Quantum mechanics of the free Dirac electrons and Einstein photons, and the Cauchy process

TL;DR

This paper constructs fundamental solutions for free Dirac and Einstein equations, revealing their relation via a unitary transform involving the Cauchy distribution, and explores their connection to classical relativistic mechanics.

Contribution

It introduces a novel construction of fundamental solutions for Dirac and Einstein equations and links quantum solutions to classical mechanics through a Cauchy distribution-based transform.

Findings

Fundamental solutions are matrix valued functionals on bump functions.

Solutions are related by a unitary transform via the Cauchy distribution.

Classical relativistic mechanics emerges from quantum Dirac electron mechanics.

Abstract

Fundamental solutions for the free Dirac electron and Einstein photon equations in position coordinates are constructed as matrix valued functionals on the space of bump functions. It is shown that these fundamental solutions are related by a unitary transform via the Cauchy distribution in imaginary time. We study the way the classical relativistic mechanics of the free particle comes from the quantum mechanics of the free Dirac electron.