.. title: magnetic monopole mass
.. slug: magnetic-monopole-mass
.. date: 2017-07-12 19:43:37 UTC+02:00
.. author: keith
.. tags: mathjax, tachyons, monopoles, dirac, quantization, recami, extended relativity
.. category:
.. link:
.. description:
.. type: text
.. figure:: /images/monopole.png
:figwidth: 180 px
:width: 180 px
:align: left
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.
.. TEASER_END
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 :math:`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
.. math::
g = \dfrac{n\hslash c}{2e} = ng_D \quad\quad n = (0,\pm1,\pm2,\pm3,...),\label{DQC}\tag{1}
shows the value of magnetic charge, :math:`g` in terms of the electric
charge, :math:`e`, Plank's constant, :math:`\hslash`, and the speed of
light, :math:`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
.. math::
r_{0e} < ƛ_{ce} < a_{0e},\label{H1}\tag{2}
monopoles have a distinct set of fundamental lengths
.. math::
a_{0m} < ƛ_{cm} < r_{0m},\label{H2}\tag{3}
*reversed* from the original. Here :math:`m` subscripts stand for
monopole and :math:`ƛ` denotes the *reduced* Compton wavelength,
which I have substituted for Datta's non-reduced Compton wavelength.
Datta substituted the :math:`n=1` values for magnetic
charge from Eq. (:math:`\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 (:math:`\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 :math:`n=2` is used in Eq. (:math:`\ref{DQC}`), rather than
:math:`n=1` used by Datta, it turns out to match very well indeed. From
here on :math:`n=2` in Eq. (:math:`\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 :math:`n`
.. math::
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}
where
:math:`ƛ_{ce} \alpha^0=ƛ_{ce}`,
:math:`\alpha = k_e e^2/\hslash c` and
:math:`k_e \equiv (4\pi\varepsilon_0)^{-1}`. We note that the
terms in the sequence are incremented by :math:`\alpha` and
include hierarchy (:math:`\ref{H1}`) and hierarchy (:math:`\ref{H2}`).
(where :math:`n=2` in the Dirac quantization condition)
We also note that the velocity sequence
.. math::
(v_{0e}, c, v_{0m}) = c \alpha^n \quad\quad n = (1, 0, -1),\label{S2}\tag{5}
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 :math:`\alpha`.
If we align sequence (:math:`\ref{S1}`) with sequence (:math:`\ref{S2}`)
by matching the ground state velocity of the electron, :math:`v_{0e}`,
with the Bohr radius, :math:`a_{0e}`, of the electron,
we can visualize the mirrored spacetime structure as shown in Table 1.
.. figure:: /images/table1.png
:width: 500
:align: center
Table 1. Alignment of velocities and lengths.
As can be seen in Table 1., the reduced Compton wavelength of the
magnetic monopole, :math:`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
.. math::
\dfrac{\hslash \alpha^{-4}}{m_e c} = \dfrac{\hslash}{m_m c},
and finally,
.. math::
m_m = m_e\alpha^4 = 1.45 \times 10^{-3} \;\textrm{eV/c}^2.\label{mass}\tag{6}
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 :math:`g = -e`, whereas here we
have (with our :math:`n=2` assumption) :math:`g = 137e`. This requires
some further research, but a possible way to recover symmetry in this
case would be
.. math::
\alpha_{\text{EM}} =
\begin{cases}
\alpha, & \text{if} \;\;v < c \\
\alpha^{-1}, & \text{if} \;\;v > c
\end{cases} \;\;.
With the result in Eq. (:math:`\ref{mass}`), the fine structure constant can
be seen as the
ratio of the subluminal electron mass to the observed superluminal
electron mass
.. math::
\alpha = (m_m/m_e)^{1/4}
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 :math:`\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] P. A. M. Dirac, `Proc. Royal Soc. A 133, 60 (1931). `_
.. [2] T. Datta, `Lett. Nuovo Cimento 37, 51 (1983). `_
.. [3] K. A. Fredericks, `Physics Essays 30, 269 (2017). `_
.. [4] E. Recami, `Riv. Nuovo Cimento 9, 1 (1986). `_
.. [5] H. C. Corben, `Int. J. Theor. Phys. 15, 703 (1976). `_
.. [6] R. Foot, `Shadowlands, Quest for Mirror Matter in the Universe, Universal Publishers, Boca Raton (2002). `_
.. [7] G. Lochak, `Z.Naturforsch. A62, 231 (2007). `_
.. [8] L. I. Urutskoev, V. I. Liksonov, and V. G. Tsinoev, `Prikladnaya Fizika (Applied Physics, in Russian), 4, 83 (2000). `_
.. [9] D. Priem et al., `Ann. Fond. L. de Broglie (in French) 34, 103 (2009). `_
.. [10] K. Fredericks, `Engineering Physics (No. 6), 15 (2013). `_