Fermion coupling to loop quantum gravity: canonical formulation

TL;DR

This paper develops a canonical formulation for fermion fields coupled to loop quantum gravity, detailing the constraints, their solutions, and a covariant Hamiltonian operator that describes fermion dynamics within quantum spacetime.

Contribution

It introduces a new regularization and quantization approach for fermions in loop quantum gravity, including a graph-changing Hamiltonian operator that captures fermion motion and backreaction.

Findings

Fermion spins contribute to spin network vertex volumes.

A diffeomorphism covariant Hamiltonian operator is constructed.

The operator describes fermion dynamics and their influence on quantum geometry.

Abstract

In the model of a fermion field coupled to loop quantum gravity, we consider the Gauss and the Hamiltonian constraints. According to the explicit solutions to the Gauss constraint, the fermion spins and the gravitational spin networks intertwine with each other so that the fermion spins contribute to the volume of the spin network vertices. For the Hamiltonian constraint, the regularization and quantization procedures are presented in detail. By introducing an adapted vertex Hilbert space to remove the regulator, we propose a diffeomorphism covariant graph-changing Hamiltonian constraint operator of the fermion field. This operator shows how fermions move in the loop quantum gravity spacetime and simultaneously influences the background quantum geometry.