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

Waldyr A. Rodrigues Jr

Published 2017-01-02Version 1

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 timelike or even spacelight photon trajectories in free space. We briefly discuss our findings in relation to a recent experiment.