A General Form of the Constraints in the Path Integral Formula

TL;DR

This paper explores a generalized form of constraints in path integral formulations for manifolds embedded in higher-dimensional Euclidean spaces, analyzing their validity and operator interpretations.

Contribution

It introduces a generalized constraint formulation in path integrals and discusses their validity and operator representations beyond the specific sphere case.

Findings

Mid-point prescription is privileged for spheres but complex for general manifolds.

A new interpretation of the path integral's validity in operator formalism.

Operators derived from the generalized path integral are discussed.

Abstract

A form of the constraints, specifying a $D$-dimensional manifold embedded in $D+1$ dimensional Euclidean space, is discussed in the path integral formula given by a time discretization. Although the mid-point prescription is privileged in the sphere $S^D$ case, it is more involved in generic cases. An interpretation on the validity of the formula is put in terms of the operator formalism. Operators from this path integral formula are also discussed.