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arXiv:2505.08369 · Advanced Photonics

On the first quantization and quantum diversity of photons

Boris N. Chichkov

Published Aug 18, 2025• 1 citation
arXivHTMLPDF

Abstract

Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.

DOI
10.1117/1.ap.7.5.055001

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Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.
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.
We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)
The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
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.