On the Quantum-Mechanics of a Single Photon

Abstract

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field 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 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 $E=\hslash \omega$ and $\mathbf{p}=\hslash \mathbf{k}$, 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.

40 pages; minor revisions of the presentation (in the Introduction, the Riesz tensor section, and the summary; the content of the previous section 2 has been moved into the appendix). A few typos have been corrected, too. To appear in Journal of Mathematical Physics

References in corpus (6)

• Photon wave functions, wave-packet quantization of light, and coherence theory
• The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism
• Photon wave function
• The "Unromantic Pictures" of Quantum Theory
• The Photon Wavefunction: a covariant formulation and equivalence with QED
• The de Broglie-Bohm pilot-wave interpretation of quantum theory

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