Theoretical studies on the creation of artificial magnetic monopoles

In this study, we discuss the theoretical studies on the creation of artificial magnetic monopole, and new electromagnetic equations. Employing Lorentz transformation, radial electrostatic fields, and a stationary wave derived from a superconducting loop, we demonstrate the existence of a magnetic monopole whereby the divergence of the magnetic field is not zero. We develop a device wherein a condenser provides electrostatic fields along the radial direction to the superconducting loop and discuss the nodes of the resulting stationary wave along the superconducting loop. We employ the Lorentz transformation with respect to the vector and electrostatic potentials. Then, because the nodes have no three-dimensional vector potential and have zero magnetic field rotation, the conserved energy is converted into new form that is associated with the magnetic field potential to yield the Lorentz transformation. As a result, we derived the relationship between the electric and the magnetic fields. This dependent relationship involves the exchange of the distribution characteristics of the static electric and static magnetic fields, and new electromagnetic equations of both electric and magnetic fields are obtained. We also analyzed the magnetic field from the magnetic monopole whose result assists the theory.


Introduction
In this study, we discuss the artificial creation of a magnetic monopole and new electromagnetic equations under a specific condition. The first description of a monopole was presented by Dirac [1].
Based on his suggestions, some experiments were conducted [2,3]. The magnetic monopole in spin ice [4,5] and other systems [6][7][8] have also been discussed. Recently, by employing the concepts of spin and Bose-Einstein condensation, a potential artificial magnetic monopole was created [9,10].
Moreover, the emergent magnetic monopoles were reported [14], and the quantum hall effect [13] is related to create magnetic monopoles [15].
In this paper, we propose another method for creating a magnetic monopole and present its theoretical basis, which includes new equations of the relationship between electric and magnetic fields regarding both static and time-dependent fields.
Dirac's paper described the monopole using the gauge. However, as long as the spatially dependent vector potential A is not zero, the monopole cannot be defined because a nonzero A results in the existing Maxwell's second equation, whereby a nonzero A eventually leads to a loop magnetic flux.
By employing Lorentz conservations, in this paper, however, radial electrostatic fields, and a stationary are derived along a superconducting loop, we demonstrate that Maxwell's second equation can be modified, such that the divergence of the magnetic field is not zero.
Recently the monopole-creation method implemented by replacing the anti-Helmholtz coils with larger coils carrying parallel currents [16]. However, because the vector potential A is employed, new electromagnetic equations are not presented. Indeed, the Maxwell equations should be symmetric for an electric field and a magnetic field vectors. In short, many magnetic monopole discussions have been exited, but the new electromagnetic equations, which explain the symmetry for electric and magnetic fields, are not yet reported.
In the cylindrical coordinates, in this paper, a condenser generates electrostatic fields along the radial direction to a superconducting loop. Given these conditions, we discuss the nodes in the stationary wave along the superconducting loop and consider the Lorentz conservations. In this process, the conserved energy is converted into new form that is associated with a magnetic field potential to yield and hold the Lorentz conservations. As a result, we obtain the specific vectors' dependent relationship between electric and magnetic fields. Because this is a dependent relationship, Maxwell's first equation becomes a modified and proposed Maxwell's second equation, which implies the divergent of a magnetic field B is not zero.
We then derived new electromagnetism equations assembly, which describe the exchange of the distribution characteristics of the electric and magnetic fields. Herein the exchanged Maxwell equations' assembly including time-dependent equations are derived. Using these equations, we obtain the electromagnetic wave whose speed is c (=3.0 × 10 8 / ). Moreover, as the result of consideration, the magnetic monopole is found to be the combination between Cooper pair and Cooper pair.
Furthermore, this paper determines the wave function that follows a monopole and the state of the condensation of monopoles. Finally, we numerically analyze the magnetic field distribution from monopole. The results demonstrate the divergence of the distribution of the magnetic fields. The significance of the paper is that it obtained theoretically new equations of both electric and magnetic fields under a specific condition. Moreover, this paper presents easier method to gain an artificial magnetic monopole. Fig. 1 shows the schematic of a superconducting loop and cylindrical condenser setup. In this study, the poles of the cylindrical condenser are charged in advance. Using the abovementioned method, we can apply the radial static electric fields to the superconducting loop. Because of these electric fields (i.e., centripetal forces), Cooper pairs move along the loop both clockwise and counterclockwise, and eventually form a stationary wave. This is because, due to the conservation of the total momenta, rotations must be both clockwise and counterclockwise. As mentioned later, this conservation of the total momenta still continues during and after the transition to the monopole phase.

Principle
As will be discussed later, creation of a monopole requires nodes along the superconducting stationary wave, and Cooper pairs in this node constitutes the monopole.

