5.4  Symmetrization of the Maxwell Equations

Albert Einstein opens his work of 1916 «A new formal interpretation of electrodynamic equations of Maxwell» with the following estimation of the accepted form of the record of Maxwell's equations:

The covariant-theoretic interpretation of the equations of electrodynamics goes back to Minkowski. It can be characterized in the following way. The components of an electromagnetic field form a 6-vector (antisymmetric tensor of the 2nd rank). It is compared to the second 6-vector, dual to the first, which in case of the special relativity differs from the first one not by the values of components, but only by how these components are compared to four coordinate axes. Both systems of equations of Maxwell can be received if to assign the divergence of one of these 6-vectors equal to zero, and the divergence of the second equal to the 4-vector of electric current. [33]

In such formulation the equation of Maxwell become:

Such «simplified» designations were used by Wolfgang Pauli in §28. Covariance of the basic equations of the electronic theory [1]. Here, the way of writing of a 4-vector (density) of electric current (speed) is additionally changed from sⁱ=ρ₀uⁱ to ρ₀U, in full conformity to the way accepted in 5.2 Maxwell Equations single out Hyperbolic Motion of Field Sources.

Einstein, actually, rediscovers in §1. The equations of a field of the quoted work [33] the well known to Minkowski form of record of Maxwell's «magnetic» equations, replaces the equations  div*F=0  to  rotF=0  and writes down the equations of Maxwell as:

Pauli gave us the following description of the situation existing then around the «natural» interpretation of the «magnetic» equations of Maxwell and (physically or mathematically) of the preferable form of their record:

As is well known, in ordinary space E is a polar, and H – an axial vector, but not the other way. We therefore should count the description of an electromagnetic field with the help tensor F natural, and the description of a field with the help tensor *F – artificial. Minkowski (Minkowski I, see [2.I]) had both records of the equations of a field. First of them [rotF=0], in many cases, in particular, in the general relativity, more evident and convenient, subsequently appeared forgotten and, for example, was not used by Sommerfeld [4]. The person who paid attention to it again was Einstein [33] in 1916. [1,§28]

Let us write down the «symmetrized» equations of Maxwell just as it was made in 5.2 in the formulas (M.div) and (M.rot), but now, – with the help of a 4-vector potential A and *A – antisymmetric pseudo-tensor of the 3rd rank, dual to the 4-vector of potential A:

At that, the equations remain fair (together with the «dual» to them ones), that connect a pair of potentials of a field (A,*A) with the pair of tensors of Maxwell (F,*F), as well as the for a long time well known «mysterious» pair of forms of record of the «magnetic» equations (in the second column):

The «mysteriousness» of the fact of existence of this very pair of forms of record of Maxwell's magnetic equations rotF=0 and div*F=0, first of all, in their seeming redundancy. If the pseudo-tensor *F, really, appears only into the equation div*F=0, and this equation is physically «equivalent» (by its action on the appropriate components of a field) to the equation rotF=0, then it is possible to limit ourselves to the choice of Einstein stated by him in [33]. Accordingly, – to «rightly» wave away from any searches of not yet revealed physical filling of the observable «duplication» of the equation rotF=0 also by the equation div*F=0. Even more so as both forms of record of Maxwell's «magnetic» equations result, finally, in the same system of equations for the components of an electromagnetic field:

where: differential 3-operators rot and div force on the components of the electric E and magnetic H fields.

More precisely, – in both cases there appear exactly four independent equations with regard to the components of fields E and H, which can be written down in the coincident 3-vector form (7). The operator of 4-divergence, forcing on antisymmetric pseudo-tensor of the 2nd rank *F, transforms it into the pseudo-tensor of the 1st rank, that has four components. In its turn, – the operator of 4-rotor, forcing on antisymmetric tensor of the 2nd rank F, transforms it into an antisymmetric tensor of the 3rd rank, that also has four independent components. Both these «fours» of the equations, presented by tensors of the 1st and the 3rd rank, allow the uniform 3-vector form of record (7). Such coincidence of number of the independent components of tensors of the 1st and the 3rd rank in 4-dimensional space, occurs only for 4-tensors of the 3rd rank, antisymmetric in all their pairs of indexes.

After Minkowski, mostly, both the acuity of experiencing the «mysteriousness» of the fact of existing of two forms of record of Maxwell's «magnetic» equations at once, and the persistence in efforts on revealing physical as well as heuristic value of this fact were lost. To the forefront were brought the «redundancy» and the necessity to choose one of the record forms as the basic one, (more adequate) reflecting a physical nature of a field better.

For the following generations of physicists the basic source, that was feeding their steady interest in the search of true («symmetrized») equations of a field, laid in the absolutely other place. It was the more easy for seizing and inheriting, more easily transmitted during «physical contacts», assumption-forefeeling about the «magnetic» equations of a field «being crippled» by the absence in them of magnetic monopoles as the sources of a magnetic field.

The imitation pattern was the second pair of Maxwell's equations – the «electric» equations divF=ρ₀U, which in the regions of space «deprived» of electric charges, can be simplified until homogeneous equations relative to the fields E and H:

ATTENTION! A physical postulate about the existence of such regions of space, «deprived» of vacuum (ether) sources of a field, – is only historically caused HYPOTHESIS, that simplifies extremely the physical situation. On the basis of such extreme simplification the classical theory of a free field of radiation was created. This very simplification of an initial theoretical model is responsible for the problems emerging during the description of radiation events and those of interaction of this radiation with the matter. Another, «dual» to it, simplifying postulate was the *HYPOTHESIS about a static character (absence of motion) of field sources of an electron at rest of Lorentz's electronic theory. This PAIR of the simplifying postulates, that were transformed in the course of time into stable «axiomatic» dogmas, – has resulted in «unremovable» gap between the classical field theory received in such a way and the facts observed during physical experiments, – has resulted in the whole century of domination of the phenomenology of the quantum theory. The Program of the Subquantum Field Theory – SubQFT – deprives this so long trusted pair of postulates of their status of the unconditional fundamental physical bases of the field theory. It is impossible to build the high-grade Field Theory within the framework of this pair of dogmas. Either we stay within the framework of quantum phenomenology or – build SubQFT. This is a fundamentally important point, that one has to keep in mind and to considered while reading this article, as well as other articles on this site, during each act of an independent analysis and estimation of the material offered here.

These very in a determined way simplified Maxwell's equations for a free field in the regions of space, «deprived» of sources, represented there by the equations (7) and (8), very evidently and successfully showed the symmetry of the fields E and H in the record of the «magnetic» (7) and homogeneous «electric» (8) equations of a field. Proceeding, mainly, from this symmetry peculiar to a man-made free field of radiation of electronic theory of Lorentz, the quantum «program» of Dirac became popular, that aimed at searching magnetic monopoles and the «restoration» of the «magnetic» equations up to non-uniform (in the image and likeness of «electric») ones. The classical variant of an attempt of this «program's» realization we find in the solution offered to the Problem 4.21. from «Problem book in Relativity and Gravitation». [32]

where: K^μ is «a conduction current of a magnetic charge» (taken with the minus sign); μ₀ is (Lorentz-invariant) scalar density of «a magnetic charge» (or of magnetic monopoles).

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