Shortly after the publication of Dirac's seminal paper [1] on the quantization of electric charge in 1931, experimenters began the search for magnetic monopoles. Dirac's result indicated a value for magnetic charge of a monopole, but the mass was not predicted and could be any value.

Over the years various estimates of monopole mass were made and experiments established limits on the mass. By the 1970s the lower limit on mass was established to be on the order of 5 GeV. Meanwhile in 1974 both t'Hooft and Polyakov found that grand unification involved magnetic monopoles and predicted a mass on the order of \(10^{16}\) GeV. Since then studies have focused on monopoles in the GeV range and greater.

Dirac's famous relation now known as the Dirac quantization condition

\begin{equation*} g = \dfrac{n\hslash c}{2e} = ng_D \quad\quad n = (0,\pm1,\pm2,\pm3,...),\label{DQC}\tag{1} \end{equation*}

shows the value of magnetic charge, \(g\) in terms of the electric charge, \(e\), Plank's constant, \(\hslash\), and the speed of light, \(c\). Datta [2] used this relation in 1983 to show that corresponding to the well-known hierarchy of fundamental lengths comprised of the classical electron radius, the Compton wavelength and the Bohr radius

\begin{equation*} r_{0e} < ƛ_{ce} < a_{0e},\label{H1}\tag{2} \end{equation*}

monopoles have a distinct set of fundamental lengths

\begin{equation*} a_{0m} < ƛ_{cm} < r_{0m},\label{H2}\tag{3} \end{equation*}

reversed from the original. Here \(m\) subscripts stand for monopole and \(ƛ\) denotes the reduced Compton wavelength, which I have substituted for Datta's non-reduced Compton wavelength. Datta substituted the \(n=1\) values for magnetic charge from Eq. (\(\ref{DQC}\)), the Dirac quantization condition, into the equations for the classical electron radius, the Compton wavelength and the Bohr radius. Datta communicated that this relation makes monopoles inherently relativistic, but that the required superluminal velocities of the ground states of monopoles in monopole atoms makes these states unphysical. Datta concluded that this argues against monopole bound states and the existence of the magnetic monopole.

There is another approach, which I have shown in my latest paper [3], that starts where Datta left off by identifying hierarchy (\(\ref{H2}\)), the implied mirrored spacetime structure of his inverted hierarchy of fundamental lengths, with the mirrored spacetime structure of extended relativity as explained by Parker, Corben, and mainly by Recami [4] and others. If \(n=2\) is used in Eq. (\(\ref{DQC}\)), rather than \(n=1\) used by Datta, it turns out to match very well indeed. From here on \(n=2\) in Eq. (\(\ref{DQC}\)) is assumed.

Now that we have the mirrored fundamental lengths and the assertion of faster-than-light velocities, we find that by aligning velocity and length sequences, it is possible to find a value for magnetic monopole mass.

mass

We define a sequence multiplying the reduced Compton wavelength by the fine structure constant raised to a power \(n\)

\begin{equation*} ƛ_{ce} \alpha^n = \dfrac{\hslash \alpha^n}{m_e c} \quad\quad n = (1, 0, -1, -2, -3, -4, -5),\label{S1}\tag{4} \end{equation*}

where \(ƛ_{ce} \alpha^0=ƛ_{ce}\), \(\alpha = k_e e^2/\hslash c\) and \(k_e \equiv (4\pi\varepsilon_0)^{-1}\). We note that the terms in the sequence are incremented by \(\alpha\) and include hierarchy (\(\ref{H1}\)) and hierarchy (\(\ref{H2}\)). (where \(n=2\) in the Dirac quantization condition)

We also note that the velocity sequence

\begin{equation*} (v_{0e}, c, v_{0m}) = c \alpha^n \quad\quad n = (1, 0, -1),\label{S2}\tag{5} \end{equation*}

comprised of the ground state velocity of the electron, the speed of light and the ground state velocity of the monopole is also incremented by \(\alpha\).

