A Euclidean formulation of relativistic quantum mechanics

TL;DR

This paper presents a Euclidean-based formulation of relativistic quantum mechanics that allows direct calculation of physical quantities from Green functions, avoiding analytic continuation, and demonstrates its application with a toy model.

Contribution

It introduces a novel Euclidean formalism for relativistic quantum mechanics that simplifies calculations and connects closely with quantum field theory principles.

Findings

Direct computation of matrix elements from Euclidean Green functions

Application to bound states and scattering without analytic continuation

Use of a toy model to demonstrate the formalism's effectiveness

Abstract

In this paper we discuss a formulation of relativistic quantum mechanics that uses Euclidean Green functions or generating functionals as input. This formalism has a close relation to quantum field theory, but as a theory of linear operators on a Hilbert space, it has many of the advantages of quantum mechanics. One interesting feature of this approach is that matrix elements of operators in normalizable states on the physical Hilbert space can be calculated directly using the Euclidean Green functions without performing an analytic continuation. The formalism is summarized in this paper. We discuss the motivation, advantages and difficulties in using this formalism. We discuss how to compute bound states, scattering cross sections, and finite Poincare transformations without using analytic continuation. A toy model is used to demonstrate how matrix elements of exp(-beta H) in normalizable states can be used to construct-sharp momentum transition matrix elements.