A new representation for non--local operators and path integrals

TL;DR

This paper introduces an analytical representation for non-local operators and path integrals, enabling precise solutions to relativistic quantum problems and offering a new tool for non-local Hamiltonian analysis.

Contribution

The authors develop a novel analytical representation for non-local operators and path integrals, facilitating accurate solutions without local operator expansions.

Findings

Achieved arbitrarily precise results for the relativistic harmonic oscillator

Derived a new Green's function representation for quantum mechanics

Validated the path integral approach with free particle in a box

Abstract

We derive an alternative representation for the relativistic non--local kinetic energy operator and we apply it to solve the relativistic Salpeter equation using the variational sinc collocation method. Our representation is analytical and does not depend on an expansion in terms of local operators. We have used the relativistic harmonic oscillator problem to test our formula and we have found that arbitrarily precise results are obtained, simply increasing the number of grid points. More difficult problems have also been considered, observing in all cases the convergence of the numerical results. Using these results we have also derived a new representation for the quantum mechanical Green's function and for the corresponding path integral. We have tested this representation for a free particle in a box, recovering the exact result after taking the proper limits, and we have also found that the application of the Feynman--Kac formula to our Green's function yields the correct ground state energy. Our path integral representation allows to treat hamiltonians containing non--local operators and it could provide to the community a new tool to deal with such class of problems.