London-Proca-Hirsch Equations for Electrodynamics of Superconductors on Cantor Sets

. In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. However, so far there is no derivation of equations for electrodynamics of superconductor on Cantor sets. Therefore, in this paper I present for the first time a derivation of London-Proca-Hirsch equations on Cantor sets. The name of London-Proca-Hirsch is proposed because the equations were based on modifying Proca and London-Hirsch’s theory of electrodynamics of superconductor. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, I suggest that the proposed London-Proca-Hirsch equations on Cantor sets can describe electromagnetic of fractal superconductors. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too


INTRODUCTION
According to J.E. Hirsch, from the outset of superconductivity research it was assumed that no electrostatic fields could exist inside superconductors and this assumption was incorporated into conventional London electrodynamics. [2] Hirsch suggests that there are difficulties with the two London equations. To summarize, London's equations together with Maxwell's equations lead to unphysical predictions. [1] Hirsch also propose a new model for electrodynamics for superconductors. [1] [2] The present paper is intended to be a follow-up paper of our four recent papers: one paper reviews Shpenkov's interpretation of classical wave equation and its role to explain periodic table of elements and other phenomena [11], and the second one presents a derivation of GravitoElectroMagnetic Proca equations in fractional space [12], the third one presents an outline of cosmology based on the concept of fractal vibrating string [13], and the fourth one presents a derivation of Proca equations on Cantor sets [14].
In this regard, in a recent paper Yang Zhao et al. derived Maxwell equations on Cantor s ets from the local fractional vector calculus. [3] It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. However, so far there is no derivation of equations for electrodynamics of superconductor on Cantor sets. Therefore, in this paper I present for the first time a derivation of London-Proca-Hirsch equations on Cantor sets. The name of London-Proca-Hirsch is proposed because the equations were based on modifying London equations, Proca equations and Hirsch's theory of electrodynamics of superconductor. table of elements and other phenomena [11], and the second one presents a derivation of GravitoElectroMagnetic Proca equations in fractional space [12], the third one presents an outline of cosmology based on the concept of fractal vibrating string [13], and the fourth one presents a derivation of Proca equations on Cantor sets [14].
In this regard, in a recent paper Yang Zhao  magnetic fields. However, so far there is no derivation of equations for electrodynamics of superconductor on Cantor sets. Therefore, in this paper I present for the first time a derivation of London-Proca-Hirsch equations on Cantor sets. The name of London-Proca-Hirsch is proposed because the equations were based on modifying London equations, Proca equations and Hirsch's theory of electrodynamics of superconductor.
Therefore the aim of the present paper is to propose a combined version of London-Proca-Hirsch model for electrodynamics of superconductor. Then I extend further this proposed model for electrodynamics of superconductor on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor [4]- [8], I suggest that the proposed London-Proca-Hirsch equations on Cantor sets can describe electromagnetic of fractal superconductors. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

HIRSCH'S MODEL TO REVISE LONDON'S EQUATIONS
According to J.E. Hirsch, from the outset of superconductivity research it was assumed that no electrostatic fields could exist inside superconductors and this assumption was incorporated into conventional London electrodynamics. [2] Hirsch suggests that there are difficulties with the two London equations. Therefore he concludes that London's equations together with Maxwell's equations lead to unphysical predictions. [1] However he still uses four-vectors J and A according to Maxwell's equations: Therefore Hirsch proposes a new fundamental equation for electrodynamics for superconductors as follows: [1] -And d'Alembertian operator is defined as: [1] 2 ETET Volume 4 Then he proposes the following equations: [1] where F is the usual electromagnetic field tensor and F 0 is the field tensor with entries and 0 from respectively when expressed in the reference frame at rest with respect to the ions.
In the meantime, it is known that Proca equations can also be used to described electrodynamics of superconductors, see [4]- [8]. The difference between Proca and Maxwell equations is that Maxwell equations and Lagrangian are based on the hypothesis that the photon has zero mass, but the Proca's Lagrangian is obtained by adding mass term to Maxwell's Lagrangian. [17] Therefore, the Proca equation can be written as follows: [17] Similarly, according to Kruglov [15] the Proca equation for a free particle processing the mass m can be written as follows: (6) Now, the similarity between equations (1) and (6b) are remarkable with exception that equation (1) is in quadratic form. Therefore I propose to consider a modified form of Hirsch's model as follows: The relevance of the proposed new equations in lieu of (4)-(5) should be verified by experiments with superconductors [16]. For convenience, the equations (8a)-(8b) can be given a name: London-Proca-Hirsch equations.

LONDON-PROCA-HIRSCH EQUATIONS ON CANTOR SETS
It can be shown that Proca equations can be derived from first principles [6], and also that Proca equations may have link with Klein-Gordon equation [7]. However, in this paper I will not attempt to re-derive Proca equations. Instead, I will use Proca equations as described in [6]. Then I will derive the London-Proca-Hirsch equations on Cantor Sets, in accordance with Zhao et al.'s approach as outlined above [3].

ETET Volume 4
Therefore, by using the definitions in equations (14)- (17), we can arrive at Proca equations on Cantor sets from (18) through (23), as follows: Since according to Blackledge, the Proca equations can be viewed as a unified wavefield model of electromagnetic phenomena [7], therefore we can also regard the Proca equations on Cantor sets as a further generalization of Blackledge's unified wavefield model. Now, having defined Proca equations on Cantor Sets, we are ready to write down London-Proca-Hirsch on Cantor sets using the same definition, as follows: As far as I know, the above London-Proca-Hirsch equations on Cantor Sets have never been presented elsewhere before. Provided the above equations can be verified with experiments, they can be used to describe electrodynamics of fractal superconductors on Cantor sets.
As a last note, it seems interesting to remark here that Kruglov [15] has derived a square-root of Proca equations as a possible model for hadron mass spectrum, therefore perhaps equations (32)-(34) may be factorized too to find out a model for hadron masses (on Cantor sets). However, this problem is left for other paper.

CONCLUDING REMARKS
In a recent paper Yang Zhao  Therefore the aim of the present paper is to propose a combined version of London-Proca-Hirsch model for electrodynamics of superconductor. Then I extend further this proposed model for electrodynamics of fractal superconductor on Cantor sets. Considering that Proca equations may be used to explain electrodynamics in superconductor, the proposed London-Proca-Hirsch equations on Cantor sets may be able to describe electromagnetic of fractal superconductors. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.