On the mathematical necessity of the co-existence of electromagnetic fields of the non-standard polarity with the standard electromagnetic fields in the case of the existence of Dirac monopoles

In this brief note we adduce the mathematical rationale of the necessity of the existence of the electromagnetic field of the non-standard polarity if the magnetic charges (Dirac monopoles) exist in reality.


INTRODUCTION
The present paper was inspired by our careful perusal of newly published brilliant work [1] about observation of Dirac monopoles which were identified, in both experiments and matching numerical simulations. Nevertheless the authors of [1] note that Maxwell´s equations refer neither to magnetic monopoles nor to the magnetic currents that arise from their motion. Indeed, the presence of magnetic charges implies the existence of a vectorial magnetic field (polar vectors) rather than a pseudo-vectorial one. But it is well known that from the system of complete Maxwell equations (CME) (i.e., equations with non-zero density of an electric charge and non-zero density of a conduction current ) it follows that any solution and of this system (1) must be as a polar vector and as an axial vector in the microscopic consideration. One can satisfy oneself that this fact directly follows from the equation ∇ • = 4 and from other equations of CME. Actually, seeing that is a scalar rather than pseudo-scalar with respect to the spatial inversion transformation ( -system → ′ -system) for coordinates and the operator ∇ changes its sign, the vector is a polar vector because it behaves as → ′ = − , and thereagainst the vector is an axial vector because it behaves as → ′ = . Let us call this kind of the polarity of the electromagnetic fields as a standard polarity.
If one follows this claim, then it is obvious that so called free Maxwell equations (FME) (i.e., equations with zero density of an electric charge and zero density of a conduction current ) must, correspondingly, have only solutions of such polarity.
However, if we consider FME from a purely formal mathematical point of view, then this condition ( is a polar vector and is an axial vector) is not obligatory. But have we a right to do such consideration? The point is that one of the authors of the present work argued [2] that a rigorous application of Gauss' law to constructing CME leads to realizing the fact that FME (2) and, correspondingly, so-called free electric and magnetic fields are not a consequence of CME: one just may postulate them. Jefimenko ([3] p. 17), nay, claims that sourceless (free) fields are impossible.

