A note on the spin connection representation of gravity

TL;DR

This paper explores the spin connection formulation of gravity, demonstrating its classical and quantum equivalence to the vierbein approach through constraint analysis and path integral formulation.

Contribution

It provides a detailed analysis of the constraints and Dirac brackets in the spin connection formulation, establishing its classical and quantum equivalence to the vierbein approach.

Findings

Dirac brackets for second class constraints are evaluated.

The Dirac algebra matches the usual space-time reparametrizations.

The path integral form confirms quantum equivalence with the e-frame.

Abstract

The formulation of gravity in 3+1 dimensions in which the spin connection is the basic field ($\omega $-frame) leads to a theory with first and second class constraints. Here, the Dirac brackets for the second class constraints are evaluated and the Dirac algebra of first class constraints is found to be the usual algebra associated to space-time reparametrizations and tangent space rotations. This establishes the classical equivalence with the vierbein approach (e-frame). The explicit form of the path integral for this theory is given and the quantum equivalence with the e-frame is also established.