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

Abstract

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 µ (τ) as τ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time x 0 = ct (measured by laboratory clocks) and chronology τ (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 τ-dependent and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width λ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As λ becomes very large, the photon mass goes to zero and the field equations become τ-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which λ 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.

Comparative Analysis of Jüttner’s Calculation of the Energy of a Relativistic Ideal Gas and Implications for Accelerator Physics and Cosmology

“…The energy that is being achieved by modern accelerators is making it possible to study the behavior of matter during the earliest, and hottest, moments of the big bang. Parametrized electrodynamics [26,27] may need to be extended to study matter in a plasma state.…”

Evolution equations from an epistemic treatment of time

“…An event e k and the corresponding temporal update n − 1 → n is always associated with the change of the state of an object O according to (14). Its state S OO (n) just after the event corresponds to the present state of the object at time n. We may also have memories at time n of its preceding state S OO (n − 1).…”

“…Several researchers have developed Stueckelberg's formalism further [10][11][12][13][14]. However, there is still no generally accepted physical motivation for the introduction of the additional evolution parameter λ .…”

“…The parameter λ is commonly related to the proper time along the world line of a given particle [11,16]. Some researchers try to generalize this notion so that the same invariant λ can describe the evolution of many particles, making it similar to the external, absolute time of Newtonian mechanics [12][13][14]17]. Some authors argue that it is possible to construct a clock that measures the value of λ [16], whereas others argue that λ is not observable [17].…”

“…In so doing we are able to get a new perspective on the Dirac equation and the operators that are associated with observables in quantum mechanics. The parameter σ plays a similar role in the present formalism as the parameter λ in Stueckelberg's theory [8][9][10][11][12][13][14][15][16][17].…”