Dirac and Lagrangian reductions in the canonical approach to the first order form of the Einstein-Hilbert action

TL;DR

This paper compares Lagrangian and Dirac reduction methods in the canonical formulation of the first order Einstein-Hilbert action, revealing differences and proposing a more suitable form for Dirac's approach to constraints.

Contribution

It introduces a reformulation of the Einstein-Hilbert action that aligns better with Dirac's constrained system methodology and clarifies inconsistencies in previous treatments.

Findings

Lagrangian reduction yields different results from Dirac's method for first class constraints.

A new form of the first order Einstein-Hilbert action is proposed for better compatibility with Dirac's approach.

Explicit comparison using a simple model demonstrates the differences and highlights issues in prior quantization attempts.

Abstract

It is shown that the Lagrangian reduction, in which solutions of equations of motion that do not involve time derivatives are used to eliminate variables, leads to results quite different from the standard Dirac treatment of the first order form of the Einstein-Hilbert action when the equations of motion correspond to the first class constraints. A form of the first order formulation of the Einstein-Hilbert action which is more suitable for the Dirac approach to constrained systems is presented. The Dirac and reduced approaches are compared and contrasted. This general discussion is illustrated by a simple model in which all constraints and the gauge transformations which correspond to first class constraints are completely worked out using both methods in order to demonstrate explicitly their differences. These results show an inconsistency in the previous treatment of the first order Einstein-Hilbert action which is likely responsible for problems with its canonical quantization.