Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories

TL;DR

This paper develops a new kinematical framework for quantum field theories that include fermions and Higgs fields, extending quantum gravity models to incorporate matter fields with a non-Fock-like representation.

Contribution

It introduces a novel representation for matter fields in diffeomorphism invariant quantum theories, utilizing Grassman-valued half-densities to solve fermionic adjointness relations.

Findings

Constructed a non-Fock-like Hilbert space for matter and gravity.

Successfully promoted classical reality conditions to quantum adjointness relations.

Established a unique gauge and diffeomorphism invariant probability measure.

Abstract

We extend the recently developed kinematical framework for diffeomorphism invariant theories of connections for compact gauge groups to the case of a diffeomorphism invariant quantum field theory which includes besides connections also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity. The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one : the elementary excitations of the connection are along open or closed strings while those of the fermions or Higgs fields are at the end points of the string. Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which in turn uniquely fixes the gauge and diffeomorphism invariant probability measure that underlies the Hilbert space. Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tec\'otl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations.