Photon Wave Equation Research Articles

Superluminal Speed of Photons in the Electromagnetic Near-Field

The possible existence of superluminal particles, which are forbidden by well-known laws of physics, has been studied by many physicists. Some of them confirmed the superluminal speed by their experiments. By using Klein-Gordon wave equation for photons, the author shows that the photon travels at a superluminal speed in an electromagnetic near field of the source and they reduce to the speed of light as they propagate into the far field.

On the quantum-mechanics of a single photon

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field ψph on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a quantum-mechanical photon wave equation. In particular, it implies that the photon wave function ψph yields conserved non-negative Born-rule-type quantum probabilities and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations E = ℏω and p = ℏk, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.

Duality and helicity: the photon wave function approach

The photon wave equation proposed in terms of the Riemann–Silberstein vector is derived from a first-order Dirac/Weyl-type action principle. It is symmetric w.r.t. duality transformations, but the associated Noether quantity vanishes. Replacing the fields by potentials and using instead a quadratic Klein–Gordon-type Lagrangian allows us to recover the double-Chern–Simons expression of conserved helicity and is shown to be equivalent to recently proposed alternative frameworks. Applied to the potential-modified theory the Dirac/Weyl-type approach yields again zero conserved charge, whereas the Klein–Gordon-type approach applied to the original setting yields Lipkin's “zilch”.

Photon reflection by a quantum mirror: A wave-function approach

We derive from first principles the momentum exchange between a photon and a quantum mirror upon reflection, by considering the boundary conditions imposed by the mirror surface on the photon wave equation. We show that the system generally ends up in an entangled state, unless the mirror position uncertainty is much smaller than the photon wavelength, when the mirror behaves classically. Our treatment leads us directly to the conclusion that the photon momentum has the known value hk/2{\pi}. This implies that when the mirror is immersed in a dielectric medium the photon radiation pressure is proportional to the medium refractive index n. Our work thus contributes to the longstanding Abraham-Minkowski debate about the momentum of light in a medium. We interpret the result by associating the Minkowski momentum (which is proportional to n) with the canonical momentum of light, which appears naturally in quantum formulations.

Maxwell's equations, quantum physics and the quantum graviton

Quantum wave equations for massless particles and arbitrary spin are derived by factorizing the d'Alembertian operator. The procedure is extensively applied to the spin one photon equation which is related to Maxwell's equations via the proportionality of the photon wavefunction Ψ to the sum E + iB of the electric and magnetic fields. Thus Maxwell's equations can be considered as the first quantized one-photon equation. The photon wave equation is written in two forms, one with additional explicit subsidiary conditions and second with the subsidiary conditions implicitly included in the main equation. The second equation was obtained by factorizing the d'Alembertian with 4×4 matrix representation of "relativistic quaternions". Furthermore, scalar Lagrangian formalism, consistent with quantization requirements is developed using derived conserved current of probability and normalization condition for the wavefunction. Lessons learned from the derivation of the photon equation are used in the derivation of the spin two quantum equation, which we call the quantum graviton. Quantum wave equation with implicit subsidiary conditions, which factorizes the d'Alembertian with 8×8 matrix representation of relativistic quaternions, is derived. Scalar Lagrangian is formulated and conserved probability current and wavefunction normalization are found, both consistent with the definitions of quantum operators and their expectation values. We are showing that the derived equations are the first quantized equations of the photon and the graviton.

KLEIN PARADOX FOR OPTICAL SCATTERING FROM EXCITED TARGETS

The well-known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible "negative probabilities" induced by certain potentials in some regimes of energy. The paradox may be resolved by employing the notion of electron–positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in the Maxwell's equations viewed as the wave equation for photons. In a medium containing "inverted energy populations" of excited atoms, e.g. in a LASER medium, one may again compute possible "negative probabilities." The resolution of the electromagnetic Klein paradox is that when the atoms decay, the final state may contain more photons then were contained the initial state. The optical theorem total cross-section for scattering photons from excited state atoms may then be computed as negative within a frequency band with matter induced amplification.

Derivation of the Photon Wave Equation in a Medium via a Poisson Process

We construct a general photon wave equation in a nondispersive and homogeneous medium. Then, we demonstrate the relation between the photon wave equation and a Poisson process. Through this formalism, the passage time of photons in the medium can be defined. Furthermore, we derive a. photon wave equation in a dispersive medium from the general formalism of the Poisson process by introducing a time shift generator.

Derivation of relativistic wave equation from the Poisson process

A Poisson process is one of the fundamental descriptions for relativistic particles: both fermions and bosons. A generalized linear photon wave equation in dispersive and homogeneous medium with dissipation is derived using the formulation of the Poisson process. This formulation provides a possible interpretation of the passage time of a photon moving in the medium, which never exceeds the speed of light in vacuum.

Symmetrical formulation of electron–photon scattering in relativistic quantum mechanics

In this paper, electron–photon scattering in relativistic quantum mechanics is formulated in a symmetrical form by making use of a modified form of the photon wave equation proposed by the author in a previous article, together with the usual form of the Dirac wave equation. By solving the coupled wave equations two S matrices are deduced, one with the photon wave function and the other with the Dirac wave function. For a given scattering process, either one may be used to give the correct result, if it is obtainable from both. Thus, the symmetry of quantum electrodynamics is explicitly revealed in this formulation.

Propagation of coherent excitons and photons in a crystal

An investigation is made of the possibility of the existence of self-induced transparency due to excitons in semiconductors. Solutions are obtained of a nonlinear material equation for excitons and of the wave equation for photons in the form of steady-state wave packets (solitons and antisolitons) of a highfrequency carrier. Conditions for the existence of bleaching and darkening waves are determined as a function of the carrier wave vectors for each of the two polariton branches and as a function of the sign of the exciton-exciton interaction constant and exciton concentration.

Alternate formulation of electron–photon interactions in relativistic quantum mechanics

In this paper, we develop an alternate formulation of the electron–photon interaction in the framework of ordinary relativistic quantum mechanics (without second quantization). We use the Dirac equation of motion to recast the current density in a form linear in the photon wave function. This enables us to take the photon as the system and, regarding the electron as an ’’external field,’’ to iterate the photon wave equation by the standard method. In particular, we deduce the second-order matrix elements of Compton scattering, pair annihilation, and pair creation. The results so obtained are, of course, identical with those of the usual formulation, where the electron is taken as the system and the radiation is treated as the external field. The present approach complements the usual one; together, the two approaches reveal the symmetry of quantum electrodynamics under interchange of source and field.