Unification of Maxwell’s and Dirac’s equations, of electrons and light

7.1 Unification of Maxwell’s and Dirac’s equations, of electrons and light

The appearance of these so-called “wave equations” for some seemed to show a kinship between the photon and the electron; this led to attempts to obtain a common representation for both kind of particles (waves) [199 , 200 , 198 , 286 ]. Some of the motivation for these papers came from formal considerations, i.e., the wish to replace the not yet well-understood new spinor representation of the Lorentz group by the old tensor representation; a working knowledge about non-commuting objects like matrices was not yet available to everyone:

“There are probably readers who will share the present writer’s feeling that the methods of non-commutative algebra are harder to follow, and certainly much more difficult to invent, than are operations of types long familiar to analysis.” ([44 ], p. 654)

More interesting is Frenkel’s remark about Darwin’s presentation of Dirac’s equations in a form analogous to Maxwell’s equations [44 ]:

“This relation between the wave-mechanical equations of a ‘quantum of electricity’ and the electromagnetic field equations, which may be looked at as wave-mechanical equations for photons, ought to have a fundamental physical meaning. Therefore, I do not think it is superfluous to win the wave equation of the electron as a generalisation of Maxwell’s equations.”²²¹ ([140 ], p. 357)

H. T. Flint of King’s College in London aimed at describing the electron in a Maxwell-like way within a five-dimensional approach. He saw two “unsatisfactory points” in Dirac’s approach, the introduction of the operator $-h--∂k − eϕk 2πi∂x$ , and the mass term $mc$ . In order to mend these thin spots he wrote down two Maxwell’s equations,

where $λμ F$ and $λμ G$ are two asymmetric field tensors, and $-∂ψ Jλ = ∂xλ$ and $-∂𝜃 Lλ = ∂xλ$ are two current vectors ( $μ,λ,ν = 1,2,...,5$ ). $ψ$ is the electron’s wave function; although not provided by Flint, the interpretation of $𝜃$ points to a second kind of wave function. Despite the plentitude of variables introduced, Flint’s result was meagre; his second order wave equation contained the correct mass term and two new terms; he did not write up all the equations occurring [129 ]. His approach for embedding wave mechanics into a Maxwell-like was continued in further papers, in part in collaboration with J. W. Fisher; to them it appeared

“unnecessary to introduce in any arbitrary way terms and operators to account for quantum phenomena.” ([128 ], p. 653; [127 ])

By adding four spinor equations at his choosing to Dirac’s equation, Wisniewski in Poland arrived at a “system of equations similar to Maxwell’s”. His conclusion sounds a bit strange:

“These equations may be interpreted as equations for the electromagnetic field in an electron gas whose elements are electric and magnetic dipoles.” [388 ]

In this context, another unorthodox suggestion was put forward by A. Anderson who saw matter and radiation as two phases of the same substrate:

“We conclude that, under sufficiently large pressure, even at absolute zero normal matter and black-body radiation (gas of light quanta) become identical in every sense. Electrons and protons cannot be distinguished from quanta of light, gas pressure not from radiation pressure.”²²²

Anderson somehow sensed that charge conservation was in his way; he meddled through by either assuming neutral matter, i.e., a mixture of electrons and protons, or by raising doubt as to “whether the usual quanta of light are strictly electrically neutral” ([3 ], p. 441).

One of the German theorists trying to keep up with wave mechanics was Gustav Mie. He tried to reformulate electrodynamics into a Schrödinger-type equation and arrived at a linear, homogeneous wave equation of the Klein–Gordon-type for the $ψ$ -function on the continuum of the components of the electromagnetic vector potential [232 ]. Heisenberg and Pauli, in their paper on the quantum dynamics of wave fields, although acknowledging Mie’s theory as an attempt to establish the classical side for the application of the correspondence principle, criticised it as a formal scheme not yet practically applicable [158 ].