Derivation of the conclusive equation
As discussed in Principle section, directions of superconducting currents are formed as z r +Q Superconducting coil -Q counterclockwise and clockwise rotations. This is because initially the total momenta are zero and thus the conservation of the momentum gives rise to both directions of the superconducting currents.
Note that the applied electric field along the radial direction in Principle section works as a centripetal force.
Considering these facts, we can see a stationary wave form along the coil loop. According to [17], solving Schrödinger equation results in the following wave function: where and n is an integer.
At this point, the following Lorentz conservations are considered. And where k, ε, c, A and φE denote the wave number, the energy, speed of light, vector potential and electric potential, respectively.
Importantly each node along the stationary wave has no momentum because the nodes have no phases, assuming that each rotation results in a plane wave.
The above equation also implies that That is, where μ0 denotes the magnetic permeability in the vacuum.
Considering dimension of the above equation, the scalar potential φH has the unit of a magnetic potential. Therefore, the following conclusive equation is derived. where This relation is substituted to the existing Gauss equation: where ρ and ε0 denote the charge density and the permittivity in the vacuum.
The above equation (11) does not already imply the existing Gauss equation relationship regarding the electric fields; hence, the distribution of the electric fields must be approached in another way.
Because Eq. (11) is not consistent with the existing Maxwell's second equation, the following equation must be derived from Eq. (9): Eqs. (11) and (12) imply that the distribution characteristics of the electric and magnetic fields have been exchanged. We know that the looped magnetic flux is generally distributed when a current is supplied along a coil. In this case, however, we can infer that the looped electric flux is created; combining the μ0 and c, a magnetic current can be considered in this case. Because the rotation of the electric field is not zero, we can consider a modified Ampere's law, using the conclusive equation (9) rot ⃗⃗ = μ 0 ⃗⃗ → .
The right-hand side is the magnetic current density iB.
In Table 1, the differences are shown between the existing and proposed Maxwell equations regarding static fields.

Derivation of time-dependent equations and electromagnetic wave
Let us consider time-dependent Maxwell equations and the derivation of an electromagnetic wave.
That is, to demonstrate that our derived time-dependent Maxwell equations are valid, it is imperative to derive the electromagnetic wave from them.
As everyone knows, the existing time-dependent Maxwell equations are and where H and D denote the magnitude of a magnetic field and the electric flux density, respectively.
To the above two equations, the following exchanges are made to be subjected according to the conclusive equation (9). Taking a rotation-operator for the both sides of Eq. (17), To this equation, Eq. (18) is substituted.
From the vector analysis formula, a wave equation for electric fields is derived: Thus velocity v of the wave equation becomes Similarly, a wave equation for magnetic fields can be derived by the dual calculations; Thus velocity for the wave equation of magnetic fields is obtained as the same form.
In Table 2, the differences of time-dependent Maxwell equations are summarized.

Structure of a monopole
As discussed in the previous section 3.1, in the process of the conservation of the Lorentz equation (4-2) and to create a monopole, the photon ħω (=2mc 2 ) which is defined by the Dirac equation must be consumed. Note that right-hand side of Eq. (4-2) contains the photon energy ħω . Considering the stationary wave derived from both the clockwise and the counterclockwise waves, it can be assumed that 2 Cooper pairs (i.e., 4 electrons) gather at each node. At each node, as indicated in Fig. 2 where m and ω denote the electron mass and the angular frequency, respectively.

Condensation of wave functions of monopoles
It is assumed that both before and after the transition to monopole phase, the total momenta should be conserved. Before the transition, because the motions of Cooper pairs have both clockwise and counterclockwise rotations, the total momenta are zero. It is also assumed that each wave function along each rotations is approximated as a plane wave which behaves as a boson. Considering both wave functions as simple product and after the transition, therefore, the net wave function at each node can be derived as where the normalization is satisfied as From the Bloch's theorem, where j and k denote the imagery unit and a wave number, respectively.
Thus the total wave function Φ results in In this equation, the wavenumber k is defined as where π/Ri denote the Brillouin zone Therefore, the above total wave function is calculated as Considering the property of a general phase, α ≡ 2 must be held.
At this time, the total wave function becomes conclusively Then consider the normalization of the total wave function; On the other hand, and as mentioned, the normalization of an element wave function is Accordingly,

Preparation of simulating equations
This section now considers simulating and macroscopic equations. Let us consider simulating equations in view of the retraced calculations.
From a Lorentz conservation, where p denotes a macroscopic momentum.
That is, then v = ± .
Thus, a current density is where ns is the concentration of Cooper pairs that can be derived as where ξ denotes the coherence of a Cooper pair.
Consequently, the magnetic field components along z-direction becomes where R denotes a radius of coil in the proposed device (see section 2).
Next let us consider the equations of the divergence of magnetic field. First, the relationship between an electric potential and an electric field is simply This equation is substituted to the abovementioned equation (45) of magnetic field component.
Differentials are taken to both sides, and thus To implement of this equation, it is necessary to obtain the electric field component Ez.

