Universal Procedure for Enforcing Quantum Constraints

TL;DR

This paper introduces a universal, abstract method for implementing quantum constraints that simplifies the projection onto physical states using a single integration process, applicable to various constraint types.

Contribution

It presents a novel, unified approach for quantum constraint enforcement that is independent of the specific constraints, utilizing a common integration procedure for the projection operator.

Findings

Projection operator derived via a single integration over Lagrange multipliers

Second-order expansion in lattice spacing needed for constraint operators

Coherent state path integrals used to illustrate the method

Abstract

An abstract formulation of quantum dynamics in the presence of a general set of quantum constraints is developed. Our constructive procedure is such that the relevant projection operator onto the physical Hilbert space is obtained with a single, common integration procedure over the original Lagrange multiplier variables that is completely independent of the general nature of the constraints. In the associated lattice-limit formulation it is demonstrated that expansion of the constraint operator contribution to second order in the lattice spacing is necessary while, as usual, only a first-order expansion is needed for the dynamical operator contribution. Among various possibilities, coherent state path integrals are used to illustrate a completely functional representation of the abstract quantization procedure.