Quantization of Diffeomorphism-Invariant Theories with Fermions

TL;DR

This paper extends loop quantum gravity techniques to include fermions in diffeomorphism-invariant gauge theories, constructing a quantum configuration space, Hilbert space, and basis for gauge-invariant states.

Contribution

It introduces a novel framework for quantizing diffeomorphism-invariant theories with fermions, including explicit constructions of quantum states and operators.

Findings

Constructed the quantum configuration space as a completion of classical fields.

Developed a gauge-invariant Hilbert space with a spin network basis.

Represented holonomy and fermionic operators on the quantum space.

Abstract

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A x F, where A is the space of connections on P and F is the space of sections of F, regarded as a collection of Grassmann-valued fermionic fields. We construct the `quantum configuration space a x f as a completion of A x F. Using this we construct a Hilbert space L^2(a x f) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit `spin network basis' of the subspace L^2((a x f)/G) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L^2((a x f)/G). We also construct a Hilbert space H_diff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourao and Thiemann.