The Canonical Structure of the First Order Einstein-Hilbert Action

TL;DR

This paper uses the Dirac constraint formalism to analyze the first order Einstein-Hilbert action in higher dimensions, revealing a complex structure of first class constraints and gauge invariances without eliminating fields prematurely.

Contribution

It provides a detailed canonical analysis of the first order Einstein-Hilbert action, identifying tertiary constraints and clarifying gauge generators without prior field elimination.

Findings

Identification of primary, secondary, and tertiary first class constraints.

Determination of d(d - 3) degrees of freedom in phase space.

Analysis of gauge invariance and gauge generators.

Abstract

The Dirac constraint formalism is used to analyze the first order form of the Einstein-Hilbert action in d > 2 dimensions. Unlike previous treatments, this is done without eliminating fields at the outset by solving equations of motion that are independent of time derivatives when they correspond to first class constraints. As anticipated by the way in which the affine connection transforms under a diffeomorphism, not only primary and secondary but also tertiary first class constraints arise. These leave d(d - 3) degrees of freedom in phase space. The gauge invariance of the action is discussed, with special attention being paid to the gauge generators of Henneaux, Teitelboim and Zanelli and of Castellani.