On the Relation between Rigging Inner Product and Master Constraint Direct Integral Decomposition

TL;DR

This paper explores the connection between the Rigging Inner Product and the Master Constraint method in the quantization of constrained systems, establishing conditions for their relation and path integral formulation.

Contribution

It provides conditions under which the Master Constraint's direct integral decomposition relates to the Rigging Map, especially through Abelian constraints.

Findings

Established conditions for relating Master Constraint DID to Rigging Map.

Showed how Abelian constraints facilitate the path integral formulation.

Connected Rigging Inner Product with path integrals in constrained quantization.

Abstract

Canonical quantisation of constrained systems with first class constraints via Dirac's operator constraint method proceeds by the thory of Rigged Hilbert spaces, sometimes also called Refined Algebraic Quantisation (RAQ). This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the Master Constraint Method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition methods (DID), which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the Rigging Inner Product to the path integral that one obtains via reduced phase space methods. However, for the Master Constraint this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the Master Constraint DID for those Abelian constraints can be directly related to the Rigging Map and therefore has a path integral formulation.