From the Poisson's equation,
where ε0 denotes the permittivity of the vacuum.

Thus
Here z is assumed to be the coherence ξ.
Consequently,         This implies that, the larger superconducting critical temperature is, the larger the magnetic field component Bz.

Review of a quantized space-time (The zero-point energy and concept of a quantized space-time)
Let us review a concept of space-time which was many times employed in our previous paper [18][19][20].
The reason of this review is to introduce radius of a monopole and to consider why our universe has only the existing Maxell equations generally.
The concept of quantized space-time, as well as the zero-point energy, can be especially elaborated in a vacuum condition. As such, we begin by describing each concept by solving the Dirac equation. The equation shows that inside a vacuum, during formation of electron and positrons, a mass gap that exists between them can be represented by the relation in which ω0, me, and c are the angular frequency, electron mass, and speed of light, respectively. As we have discussed in [19], the mass of an electron is the most basic parameter; thus, Eq. (55) provides a constant quantized space λ0 and a quantized time t0, which are defined as follows:

Equation (54) can be interpreted in the form
Consequently, we can derive a more general equation of constant quantized time-space length and time as follows: Accordingly, we have described in [19] that the gravitational force or Lorentz force from the magnetic fields comes from the fact that up-and down-leptons that are embedded in a quantized space-time λc form rotations and then combine with each other as paired quantized space-time because of the attractive force from the Lorentz force or gravity. This paired quantized space-time behaves like a boson. A schematic representation of this phenomenon is presented in Fig. 10.

Properties of a monopole particle
In general, every particle must have a spin angular momentum. However, if a magnetic monopole had a spin angular momentum, that magnetic flux would form a loop. Thus, a monopole must not have a spin angular momentum. This fact can be described as follows: Remind that our monopole is created by the fact that two Cooper pairs (i.e., four electrons) are combined. Thus, in principle, the following balance between angular momentum and spin angular momentum should be formed in order that the monopole has no spin angular momentum.
where l and s denote the angular momentum in that two Cooper pairs take their self-rotation and the normal spin angular momentum, respectively.
The left-hand side is calculated as where a and ω denote the radius of the self-rotation, and the angular frequency, respectively.
Using the following equation, the angular frequency is deleted. Thus, This is the value of radius of a monopole, which is importantly equal to the quantized space-time in the previous reviewed section. See eq. (57) in the reviewed section.

Symmetry of an electric field E and a magnetic field B
This paper described that there are two types assemblies in terms of Maxwell equations. The crucial point is that the roles between a magnetic field B and an electric field E vectors are completely altered.
This implies that originally a magnetic field B and an electric field E are symmetric. Although these are symmetric, the reason why we always measure the existing Maxwell equations is based on the quantized space-time structure as reviewed in the previous reviewed section. This implies that the condition that our newly proposed Maxwell equations must be rare and that, to measure our newly proposed Maxwell equations, it is required to prepare the specific setup, which implies that a monopole does not exist in our normal nature and universe but can be created artificially.

Interpretation of results of simulations
The result figures from Fig. 6 to Fig. 9 indicate the macroscopic magnetic field distribution by many monopoles. As shown, besides the Br component, ± magnetic field components are depicted. As shown in the schematic figure 11, this implies that, as a condition of existences of the monopoles, it is imperative to measure ± magnetic field components as well as Br component.

Significances of the present paper
It is believed that a natural monopole does not exist. However, this paper described that it is possible to create a magnetic monopole artificially. The significances are that a new elementary particle has Bz been predicted and that Maxwell equations are symmetric between an electric field E and a magnetic field B. In this paper, these alternative equations were derived as both static ones and time-dependent ones. Although these alternative Maxwell equations exist only under a specific condition, the fact that there are another assembly of Maxwell equations implies that a new insight is provided to electromagnetism field of physics. Moreover, concerning quantum mechanics, we succeeded in describing the wave function that describes a monopole.

Conclusion
In this paper, we described the potential existence of an artificial magnetic monopole and proposed new electromagnetism equations. That is, we proposed the new method of creating magnetic monopoles. Based on this device and considering the stationery wave and the Lorentz conservations, we obtained the existence of a magnetic monopole. In the process of this theory, the new and alternative Maxwell equations were obtained as well as obtaining the meaning of quantum mechanics of monopoles. Moreover, the numerical calculations assisted them. As a result, we could obtain the divergent distribution of the magnetic fields theoretically.
The measurements of macroscopic divergent magnetic field are simply requirements condition.
That is, it is further required to test whether truly a monopole particle is created. As a follow-up, we wish to consult this point with experts of experiments of the condensed matter physics. Moreover, it is important to catch the electromagnetic wave regarding magnetic monopoles as a follow-up.