Fermions coupled to the Palatini action in $n$ dimensions

TL;DR

This paper investigates how fermions couple to the Palatini action in various dimensions, analyzing the Lagrangian and Hamiltonian formulations, and identifying conditions for equivalence to Einstein-Dirac theory, with implications for quantum gravity.

Contribution

It provides a Hamiltonian analysis of fermion coupling to the Palatini action in n dimensions, including covariant phase-space variables and symplectomorphisms, extending previous formulations.

Findings

First-order action equivalent to Einstein-Dirac in 4D under specific parameters

Hamiltonian formulation with Lorentz-covariant variables and first-class constraints

Multiple Hamiltonian formulations via symplectomorphisms, including half-densitized fermions

Abstract

We study minimal and nonminimal couplings of fermions to the Palatini action in $n$ dimensions ($n\geq 3$) from the Lagrangian and Hamiltonian viewpoints. The Lagrangian action considered is not, in general, equivalent to the Einstein-Dirac action principle. However, by choosing properly the coupling parameters, it is possible to give a first-order action fully equivalent to the Einstein-Dirac theory in a spacetime of dimension four. By using a suitable parametrization of the vielbein and the connection, the Hamiltonian analysis of the general Lagrangian is given, which involves manifestly Lorentz-covariant phase-space variables, a real noncanonical symplectic structure, and only first-class constraints. Additional Hamiltonian formulations are obtained via symplectomorphisms, one of them involving half-densitized fermions. To confront our results with previous approaches, the time gauge is imposed.