representations do not yield gauge-invariant wave functionals in gauge theories, and that mixed representations are required, using geometric quantization.

Findings

Pure coordinate representations fail to produce gauge-invariant states.

Mixed momentum/coordinate representations achieve gauge invariance.

Results apply to linear spin-two theory and General Relativity.

Abstract

Starting from the observation that in Yang-Mills theory the Schroedinger state functional in the momentum representation is not gauge invariant, we investigate the reversed question: Which are the representations for the operators of a gauge theory that lead to an invariant wave functional once the quantum constraints have been imposed upon it? Stated otherwise: Which representation do we have to use if we wish the constraints of the theory to eliminate the non-physical degrees of freedom from the states? We use the framework of geometric quantization to attack this question. In particular, it is found that in the linear spin-two theory as well as in General Relativity, gauge invariance cannot be achieved by a pure coordinate (i.e., field) representation, but that one has to use mixed momentum/coordinate representations instead. Our results are illustrated by the example of the free relativistic point-particle as well as by simple cosmological mini-superspace models in the framework of General Relativity.