From the Photon to Maxwell Equation. Ponderations on the Concept of Photon Localizability and Photon Trajectory in a de Broglie-Bohm Interpretation of Quantum Mechanics

TL;DR

This paper explores the photon concept using Clifford algebra, deriving Maxwell's equations from a relativistic Hamilton-Jacobi framework, and discusses the possibility of photon trajectories within a de Broglie-Bohm interpretation, including quantum potential effects.

Contribution

It introduces a novel approach connecting photon theory, Maxwell equations, and quantum trajectories using Clifford bundles and Hamilton-Jacobi formalism.

Findings

Maxwell equations derived from photon Hamilton-Jacobi equation

Photon trajectories may be non-lightlike due to quantum potential

Pulse reshaping phenomena observed in vacuum

Abstract

In this paper using the Clifford bundle formalism we show how starting from the photon concept and its relativistic Hamilton-Jacobi equation (HJE) we immediately get (with a simple hypothesis concerning the form of the photon canonical momentum) Maxwell equation (ME) satisfied by a null $2$-form field $\boldsymbol{F}$ which is a plane wave solution (PWS) of ME. Moreover, we show how introducing a potential $1$-form $\boldsymbol{A}$ such that $\boldsymbol{F}=d\boldsymbol{A}$ we can see how a duality rotation changed in a \ spatial rotation transformation besides showing how $\mathrm{i}=\sqrt{-1}$ enters Maxwell theory thus permitting the writing of a representative of ME as Schr\"{o}dinger like equation which plays a key role in answering one of the main questions addressed in this paper, namely: is there any sense in talking about photon trajectories in de Broglie-Bohm like theories? To this end we investigate first the nature of the energy-momentum \emph{extensor} field of the Maxwell field $\mathbf{T}(n)$ in some special situations showing that for some of those cases $\mathbf{T}_{0}=\mathbf{T}(\boldsymbol{\gamma}_{0})$ even cannot describe the flow of energy. A proposed solution is offered. We also prove that there exists a pulse reshaping phenomenon even in vacuum. Finally we discuss the Schr\"{o}dinger equation for a photon that follows from quantum field theory and investigate solutions that some authors think imply in photon localization. We discuss if such an idea is meaningful. Moreover, we show that for such solutions it is possible (once we accept that photon wave functions are extended in the space and also in the time domains) to derive a generalized HJE containing a quantum potential and which may lead to non lightlike photon trajectories in free space. We briefly discuss our findings in relation to a recent experiment.