Hamiltonian Analysis of Plebanski Theory

TL;DR

This paper performs a Hamiltonian analysis of Plebanski theory, identifying constraints, sectors, and measures relevant for quantum gravity, with implications for both Euclidean and Lorentzian signatures.

Contribution

It provides a detailed Hamiltonian formulation of Plebanski theory, clarifying the structure of constraints and the measure for path integral quantization.

Findings

Identification of regular and non-regular sectors in phase space.

Explicit form of first and second class constraints.

Derivation of the path integral measure analogous to quantum gravity measures.

Abstract

We study the Hamiltonian formulation of Plebanski theory in both the Euclidean and Lorentzian cases. A careful analysis of the constraints shows that the system is non regular, i.e. the rank of the Dirac matrix is non-constant on the non-reduced phase space. We identify the gravitational and topological sectors which are regular sub-spaces of the non-reduced phase space. The theory can be restricted to the regular subspace which contains the gravitational sector. We explicitly identify first and second class constraints in this case. We compute the determinant of the Dirac matrix and the natural measure for the path integral of the Plebanski theory (restricted to the gravitational sector). This measure is the analogue of the Leutwyler-Fradkin-Vilkovisky measure of quantum gravity.