Quantization of Diffeomorphism-Invariant Theories with Fermions
John C. Baez, Kirill V. Krasnov
TL;DR
The paper develops a loop-quantization-inspired framework for diffeomorphism-invariant gauge theories coupled to fermions by constructing a quantum configuration space $\overline{\cal A} \times \overline{\cal F}$ and a kinematical Hilbert space $L^2(\overline{\cal A} \times \overline{\cal F})$. It then builds a gauge-invariant basis of fermionic spin networks for $L^2((\overline{\cal A} \times \overline{\cal F})/{\cal G})$ and introduces fermionic path observables represented as operators on this space, including Berezin-derivative based momentum operators with a normal-ordering prescription. A diffeomorphism-invariant Hilbert space ${\cal H}_{\rm diff}$ is obtained via the ALMMT group-averaging procedure, yielding a robust kinematical and gauge-invariant scaffold for quantization. The work focuses on representing holonomy- and fermion-based observables and discusses reality conditions and potential superselection sectors, establishing a foundation for incorporating dynamics (the Hamiltonian constraint) in these theories. Overall, the paper extends loop-quantization concepts to diffeomorphism-invariant theories with fermions, enabling gauge- and diffeomorphism-invariant states and observables suitable for future dynamical analysis.
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.