On the Wave Function of the Photon

It is believed that certain matrix elements of the electromagnetic field operators in quantum electrodynamics, in close analogy with nonrelativistic quantum theory of massive particles, may be treated as photon wave functions. In this paper, I would like to push this interpretation even further by arguing that one can set up a consistent wave mechanics of photons that could be often used as a convenient tool in the description of electromagnetic fields, independently of the formalism of second quantization. In other words, in constructing quantum theories of photons we may proceed, as in quantum theory of massive particles, through two stages. At the first stage we introduce wave functions and a wave equation obeyed by these wave functions. At the second stage we upgrade the wave functions to the level of field operators in order to deal more effectively with the states involving many particles and to allow for processes in which the number of particles is not conserved. An important additional consequence of having a genuine wave function for the photon is a possibility to define an analog of the Wigner function for the photon with its semiclassical interpretation as a (quasi) distribution function in the phase space. The very concept of the photon wave function is not new, but strangely enough it has never been systematically explored. Some textbooks on quantum mechanics start the introduction to quantum theory with a discussion of photon polarization measurements (cf., for example [1–4]). Dirac, in particular, writes a lot about the role of the wave function in the description of quantum interference phenomena for photons (in this context he uses the now famous phrase: “Each photon interferes only with itself”), but in his exposition the photon wave function never takes on a specific mathematical form. It is true that in the description of polarization simple prototype two-component wave functions are often used to describe various polarization states of the photon and with their help the preparation and the measurement of polarization is thoroughly explained. However, after such a heuristic introduction to quantum theory, the authors go on to the study of massive particles and if they ever return to a quantum theory of photons it is always within the formalism of second quantization with creation and annihilation operators. In some textbooks (cf., for example [5–7]) one may even find statements that completely negate the possibility of introducing a wave function for the photon. I shall introduce a wave function for the photon by reviving and extending the mode of description of the electromagnetic field based on the complex form of the Maxwell equations. This form was known already at the beginning of the century [8,9] and was later rediscovered by Majorana [10] who explored the analogy between the Dirac equation and the Maxwell equations. The complex vector that appears in this description will be shown to have the properties that one would associate with a one-photon wave function, including also an acceptable probabilistic interpretation. Photons are much different from massive particles. They are also different from neutrinos since the photon number does not obey a conservation law. These differences are a source of complications, especially when photons propagate in a medium. However, even these complications have a certain value: they teach us something new about the nature of photons. The approach adopted in this paper is distinctly different from the line of investigation started by Landau and Peierls [11] and continued more recently by Cook [13]. The Landau-Peierls and Cook wave functions are nonlocal objects. The nonlocality is introduced by operating on the electromagnetic fields with the integral operator (−∆)−1/4,