Maxwell's Equations from Quantum Mechanics III: Synthesis: Gauge Invariance, Conserved Currents, and Hydrogen Atoms as the Sources of the Electromagnetic Field
Maxwell’s equations, formulated in 1865, preceded quantum mechanics by six decades. When the logical direction is reversed, however, the electromagnetic field structure can be interpreted as the form required for a quantum theory of charged matter once three standard ingredients are imposed: local gauge invariance, conserved electric current, and a dynamical electromagnetic field. Quantum mechanics supplies the charged-particle state, probability amplitudes, quantum currents, and—through the Dirac equation—a relativistically consistent description of electron charge and spin. Gauge symmetry then organizes the electromagnetic interaction, while current conservation constrains the admissible source structure. The homogeneous Maxwell identities follow naturally from expressing the field through a four-potential, whereas the sourced Maxwell equations require an additional electromagnetic field action or equivalent dynamical assumption. The resulting framework clarifies both the strength and the limit of a “Maxwell from quantum mechanics” program: quantum theory and gauge structure strongly constrain electromagnetism, but Maxwell field dynamics are not obtained from the electron wave equation alone.
Maxwell’s equations, formulated in 1865, preceded quantum mechanics by six decades. When the logical direction is reversed, however, the electromagnetic field structure can be interpreted as the form required for a quantum theory of charged matter once three standard ingredients are imposed: local gauge invariance, conserved electric current, and a dynamical electromagnetic field. Quantum mechanics supplies the charged-particle state, probability amplitudes, quantum currents, and—through the Dirac equation—a relativistically consistent description of electron charge and spin. Gauge symmetry then organizes the electromagnetic interaction, while current conservation constrains the admissible source structure. The homogeneous Maxwell identities follow naturally from expressing the field through a four-potential, whereas the sourced Maxwell equations require an additional electromagnetic field action or equivalent dynamical assumption. The resulting framework clarifies both the strength and the limit of a “Maxwell from quantum mechanics” program: quantum theory and gauge structure strongly constrain electromagnetism, but Maxwell field dynamics are not obtained from the electron wave equation alone.
This paper examines how Maxwell electrodynamics can be recovered as the field-theoretic structure required to consistently couple quantum matter to electromagnetism, while distinguishing what follows from quantum mechanics from what must be supplied independently. Local phase invariance introduces electromagnetic potentials through minimal coupling and yields the homogeneous Maxwell equations as geometric identities, while quantum probability conservation provides a conserved electric current and Ehrenfest dynamics recover the Lorentz-force form. The inhomogeneous Maxwell equations require an additional Lorentz- and gauge-invariant electromagnetic field action, whose vacuum constants satisfy μ0ε0=1/c2\mu_0\varepsilon_0=1/c^2, linking the construction to the SQC/MSC electric–magnetic prefactor relation. Hydrogen is used as an explicit microscopic source: the 1s1s state generates a quantum charge distribution, 2p2p states provide orbital currents and magnetic moments, and 1s1s–2p2p superpositions produce radiative dipole transitions. The treatment is then extended from individual atoms to a hydrogen gas, where polarization and magnetization generate the macroscopic fields DD and HH. The resulting framework shows how quantum sources and material response fit consistently within Maxwell electrodynamics without claiming that quantum mechanics alone uniquely derives the complete electromagnetic field dynamics.
This paper extends the preceding Schrödinger-based analysis to the relativistic Dirac framework and examines how gauge covariance, spin magnetism, and quantum-vacuum effects deepen the connection between quantum theory and Maxwell electrodynamics. Local phase invariance of the Dirac equation yields the electromagnetic field tensor, while the conserved Dirac four-current provides a covariant source for Maxwell’s equations. The Pauli limit recovers g=2g=2 and the Bohr magneton, and the Gordon decomposition separates convection and spin-magnetization currents. The Foldy–Wouthuysen expansion further connects relativistic quantum dynamics to spin–orbit and Darwin terms involving the electromagnetic fields. Hydrogen fine-structure and Darwin-shift benchmarks are evaluated quantitatively, while the Dirac velocity operator and Zitterbewegung scale are compared with the magnetic dipole length appearing in the SQC/MSC M1 formulation. The analysis is then extended to the quantum vacuum through the Schwinger critical fields and Euler–Heisenberg effective theory, including vacuum birefringence. The paper distinguishes structural consequences of relativistic quantum mechanics from electromagnetic field dynamics that must still be specified independently.
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.
Electrically charged particles such as Electrons and Protons carry electric, E, and magnetic, B, fields. In addition to these fields, Quantum Mechanics (QM) endows these particles with an `arcane and spooky' field --- the wavefunction. This wavefunction of QM is not only assumed to be separate but distinct from the electromagnetic field. We herein upend this view by demonstrating otherwise. That is, we demonstrate that the four components of the Dirac wavefunction, can be shown to not only be an intimate, but, a direct function of the electromagnetic field carried by the particle in question. Insofar as unity, depth in our understanding and insight into both Dirac and Maxwell's equations as major pillars of Modern Physics, we believe that this work may very well inch us one-step-closer to the truth.
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 coordinate and momentum spinors. 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 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. Quantum mechanics describes interactions using complex wave functions that cannot be observed directly; only upon measurement do they translate into probabilities expressed as the squares of the magnitudes 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 partial derivatives on coordinates. Section 3 presents a version of quantum theory in spinor coordinate space. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. An equation of motion for the electron is proposed that is relativistically invariant. 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. 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. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations are presented, which can serve as a model of the graviton. Section 5 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.
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 coordinate and momentum spinors. 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 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 partial 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. 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. Section 4 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.