Different canonical formulations of Einstein's theory of gravity

TL;DR

This paper reviews four main canonical formulations of Einstein's gravity, detailing their differences, relations, and pedagogical aspects, to aid understanding and development of quantum gravity theories.

Contribution

It provides a comprehensive comparison of four canonical formalisms of Einstein's gravity and their canonical transformations, facilitating better understanding and further research.

Findings

Descriptions of four canonical formalisms

Canonical transformations relating the formalisms

Pedagogical presentation of the formalisms

Abstract

We describe the four most famous versions of the classical canonical formalism in the Einstein theory of gravity: the Arnovitt-Deser-Misner formalism, the Faddeev-Popov formalism, the tetrad formalism in the usual form, and the tetrad formalism in the form best suited for constructing the loop theory of gravity, which is now being developed. We present the canonical transformations relating these formalisms. The paper is written mainly for pedagogical purposes.