Abstract
Quantization of the electromagnetic field is traditionally introduced at the level of second quantization: the classical field variables are replaced by field operators. I believe that the reasons why a first-quantized theory of photons has never been fully developed are mainly historical. Had Dirac discovered his relativistic wave equation [1] prior to his quantization of the electromagnetic field [2], he would have noticed and most probably further explored a great similarity between the wave equation for the electron (or even better for the neutrino) and the Maxwell equations. As it happened, this similarity was noticed later (for the first time apparently by Majorana [3]) and played no role in the development of the quantum theory of electromagnetism because the quantized electromagnetic field has been introduced from the very beginning and accounted for all quantum properties electromagnetic radiation. Subsequently quantum electrodynamics has become so successful in explaining with utmost accuracy all experiments within its range of applicability that there was no need to search for an alternative formulation that would employ the concept of the photon wave function. Considering our trust in quantum electrodynamics and our familiarity with its formal apparatus one may even ask if there is any justification at all for, what it essentially amounts to, a reconstruction of the notion of the photon function from QED, only to face a not so familiar object whose properties are yet to be uncovered.
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Bialynicki-Birula, I. (1996). The Photon Wave Function. In: Eberly, J.H., Mandel, L., Wolf, E. (eds) Coherence and Quantum Optics VII. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9742-8_38
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