Hamiltonian gravity with fermions

TL;DR

This paper formulates a canonical Hamiltonian approach to gravity coupled with fermions, analyzing constraints, gauge symmetries, and implications for quantization and modified gravity theories.

Contribution

It introduces a detailed canonical formulation of Einstein-Cartan gravity with fermions, including new fermionic constraints and phase-space reduction methods.

Findings

Identifies fermionic contributions to gravitational constraints.

Establishes gauge transformations equivalent to spacetime diffeomorphisms and SL(2,C).

Discusses implications for canonical quantization and modified gravity theories.

Abstract

Fermions are coupled to the Einstein-Cartan system in the canonical formulation, including the cosmological, the Barbero-Immirzi, and the non-minimal coupling constants. The resulting ten first-class constraints generate gauge transformations that are on-shell equivalent to spacetime diffeomorphisms and SL(2,C) transformations. The gravitational second-class constraints receive fermionic contributions, which can be implemented by use of Dirac brackets or by solving them directly. Furthermore, we identify new fermionic (second-class) constraints that are necessary to recover the Dirac-fermion theory by relating the momenta to the configuration variables on dynamical solutions; this fermionic phase-space reduction is accomplished by use of corresponding Dirac brackets. The theory remains well-defined off the second-class constraint surfaces with ten additional degrees of freedom - six of which are gravitational and the remaining four are fermionic. Discrete (CPT) symmetries as well as implications for canonical quantization and modified theories of gravity with fermions are discussed.