If we align sequence (\(\ref{S1}\)) with sequence (\(\ref{S2}\)) by matching the ground state velocity of the electron, \(v_{0e}\), with the Bohr radius, \(a_{0e}\), of the electron, we can visualize the mirrored spacetime structure as shown in Table 1.

Table 1. Alignment of velocities and lengths.

As can be seen in Table 1., the reduced Compton wavelength of the magnetic monopole, \(ƛ_{cm}\), is defined in terms of the reduced Compton wavelength of the electron. Since the equation for reduced Compton wavelength includes a mass term, we can find the mass of the magnetic monopole using

\begin{equation*} \dfrac{\hslash \alpha^{-4}}{m_e c} = \dfrac{\hslash}{m_m c}, \end{equation*}

and finally,

\begin{equation*} m_m = m_e\alpha^4 = 1.45 \times 10^{-3} \;\textrm{eV/c}^2.\label{mass}\tag{6} \end{equation*}

mirror worlds

This gets at the heart of the relationship between tachyons and magnetic monopoles. When you extend special relativity to speeds faster-than-light, you have in effect two mirror worlds, [5] one faster-than-light and one slower-than-light. Everything is duplicated, but at a different speed. Electromagnetism of the faster-than-light world as viewed from the slower-than-light world then is flipped, that is, in the sense of electromagnetic duality. This means that the normal slower-than-light observer will detect a normal electron from the faster-than-light world as a magnetic monopole.

This theory shares some features of mirror matter theories [6] and is of interest for some of the same reasons like the search for dark matter and dark energy.

So, the mystery of electron charge may lie in the relationship between two co-existing velocity states of the electron. Magnetic charge would be just superluminal electric charge and the magnetic monopole would exist only as the manifestation of the superluminal velocity state.

The theory does not map one-to-one with Recami. Recami has proposed that the magnetic charge should be \(g = -e\), whereas here we have (with our \(n=2\) assumption) \(g = 137e\). This requires some further research, but a possible way to recover symmetry in this case would be

\begin{equation*} \alpha_{\text{EM}} = \begin{cases} \alpha, & \text{if} \;\;v < c \\ \alpha^{-1}, & \text{if} \;\;v > c \end{cases} \;\;. \end{equation*}

With the result in Eq. (\(\ref{mass}\)), the fine structure constant can be seen as the ratio of the subluminal electron mass to the observed superluminal electron mass

\begin{equation*} \alpha = (m_m/m_e)^{1/4} \end{equation*}

The idea of a sub-eV magnetic monopole is in the opposite direction from where most of the attention is now focused on monopoles, i.e. the \(\sim 10^{16}\) GeV range. It must be mentioned that there has been some work done in the area of neutrinos that could be faster-than-light and some of this was not related to the apparent equipment error related to the so-called faster-than-light neutrino anomaly with the OPERA experiment.

Lockak [7] has done work in the area of the massless magnetic monopole. This is all the more interesting due to his citations [8], [9] of particle tracks which correspond directly with my studies. [10]

references

[1]	Dirac, Proc. Royal Soc. A 133, 60 (1931).

[2]	Datta, Lett. Nuovo Cimento 37, 51 (1983).

[3]	Fredericks, Physics Essays 30, 269 (2017).

[4]	Recami, Riv. Nuovo Cimento 9, 1 (1986).

[5]	Corben, Int. J. Theor. Phys. 15, 703 (1976).

[6]	Foot, Shadowlands, Quest for Mirror Matter in the Universe, Universal Publishers, Boca Raton (2002).

[7]	Lochak, Z.Naturforsch. A62, 231 (2007).

[8]	Urutskoev, V. I. Liksonov, and V. G. Tsinoev, Prikladnaya Fizika (Applied Physics, in Russian), 4, 83 (2000).

[9]	Priem et al., Ann. Fond. L. de Broglie (in French) 34, 103 (2009).

[10]	Fredericks, Engineering Physics (No. 6), 15 (2013).

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