Polynomial Hamiltonian form of General Relativity

TL;DR

This paper reformulates General Relativity's phase space into a polynomial Hamiltonian form using an extended Poisson manifold, simplifying the action and constraints, and applies it to the Friedmann universe.

Contribution

It introduces a polynomial Hamiltonian formulation of General Relativity by extending phase space and explores its implications for the Friedmann universe.

Findings

Constraints become polynomial in new variables

Dirac brackets define a degenerate Poisson structure

Application to Friedmann universe demonstrates utility

Abstract

Phase space of General Relativity is extended to a Poisson manifold by inclusion of the determinant of the metric and conjugate momentum as additional independent variables. As a result, the action and the constraints take a polynomial form. New expression for the generating functional for the Green functions is proposed. We show that the Dirac bracket defines degenerate Poisson structure on a manifold, and a second class constraints are the Casimir functions with respect to this structure. As an application of the new variables, we consider the Friedmann universe.