# A BRIEF SURVEY OF THE VARIOUS FIELD EQUATIONS SINCE 1873

The terminology of quantum field theory (QFT) is misleading; the 'field' referred to is not a field, not the measureable kind of E- or H-field at any rate, but it is defined as a field while in reality related to classical 4-potentials, i.e. voltages. When compared to SFT, a true 'field' theory, we need to examine what a field is at the atomic level compared to dipole and coil measurements. It is instructive to survey the main equations used by physicists since Maxwell's equations were formulated in 1873. They describe the macroscopic E- and H-fields, and their associated charges and currents measured in experiments by Coulomb, Faraday, Ampere, Biot, and Savart from 1785 onwards. Several EM wave equations were derived including decoupled forms where the E- or H-fields appear in isolation; Maxwell's equations were specialized to various applications e.g. for quasistationary or radiation conditions. Hertz's potentials introduced a mixed-field substitution in terms of a Lagrangian or energy density for solving via integrals over radiation surfaces where infinite regions needed to be considered; these are known as the Hertzian vector and scalar potential wave equations.

Following theoretical and experimental demonstrations by Planck and Einstein of the existence of a quantum physics, there was a failure by physicists to find a mathematics based directly on Maxwell's equations that applied to the electron's motion in the atom. In 1926 SchrÃ¶dinger used energy conservation to obtain a quantum mechanical equation in a variable called the wave function that accurately described single-electron states such as the hydrogen atom. The wave function depended on a Hamiltonian function and the total energy of an atomic system, and was compatible with Hertz's potential formulation. The wave function depends on the sum of the squares of E- and H-fields as is seen by examining the energy density function of the electromagnetic field. In 1928 Dirac realising the wave functions were not relativistic sought a set of equations incorporating Einstein's relativity. Dirac's equations were described in terms of two 'fields', the so-called Dirac fields, and were described as 'field equations of motion'. The term "Dirac's two wave equations" was also used. Like SchrÃ¶dinger's equation, there was a mathematical smearing of the SFT fields as we shall see. The problem was now 'wave-like' instead of two uncluttered fields and Heisenberg formulated the uncertainty principle. The underlying SFT centre-of-motion fields had been lost in the potential equations. By the time the equations governing the weak and strong nuclear forces were found using modern versions of QFT, quantum electrodynamics (QED) and quantum chromodynamics (QCD), any fields, macroscopic or atomic, were a long-forgotten reality.

But why can't the potentials give us a correct picture of the E- and H-fields at atomic levels? After all we have Hertz's potential equations that give a correspondence between classical potentials and fields? The question is: do Maxwell's E- and H-fields determined between point-charges exist within the nanoscopic domain of the atom? Recently it has been demonstrated by EMSFT for the hydrogen atom that these E- and H-field forms are not applicable to sub-atomic charges. Why? The analytic solutions obtained from EMSFT for the hydrogen atom are validated by the known spectroscopy where we determine the atomic fields between centres of motion and not between charge points. This issue is at the crux of why classical vector and scalar potentials cannot obtain the correct solution; the macroscopic fields of Coulomb and Biot-Savart do not hold at atomic dimensions; the fields caused by the motions of the photons inside the atom are not correctly formulated point-charge to point-charge. The classical potentials cannot give us the correct answer, because the classical field theory as we have long known is wrong. The potential solution was in a sense chasing its tail; the classical fields and potentials are incorrect over atomic dimensions as Heisenberg had correctly determined. Reality wasn't in error; but classical field theory was and thus also quantum field theory. Coulomb's, and Biot's and Savart's famous E- and H-field forms apply to macroscopic phenomena not to atomic systems. The photons inside atoms in fact stream between electrons and nucleons. These photonic streams are not ubiquitous nor continuous, they are discrete and discontinuous. They behave like Dirac delta functions, an interesting fact in terms of their role in solving Maxwell's equations for self fields (see below on numerical methods FEM vs FDM).

Another term needs clarification: spinor. In Dirac's formulation the resulting complex matrices were capable of synthesis into various Dirac "bispinors". These are adjointly coupled 2 x 2 'unit' spinors (determinant = 1) that have a left- or right-handed helicity associated with them. In the chiral representation of Dirac's equation, the terms are 4 x 4 matrices comprised of Pauli spinors. In SFT, the term 'spinor' is used for the motions of the E- and H-fields, and for the motions of the particles, such as the electron or proton. Everything in the mathematics of SFT, both particles and their (particulate) fields, move as rotating vectors; like QFT for the atom there are two spinors, or four variables per subatomic particle.

In the following, the terms 'wave equation' and 'vector and scalar potentials' are applied to all quantum field theories that follow the heritage of Dirac's wave equations up to and including today's standard model. In this aspect SFT is indeed the only true 'field' theory, not only because it uses the term 'field' in an historically correct sense but further it applies these fields not between charge points, but (instantaneous) centres of motion.