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
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 Schrödinger 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 $\mathbf{T}_{0}=\mathbf{T}(\boldsymbolγ_{0})$ 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 Schrödinger 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.
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 F which is a plane wave solution (PWS) of ME. Moreover, we show how introducing a potential 1-form A such that F=dA we can see how a duality rotation changed in a \ spatial rotation transformation besides showing how i=−1 enters Maxwell theory thus permitting the writing of a representative of ME as Schrödinger 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 T(n) in some special situations showing that for some of those cases T0=T(γ0) 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 Schrödinger 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.
Several important questions regarding the photon wave and particle concepts have so far not been fully settled by means of conventional theory which is based on Maxwell's equations. This concerns the problems of the angular momentum and the divergence of the obtained solutions in vacuum space. In attempts to tackle these and other unclear points, a number of extended non-Maxwellian theories have been elaborated. The present analysis is due to one of these approaches, as being based on a nonzero electric field divergence in the vacuum state, under the condition of Lorentz invariance. This theory leads to models for the individual photon being based on axisymmetric wave packet solutions with imposed relevant quantum conditions. Such solutions can be confined to a limited volume in vacuum space, thereby giving rise to a nonzero angular momentum, as required for a boson particle. Two wave packed modes are obtained which have limited characteristic diameters in the direction being perpendicular to that of the propagation. For one of the modes the diameter even becomes comparable to atomic dimensions. This provides the possibility for an efficient photon-electron interaction in the widely discussed but not fully understood photoionization effect. Both modes further have features in common with an earlier theory by de Broglie, where the photon is represented by a propagating pilot wave with a superimposed particle-like part having a very small rest mass. A proposal is finally made for the photon to appear in different quantum states, and where rapid transitions can take place between these states in the form of "photon oscillations".
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
The quantum theory of relativistic particles, based on the first quantization technique similar to that used by Schroedinger and Dirac in formulating quantum mechanics, is reconsidered on the basis of a photon-like dispersion relation corresponding to the energy conservation equation of Einstein's special relativity. First, scalar quantum mechanics of particles operating with their wave functions is discussed. Using the first quantization of the photon-like dispersion relation, very simple new derivation of the Dirac equation is given. Then, vector field quantum mechanics is introduced, which defines vector fields associated with the relativistic particle. Basic equations for the vector-field quantum mechanics, similar to the source-free Maxwell equations, are derived. Following these equations, the particle's de Broglie wave can be considered as the transversal electromagnetic wave. Therefore, the wave-particle duality can be redefined as the electromagnetic wave-particle duality. Relationships between the scalar and vector field quantum mechanics are analyzed.
Pursuing the Hamiltonian formulation of the de Broglie-Bohm (deBB) theory presented in the preceding paper, the Hamilton-Jacobi (HJ) theory of the wave-particle system is developed. It is shown how to derive a HJ equationfor the particle, which enables trajectories to be computed algebraically using Jacobi's method. Using Liouville's equation in the HJ representation we find the restriction on the Jacobi solutions which implies the quantal distribution. This gives a first method for interpreting the deBB theory in HJ terms. A second method proceeds via an explicit solution of the field+particle HJ equation. Both methods imply that the quantum phase may be interpreted as an incomplete integral. Using these results and those of the first paper it is shown how Schrodinger's equation can be represented in Liouvillian terms, and vice versa. The general theory of canonical transformations that represent quantum unitary transformations is given, and it is shown in principle how the trajectory theory may be expressed in other quantum representations. Using the solution found for the total HJ equation, an explicit solution for the additional field containing a term representing the particle back-reaction is found. The conservation of energy and momentum in the model is established, and a weak form of the action-reaction principle is shown to hold. Alternative forms for the Hamiltonian are explored and it is shown that, within this theoretical context, the deBB theory is not unique. The theory potentially provides an alternative way of obtaining the classical limit.
We propose to treat Maxwell equations as a wave equation of motion for a photon (a quantum of light) and to develop the photon mechanics on this basis. Consider vectors of electric and magnetic fields E and H as a single object (a bivector) F , which corresponds to the function ψ in the wave theory of the electron. Multiplying the vacuum Maxwell equations − rotH+ 1 c ∂E ∂t = 0, rot E + 1 c ∂H ∂t = 0 (1) by hc 2πi , it is possible to rewrite them in the form HF + h 2πi ∂F ∂t = 0, (2) where H is the operator H = hc 2πi { 0 − rot rot 0 } ; (3) its meaning follows from the comparison of equation (2) with (1). Equation (2) can be interpreted as a wave equation for a photon.
We develop a structural bridge between relativistic Hamilton–Jacobi theory and the relativistic Schrödinger equation within the framework of tempered distributions and Schwartz linear algebra. For translation-invariant Hamiltonians, the principal functions Sp(x)=⟨p,x⟩ restricted to the mass shell form a complete integral of the Hamilton–Jacobi equation, while their exponential images ηp=exp(ℏiSp) constitute a Schwartz basis of the tempered state space. On each spectral fiber, both classical and quantum equations reduce to the same Einstein dispersion relation. We prove that the relativistic Schrödinger equation is precisely the Schwartz–von Neumann S–linear extension of the classical energy relation from certainty momentum states to arbitrary tempered superpositions. In the presence of scalar potentials, the Hamiltonian arises as a mixed (momentum-diagonal and position-diagonal) extension, showing that the extension principle is not restricted to the free case. We further demonstrate that exact quantum dynamics cannot, in general, be represented by a single exponential phase exp(ℏiS) unless S is affine in space. Instead, quantum evolution is obtained by S–superpositions of the principal exponential family associated with a complete integral of the Hamilton–Jacobi equation. In this sense, classical elimination of parameters is replaced by linear spectral superposition. Geometrically, the exponential mapping transforms the flat affine space of Minkowski generators into a curved manifold of principal waves on which the nonlinear Hamilton–Jacobi flow pushes forward to a linear unitary Schrödinger flow. Through de Broglie–Maxwell isomorphisms, the construction extends to complex electromagnetic-like fields, preserving translation representation, dispersion relations, and polarization geometry. The results suggest that, for translation-invariant systems, quantization may be understood as an infinite-dimensional complex linearization of a classical certainty space rather than as a semiclassical approximation. Within the tempered-distribution setting, relativistic quantum dynamics emerges as the superpositional completion of a classical complete integral.