The quantum Cauchy functional and space-time approach to relativistic quantum mechanics

TL;DR

This paper develops a space-time framework for relativistic quantum mechanics using quantum Cauchy pre-measures, linking classical relativistic mechanics with quantum theory of particles like electrons and photons.

Contribution

It introduces generalized quantum Cauchy pre-measures that extend to measures on dual Hilbert spaces, unifying quantum and classical relativistic particle dynamics.

Findings

Quantum Cauchy pre-measures are $\sigma$-additive and supported on trajectories with velocities in Hilbert space.

Electron and photon pre-measures are unitarily equivalent to Green's functions of Dirac and Maxwell equations.

The classical relativistic mechanics emerges from the quantum mechanics of free Dirac particles.

Abstract

We construct generalized quantum Cauchy pre-measures that correspond to the analytic continuation of the transition probability of the Cauchy process to imaginary time. We show that these complex pre-measures of time translations extend to a measure on the space dual to a real Hilbert space whose support is locally compact in the uniform convergence topology and with velocities in the Hilbert space. At that the quantum Cauchy-Dirac and Cauchy-Maxwell pre-measures of time translations of electrons and photons, that correspond to the retarded Green's functions of the Dirac and Maxwell equations with no sources viewed as generalized functions on bump functions, are unitary equivalent to quantum Cauchy pre-measures Therefore these pre-measures on the space dual to a real Hilbert space are $\sigma$-additive as well, but their support on the electron (respectively, photon) trajectories are compact in the uniform convergence topology with velocities in the Hilbert space. We study the way the classical relativistic mechanics of particle comes from the quantum mechanics of the free Dirac particle.