Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories
Thomas Thiemann
TL;DR
This work advances a diffeomorphism-invariant, non-Fock kinematical framework to include fermions and Higgs fields alongside gravity and gauge fields. By introducing Grassmann-valued half-densities and Higgs point holonomies, it solves the adjointness and reality-condition constraints that previously hindered fermionic and scalar matter in Ashtekar-type formalisms, yielding unique inner products and measures. It constructs fermionic and Higgs Hilbert spaces with diffeomorphism- and gauge-invariant structures, and integrates them with the gravitational and gauge sectors via group averaging to produce a complete gauge-dauge- and diffeomorphism-invariant kinematical setting. The resulting spin-colour-network and vertex-based formalisms provide a robust, background-free platform for coupling Standard-Model-like matter to quantum gravity within a rigorous canonical framework.
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-Tecótl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations.