Integrability-Nonintegrability Structures and Individual Photons' Description as Finite Field Objects

TL;DR

This paper develops a finite field model for individual photons, linking their translational and rotational properties through geometric structures and deriving key physical relations like energy and Planck's formula.

Contribution

It introduces a novel geometric framework connecting photon properties with nonintegrability and curvature of distributions on spacetime.

Findings

Photon energy-momentum characteristics are correctly modeled.

Curvature of distributions is proportional to energy density.

Planck's formula emerges naturally from the model.

Abstract

This paper presents an attempt to come to a natural field model of individual photons considered as finite entities and propagating along some distinguished direction in space in a consistent translational-rotational manner. The starting assumption reflects their most trustful property to propagate translationally in a uniform way along straight lines. The model gives correct energy-momentum characteristics and connects the rotational characteristics of photons with corresponding nonintegrability (or curvature) of some 2-dimensional distributions (or Pfaff systems) on $\mathbb{R}^4$. It is obtained that the curvature is proportional to the corresponding energy-density. The field equations are obtained through a Lagrangian and they express a consistency condition between photon's translational and rotational propagation properties. The energy tensor is deduced directly from the equations since the corresponding Hilbert energy-tensor becomes zero on the solutions. Planck's formula $E=h\nu$ is naturally obtained as an integral translational-rotational consistency relation.