The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

Martin Land

doi:10.3390/e19050234

Abstract

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg-Horwitz-Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^\mu (\tau)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0 = ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are \hbox{$\tau$-dependent} and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle.

Highlights

In developing his interpretation of antiparticles as particles travelling backward in time, Stueckelberg [1,2] hoped to demonstrate that pair creation/annihilation processes would appear naturally in a thoroughly deterministic and classical relativistic Hamiltonian mechanics, once such a formalism could be properly constructed
Stueckelberg–Horwitz–Piron electrodynamics can be approached as an abstract gauge theory, exploring the consequences of allowing the gauge transformation (12) of the quantum wave function to depend on the evolution parameter in the dynamical framework
In another sense, just as Maxwell sought to formalize the empirical results of Cavendish and Coulomb, SHP may be seen as accounting for classical Maxwell electrodynamics in light of the pair creation/annihilation phenomena observed by Anderson

Summary

Introduction

In developing his interpretation of antiparticles as particles travelling backward in time, Stueckelberg [1,2] hoped to demonstrate that pair creation/annihilation processes would appear naturally in a thoroughly deterministic and classical relativistic Hamiltonian mechanics, once such a formalism could be properly constructed. To resolve this apparent contradiction, Seidewitz has observed [12] that Haag’s proof relies on transformations generated by a Hamiltonian which is the 0-component of a four-vector and parameterized by x0 He demonstrates that the construction of standard QFT in the Stueckelberg framework leads to an interaction picture obtained by acting on free fields with a unitary transformation generated by a scalar Stueckelberg Hamiltonian, as for example in Equation (11) below, and parameterized by τ. The occurrence of event x μ (τ1 ) at τ1 is understood to be an irreversible process that cannot be changed by a subsequent event occurring at the same spacetime location, x μ (τ2 ) = x μ (τ1 ) with τ2 > τ1 This absence of closed time-like curves applies in SHP quantum electrodynamics [15] where the particle propagator G ( x2 − x1 , τ2 − τ1 ) vanishes unless τ2 > τ1 , preventing divergent matter loops, when x2 = x1. The details of this τ-dependence in the interacting fields and currents can be studied by reconciling classical SHP with classical Maxwell electromagnetic phenomenology

Classical SHP Electrodynamics

Maxwell Theory as an Equilibrium State of SHP

Conclusions