Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.
Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.
We present linear transformations between the components of the electric and magnetic fields, E = F1 (H ) and H = F2 (E ), which leave the expression for the density of electromagnetic field energy unchanged. If the coefficients occurring in these transformations are interpreted as the components of the momentum vector of the photon and use is made of the correspondence principle W →∂t and p→∂x, then these transformations turn out to be Maxwell’s equations. In a similar fashion, when starting from the probability density associated with the Dirac wave function, then one can also arrive, via appropriate transformations between the components of the 4- spinor, at the Dirac equation.
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.
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.
In this paper it is shown that Maxwell's theory of the electromagnetic field in vacuum can be stated in a form closely parallel to Dirac's theory of the electron. The electromagnetic field is described by a three-by-one column matrix whose elements are linear combinations of the three independent spinor components of the field. The Maxwell equations take a form similar to the Dirac equation for a free electron but involving three-by-three matrices. In terms of a wave function normalized so that the integral-square is the number of photons, the classical expressions for the energy, momentum, and angular momentum in the field are related to expected values of the Hamiltonian, momentum, and angular momentum operators. The angular momentum operator consists of an orbital part plus a three-by-three spin-one matrix part. As in Dirac's theory, the contributions from the states of negative Hamiltonian eigenvalue must be subtracted---these are states of opposite circular polarization.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations.
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.