THE MATHEMATICAL POINT OF VIEW
Thus in the present work we show that the system of equations (2) admits nonstandard solutions regarding the polarity of the vectors and . In other words we show that such nonstandard solutions of FME exist, when the magnetic field can be a polar vector as well as an axial vector, and the electric field in turn can be an axial vector as well as a polar vector. Of course, we omit the consideration of the trivial case when one can make the following interchange in FME (2): → ′ , → ′ .
So let us approach a solution of the problem of the estimation and in the system (2) purely mathematically: One can represent some solutions of Eq. (2) as follows: where 1 , 2 and Ω are constant and where the vectors and are the solution of the system M a r c h 0 2 , 2 0 1 5 such that is a polar vector and is an axial vector (c is the velocity of light in vacuum). This solution expressed by components (Cartesian and spherical ones) is: where is a dimension constant = 1 2 5 2 −1 ; and , , are the orts of the spherical coordinate system. From the whole class of solutions of the system (4) we have chosen the simplest non-trivial one. The detailed way of obtaining this solution one can find in [4] or [5].
Let us consider the solution (3) for some different constants 1 and 2 .
Let, for example, 1 = 1 and 2 = 0. In this case from Eqs. (3) we have the fields with the well-known standard polarity and : = sin Ω and = cos Ω .
Now let us consider the case when in Eqs. (3) 1 = 0 and 2 = 1, in this case we have the electromagnetic fields (let us call this kind of fields the electromagnetic fields of a non-standard polarity) and : We see now that this solution is already unusual in the sense of the polarity: is an axial vector and is a polar one! It is easy to satisfy oneself that the electromagnetic wave formed by this field spreads in the opposite direction with respect to the direction of the wave spreading by S-fields (7). Actually and the energy density M a r c h 0 2 , 2 0 1 5 Here, however, a certain question can arise: how are these "mathematical" fields rightful physically? Can one apply, for example, the definitions of the energy density and the energy-flux vector to fields of the non-standard polarity? It is easy to show that one can apply the mentioned definitions to these non-standard fields if the discussed fields are solutions of FME. Indeed, let us multiply both sides of ∇ × = − 1 by and both sides of ∇ × = 1 by (taking into account that and are solutions of these equations) and combine the resultant equations. Then we have Using the well-known formula of the vector analysis, we rewrite this relation in the form or after multiplying this relation by 4 Now we show that the vector = 4 × (11) and the relation 2 + 2 8 are the energy-flux vector and the energy density of the field (8). First we integrate (10) over a volume and apply Gauss' theorem to the term on the right. Then we obtain If the integral extends over all space, then the surface integral vanishes because the field is zero at infinity. 2 Then (12) becomes Thus, for the closed system consisting of the electromagnetic field of a nonstandard polarity, the quantity with dimensions of energy in brackets in this equation is conserved. Therefore we can call the quantity 2 + 2 8 the energy density of the electromagnetic field (8).
Now if we integrate over any finite volume, then the surface integral in (12) generally does not vanish, so that we can write the equation in the form 1 Recall that in this case = . 2 one can see this from Eqs. On the left the change in the total energy of field per unit time stands. Therefore the integral • must be interpreted as the flux of field energy across the surface bounding the given volume, so that the vector is a density of this flux (the Poynting vector) that is the amount of the field energy passing through unit area of the surface in unit time.
Thus we can see that expressions for the energy density and the Poynting vector were obtained without subjecting the field to any polarity conditions. We are not going to consider in detail the N-solution of FME, instead, we emphasize that the waves formed by this solution spread in the opposite direction with respect to the waves formed by the S-solution (7). Absolute magnitudes of the Poynting vectors for S and N coincide, while the distribution of the energy density is different. Now, referring to the findings of [2] namely that free electric field is not a consequence of complete Maxwell equations (CME), and remembering the phrase by Morris Kline from [6], namely "What is especially remarkable about electromagnetic wavesand reminiscent of gravitationis that we have not the slightest physical knowledge of what electromagnetic waves are. Only mathematics vouches for their existence, and only mathematics enabled engineers to invent the marvels of radio and television... Mathematicians and theoretical physicists speak of fieldsthe gravitational field, the electromagnetic field ..., and othersas though they were material waves which spread out into space and exert their effects somewhat as water waves pound against ships and shores. But these fields are fictions. We know nothing of their physical nature. They are only distantly related to observables such as sensations of light, sound, motions of objects, and the now perhaps too familiar radio and television ... But by formulating mathematically the laws of these fictional fields, which have no apparent counterparts in reality, and by deducing consequences of these laws, we obtain conclusions, which when suitably interpreted in physical terms can be checked against sense perceptions. ", we can conclude the following.
The generally accepted so called free Maxwell equations (2)  where is the magnetic current density, is the magnetic charges density and 1 4 may be some magnetic displacement current density.
We would like to mention here the system of equations describing the electromagnetic fields granting the possible existence of magnetic charges which was proposed by Griffiths in his well-known textbook [7] 3 : where represents the magnetic charges density, and the electric charges density; would be the magnetic current density, and the electric current density. One can see that if we believe that has to be a true scalar quantity rather than a pseudo-scalar one and taking into account that is the true scalar quantity by definition one can conclude that in this case Eqs. (Grif) in a certain sense are inconsistent equations from the viewpoint of the vector analysis. Let us explain this statement: the system (Grif) must have solutions without the definite polarity when = + and = + , where and are axial vectors and and are polar ones. But this system cannot have the simultaneous solutions of this type = and = for example.
So we have to conclude that in the case of the existence of magnetic charges which are the true scalars the total electromagnetic field must be described by three independent systems of equations describing electric and magnetic fields of different nature and origin. One of them corresponds to the free field of the standard polarity as well as to the free field of the non-standard polarity: Note that the set of the systems of equations (FME), (CMEN) and (CMES) admits more solutions than the system of equations (Grif). Furthermore, one can add that the simultaneous solution of these systems 4 begs for the electromagnetic field of the non-standard polarity to co-exist with the standard electromagnetic fields in the case of the existence of Dirac monopoles.
In conclusion, we would like to quote the words of Louis de Broglie: "Only the motions of elements localized in space and time are really physical" [8] he said.