On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation

TL;DR

This paper reviews the derivation of path integral measures from the reduced phase space formulation in gauge theories, demonstrating its formal equivalence with Dirac and Master constraint quantisation, and discusses measure corrections especially in theories with second class constraints.

Contribution

It provides a comprehensive review of deriving path integral measures from reduced phase space and establishes their formal equivalence with Dirac and Master constraint quantisation methods.

Findings

Reduced phase space path integral agrees with Dirac's operator constraint quantisation.

Equivalence with Master constraint quantisation holds for first class constraints, requiring Abelianisation for non-trivial structure functions.

Correct configuration space measure deviates from exponential of Lagrangian action, especially with second class constraints.

Abstract

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this article we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantisation and, more specifically, with the Master constraint quantisation for first class constraints. For first class constraints with non trivial structure functions the equivalence can only be established by passing to Abelian(ised) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.