Relativistic dynamics, Green function and pseudodifferential operators

TL;DR

This paper explores the role of pseudodifferential operators in relativistic quantum dynamics, constructing Green functions, analyzing wave equations with arbitrary spin, and proposing a non-local, Lorentz-invariant position operator.

Contribution

It introduces a new theoretical framework linking pseudodifferential operators, Green functions, and relativistic wave equations, including a novel non-local position operator.

Findings

Explicit construction of Green function kernels and their dependence on spacetime dimensions

Analysis of relativistic wave equations with arbitrary spin and causality issues

Proposal of a non-local, Lorentz-invariant position operator that avoids traditional problems

Abstract

The central role played by pseudodifferential operators in relativistic dynamics is very well know. In this work, operators as the Schrodinger one (e.g: square root) are treated from the point of view of the non-local pseudodifferential Green functions. Starting from the explicit construction of the Green (semigroup) theoretical kernel, a theorem linking the integrability conditions and their dependence on the spacetime dimensions is given. Relativistic wave equations with arbitrary spin and the causality problem are discussed with the algebraic interpretation of the radical operator and their relation with coherent and squeezed states. Also we perform by mean of pure theoretical procedures (based in physical concepts and symmetry) the relativistic position operator which satisfies the conditions of integrability : it is non-local, Lorentz invariant and does not have the same problems as the "local"position operator proposed by Newton and Wigner. Physical examples, as Zitterbewegung and rogue waves, are presented and deeply analysed in this theoretical framework.