# On the Hamiltonian formalism of the Tetrad-Gravity with fermions

## TL;DR

This paper extends the Hamiltonian formalism of tetrad-connection gravity to include fermions, demonstrating the algebra of constraints remains consistent and identifying a canonical transformation to simplify the phase space structure.

## Contribution

It introduces a Hamiltonian analysis of tetrad-connection gravity with fermions, fixing non-dynamic connections and revealing a canonical transformation to simplify the phase space.

## Key findings

- First-class constraints form a closed algebra with structural constants.
- Existence of a canonical transformation simplifies the Dirac brackets.
- The algebra of constraints remains consistent with pure gravity case.

## Abstract

We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero. Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.