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The Free Photon Wave Function's Gauge-Invariant, Lorentz-Covariant Antisymmetric-Tensor Form

Steven Kenneth Kauffmann

Published Apr 1, 2019• 0 citations
OpenAlex

Abstract

If a free photon's wave function is taken to be a four-vector function of its space-time coordinates that has vanishing four-divergence (the Lorentz condition), it isn't uniquely determined by the free-photon Schroedinger equation. This gauge indeterminacy can be eliminated by taking that wave function to be a three-vector function of its space-time coordinates -- at the expense of its Lorentz-covariant form. These conflicts are resolved by taking a free photon's wave function to be an antisymmetric-tensor function of its space-time coordinates which has vanishing four-divergence and also satisfies the Lorentz-covariant cyclic Gauss-Faraday equation that is satisfied by all antisymmetric-tensor real-valued electromagnetic fields. It is shown that for every source-free antisymmetric-tensor real-valued electromagnetic field, there exists a corresponding free-photon antisymmetric-tensor complex-valued wave function.

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If a free photon's wave function is taken to be a four-vector function of its space-time coordinates that has vanishing four-divergence (the Lorentz condition), it isn't uniquely determined by the free-photon Schroedinger equation. This gauge indeterminacy can be eliminated by taking that wave function to be a three-vector function of its space-time coordinates -- at the expense of its Lorentz-covariant form. These conflicts are resolved by taking a free photon's wave function to be an antisymmetric-tensor function of its space-time coordinates which has vanishing four-divergence and also satisfies the Lorentz-covariant cyclic Gauss-Faraday equation that is satisfied by all antisymmetric-tensor real-valued electromagnetic fields. It is shown that for every source-free antisymmetric-tensor real-valued electromagnetic field, there exists a corresponding free-photon antisymmetric-tensor complex-valued wave function.
This tutorial begins with the relationship of source-free classical electromagnetism to ultra-relativistic free-photon quantum mechanics. The linear transformation of the source-free classical-electromagnetic real-valued transverse vector potential to its corresponding free-photon Schroedinger-equation complex-valued transverse vector wave function is obtained. It is then pointed out that despite the free-photon Klein-Gordon equation's being formally identical to the source-free classical-electromagnetic vector-potential wave equation, it yields not only free-photon Schroedinger-equation wave functions but also their complex conjugates, which don't satisfy the free-photon Schroedinger equation. This is a consequence of admitting complex-valued solutions of the Klein-Gordon equation -- of course only its real-valued solutions apply to the classical vector potential. It is pointed out that solutions of the free-particle Dirac equation likewise occur in conjugate pairs, and that its Hamiltonian operator implies a variety of unphysical consequences, e.g., any Dirac free particle's speed is that of light times the square root of three.
The classical source-free electric field is a polar transverse-vector field, but the classical magnetic field is an axial transverse-vector field. A derivative-related linear transformation (which is its own inverse) of the classical axial magnetic field in fact produces an alternate polar transverse-vector representation of the classical magnetic field. The classical source-free complex-valued electromagnetic polar transverse-vector field whose real part is the classical source-free polar electric field, and whose imaginary part is the alternate polar representation of the classical source-free magnetic field, turns out to satisfy the time-dependent Schroedinger equation whose Hamiltonian operator is that of the free photon. That classical source-free complex-valued electromagnetic polar transverse-vector field can, moreover, be slightly linearly modified to become the normalized wave function of the free photon.
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.
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.
Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.