Unitary Representations of the Translational Group Acting as Local Diffeomorphisms of Space-Time

TL;DR

This paper introduces a new mathematical framework for constructing diffeomorphism-invariant quantum states in field theories like general relativity, using fibre bundles and group cohomology to achieve unitary representations of local space-time diffeomorphisms.

Contribution

It develops a novel approach linking group cohomology and fibre algebra to build invariant quantum states under local diffeomorphisms, advancing quantum gravity quantization methods.

Findings

Constructed quantum states invariant under local diffeomorphisms.

Linked group cohomology of the translational group to quantum state invariance.

Provided a mathematical foundation for diffeomorphism-invariant quantum theories.

Abstract

We develop a new mathematical approach to diffeomorphism invariant quantum states for the quantisation of general field theories such as general relativity and modified gravity. Treating quantum fields as fibre bundles, we discuss operators acting on the fibre algebra that defines a Hilbert space. The algebras of two types of operators are considered in detail, namely the observables as generic physical variables and the quantum operators suitable for describing symmetries and transformations. We then introduce generalised quantum states of these operators and examine their properties. By establishing a link between the commutativity and group cohomology of the translational group as a subgroup of the Poincare group, we show that this leads to the construction of quantum states invariant under the action of the translational group, as the local gauge group of diffeomorphisms, with unitary representations.