Source-Free Classical Electromagnetism, the Free-Photon Schroedinger Equation, and the Unphysical Conjugate-Pair Solutions of the Klein-Gordon and Dirac Equations
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.
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.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
Classical equations of motion that are first-order in time and conserve energy can only be quantized after their variables have been transformed to canonical ones, i.e., variables in which the energy is the system's Hamiltonian. The source-free version of Maxwell's equations is purely dynamical, first-order in time and has a well-defined nonnegative conserved field energy, but is decidedly noncanonical, an issue that has attracted little attention. The opposite parities of the electric and magnetic fields and consequent curl operations that typify Maxwell's equations are especially at odds with their being canonical fields. Transformation of the magnetic field into the transverse part of the vector potential helps but is not sufficient; further simple nonnegative symmetric integral transforms, which commute with all differential operators, are needed for both fields; such transforms also supplant the curls in the equations of motion. The canonical replacements of the source-free electromagnetic fields remain transverse-vector fields, but are more diffuse than their predecessors. Combined as the real and imaginary parts of a complex field, the canonical fields prove to be the transverse-vector wave function of a time-dependent Schroedinger equation whose Hamiltonian operator is the quantization of the free photon's square-root relativistic energy. Thus proper quantization of the source-free Maxwell equations is identical to second quantization of free photons that have normal square-root energy. There is no physical reason why first and second quantization of any relativistic free particle ought not to proceed in precise parallel, utilizing the square-root Hamiltonian operator. This natural procedure leaves no role for the completely artificial Klein-Gordon and Dirac equations, as accords with their grossly unphysical properties.
It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac's key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schroedinger equation for a solitary particle is laid bare, as is the invalidity of the standard "proof" that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don't exist for a massless particle. Instead, Maxwell's equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schroedinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions' relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.
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.
Classical equations of motion that are first-order in time and conserve energy can only be quantized after their variables have been transformed to canonical ones, i.e., variables in which the energy is the system's Hamiltonian. The source-free version of Maxwell's equations is purely dynamical, first-order in time and has well-defined nonnegative conserved field energy but is decidedly noncanonical. That should long ago have made source-free Maxwell equation canonical Hamiltonization a research priority, and afterward, standard textbook fare, but textbooks seem unaware of the issue. The opposite parities of the electric and magnetic fields and consequent curl operations that typify Maxwell's equations are especially at odds with their being canonical fields. Transformation of the magnetic field into the transverse part of the vector potential helps but is not sufficient; further simple nonnegative symmetric integral transforms, which commute with all differential operators, are needed for both fields; such transforms also supplant the curls in the equations of motion. The canonical replacements of the source-free electromagnetic fields remain transverse-vector fields, but are more diffuse than their predecessors, albeit less diffuse than the transverse vector potential. Combined as the real and imaginary parts of a complex field, the canonical fields prove to be the transverse-vector wave function of a time-dependent Schrodinger equation whose Hamiltonian operator is the quantization of the free photon's square-root relativistic energy. Thus proper quantization of the source-free Maxwell equations is identical to second quantization of free photons that have normal square-root energy. There is no physical reason why first and second quantization of any relativistic free particle ought not to proceed in precise parallel, utilizing the square-root Hamiltonian operator. This natural procedure leaves no role for the completely artificial Klein-Gordon and Dirac equations, as accords with their grossly unphysical properties.
It was pointed out in a previous paper that although neither the Klein-Gordon equation nor the Dirac Hamiltonian produces sound solitary free-particle relativistic quantum mechanics, the natural square-root relativistic Hamiltonian for a nonzero-mass free particle does achieve this. Failures of the Klein-Gordon and Dirac theories are reviewed: the solitary Dirac free particle has, inter alia, an invariant speed well in excess of c and staggering spontaneous Compton acceleration, but no pathologies whatsoever arise from the square-root relativistic Hamiltonian. Dirac’s key misapprehension of the underlying four-vector character of the time-dependent, configuration-representation Schrodinger equation for a solitary particle is laid bare, as is the invalidity of the standard “proof” that the nonrelativistic limit of the Dirac equation is the Pauli equation. Lorentz boosts from the particle rest frame point uniquely to the square-root Hamiltonian, but these don’t exist for a massless particle. Instead, Maxwell’s equations are dissected in spatial Fourier transform to separate nondynamical longitudinal from dynamical transverse field degrees of freedom. Upon their decoupling in the absence of sources, the transverse field components are seen to obey two identical time-dependent Schrodinger equations (owing to two linear polarizations), which have the massless free-particle diagonalized square-root Hamiltonian. Those fields are readily modified to conform to the attributes of solitary-photon wave functions. The wave functions’ relations to the potentials in radiation gauge are also worked out. The exercise is then repeated without the considerable benefit of the spatial Fourier transform.