Peculiarities of the Canonical Analysis of the First Order Form of the Einstein-Hilbert Action in Two Dimensions in Terms of the Metric Tensor or the Metric Density

TL;DR

This paper examines the canonical analysis of the first order Einstein-Hilbert action in two dimensions, comparing metric tensor and metric density formulations, and reveals their equivalence in terms of constraint algebra and gauge invariance.

Contribution

It provides a detailed comparison of the metric tensor and metric density formulations in 2D, highlighting their differences and similarities in canonical structure and gauge transformations.

Findings

Both formulations yield a closed algebra of constraints with field-independent structure constants.

The gauge transformations differ from standard diffeomorphisms in both approaches.

The metric tensor formulation is analyzed in detail and compared with previous results.

Abstract

The peculiarities of doing a canonical analysis of the first order formulation of the Einstein-Hilbert action in terms of either the metric tensor $g^{\alpha \beta}$ or the metric density $h^{\alpha \beta}= \sqrt{-g}g^{\alpha \beta}$ along with the affine connection are discussed. It is shown that the difference between using $g^{\alpha \beta}$ as opposed to $h^{\alpha \beta}$ appears only in two spacetime dimensions. Despite there being a different number of constraints in these two approaches, both formulations result in there being a local Poisson brackets algebra of constraints with field independent structure constants, closed off shell generators of gauge transformations and off shell invariance of the action. The formulation in terms of the metric tensor is analyzed in detail and compared with earlier results obtained using the metric density. The gauge transformations, obtained from the full set of first class constraints, are different from a diffeomorphism transformation in both cases.