On the mathematical formulation of the restricted Feynman path integrals through broken line paths

TL;DR

This paper rigorously formulates restricted Feynman path integrals for position measurements, proving their well-defined nature in $L^{2}$ space and their connection to non-self-adjoint Schrödinger equations, extending to quantum spin systems.

Contribution

It provides a rigorous mathematical foundation for RFPIs in quantum measurements and generalizes existing results to include quantum spin systems.

Findings

RFPIs are well-defined in $L^{2}$ space.

RFPIs solve non-self-adjoint Schrödinger equations.

Results extend to quantum spin systems.

Abstract

The restricted Feynman path integrals (RFPIs) have been proposed to study continuous quantum measurements in physics. The RFPIs are heuristically determined in terms of the usual probability amplitude multiplied by weight for each path, which contains information about the results and the resolution of the measuring device. In the present paper we will consider the RFPIs particularly for the position measurements and will prove rigorously that these RFPIs are well defined in the $L^{2}$ space and are the solutions to the non-self-adjoint Schroedinger equations. Our results in the present paper give a generalization of the results on the usual Feynman path integrals for the Schroedinger equations.Furthermore, our results are extended to quantum spin systems.