Four dimensional gravity as an almost Poisson system

TL;DR

This paper investigates the phase space structure of a noncanonical formulation of 4D gravity, revealing an almost-Poisson structure that governs the dynamics but lacks full symplectic properties.

Contribution

It introduces an almost-Poisson bracket for IRPG, providing a new perspective on its phase space structure and dynamics beyond traditional symplectic frameworks.

Findings

IRPG does not support a full Poisson bracket on the entire phase space

An almost-Poisson bracket correctly reproduces equations of motion

The almost-Poisson structure fails to satisfy Jacobi identity and antisymmetry

Abstract

In this paper we examine the phase space structure of a noncanonical formulation of 4-dimensional gravity referred to as the Instanton representation of Plebanski gravity (IRPG). The typical Hamiltonian (symplectic) approach leads to an obstruction to the definition of a symplectic structure on the full phase space of the IRPG. We circumvent this obstruction, using the Lagrange equations of motion, to find the appropriate generalization of the Poisson bracket. It is shown that the IRPG does not support a Poisson bracket except on the vector constraint surface. Yet there exists a fundamental bilinear operation on its phase space which produces the correct equations of motion and induces the correct transformation properties of the basic fields. This bilinear operation is known as the almost-Poisson bracket, which fails to satisfy the Jacobi identity and in this case also the condition of antisymmetry. We place these results into the overall context of nonsymplectic systems.