Reflection positivity in Euclidean formulations of relativistic quantum mechanics of particles

TL;DR

This paper analyzes the structure of reflection positive Euclidean covariant distributions for constructing relativistic quantum mechanical models with finite degrees of freedom, highlighting their properties and potential for model construction.

Contribution

It characterizes the general structure of reflection positive distributions in Euclidean quantum mechanics of finite systems, aiding model development without relying on locality.

Findings

Reflection positivity is less restrictive for finite systems.

Euclidean approach simplifies Poincaré invariant model construction.

Inner products can be computed without analytic continuation.

Abstract

This paper discusses the general structure of reflection positive Euclidean covariant distributions that can be used to construct Euclidean representations of relativistic quantum mechanical models of systems of a finite number of degrees of freedom. Because quantum systems of a finite number of degrees of freedom are not local, reflection positivity is not as restrictive as it is in a local field theory. The motivation for the Euclidean approach is that it is straightforward to construct exactly Poincar\'e invariant quantum models of finite number of degrees of freedom systems that satisfy cluster properties and a spectral condition. In addition the quantum mechanical inner product can be computed without requiring an analytic continuation. Whether these distributions can be generated by a dynamical principle remains to be determined, but understanding the general structure of the Euclidean covariant distributions is an important first step.