## Magnetic Monopole (Updated 2022)

### Contents

Symmetrical Maxwell's Equations
Dirac Monopole
Observations and Experiments
The GUT Version
Search in 2022

### Symmetrical Maxwell's Equations

The regular Maxwell's equations for electromagnetism is asymmetric with respect to the electric charge and magnetic charge, which is absent altoghter. Observationally, the magnetic field pervading the entire Milky Way confirms a lack of magnetic monopoles to cancel out (or short out) such field on the galactic scale. Theoretically, it is proposed that monopoles were exponentially diluted during the inflation, to such an extent that there would be little chance of even one in the Milky Way (since magnetic charge is heavier and fewer). Anyway, magnetic monopoles have never been observed (apart from a synthetic one, see topological superconductor). However, the same popped up again in the Grand Unified Theories (GUT) in an even more exotic form. Table 01 lists the symmetrical Maxwell's equations with the inclusion of magnetic charge and magnetic current.

#### Table 01 Maxwell's Equations, Symmetrical [view large image]

In a hypothetical EM space, the electric/magnetic charge qe/qm and electric field E - magnetic field B are transformed like rotation of an object in a 2 dimensional space (Figure 001). Such operation would leave the symmetrical Maxwell's equations unchanged. In particular, for the phase angle = 90o the role of qe/qm and E/B will interchange. For the real world with only electric charge, both qm and have to vanish. The time reversal operation t -t would also leave the Maxwell's equations invariant.

The effect of time reversal on some classical variables can be divided into two cases :
• T Parity = +1 - position, acceleration, force, energy, mass, electric potential, field, charge density, polarization, and all physical constants (except those associated with weak interaction).
• T Parity = -1 - time, velocity, linear momentum, angular momentum, power, electromagnetic vector potential, current, current density, magnetization, magnetic field and induction.
Thus, the fancy phrase of "time-reversal breaking" is just the effect of magnetic field on a piece of material lifting the degeneracy of electronic states. This is in contrast with non-magnetic materials which has time-reversal symmetry.

### Dirac Monopole

In the 1931 paper on "Quantised Singularities in the Electromagnetic Field", P.A.M. Dirac derived among other things the magnetic monopole from existing theory in that period. The system under consideration is a charged particle with its wave function e moving in
different paths within its own magnetic field. In particular the wave functions of two different paths splitting at the same point o and recombining at another point x would be different only in the phase, which plays no part in defining the probability. The phase difference is equated to the magnetic flux, which has been verified successfully by the "Aharonov-Bohm effect" in 1960. The following is a simplified derivation using some visual aids as shown in Figure 01.

#### Figure 01 Dirac Monople [view large image]

BTW, if the particle has electric charge Qqe, it can be shown that Q = n, i.e., an integer. Therefore, the existence of a magnetic monopole implies quantization of electric charge.

### Observations and Experiments

There were many failed attempts to search for the magnetic monopole. For example, the MACRO (Monopole, Astrophysics, and Cosmic Ray Observatory) detector located at the Gran Sasso National Laboratories in Italy had been designed to look for supermassive magnetic monopoles among other exotic particles. It had stopped collecting data by December 2000 after spending five years in the futile effort.
Figure 02 shows the flux upper limit of monopole search in cosmic ray. It indicates that the detectors are sensitive only up to that level and failed to detect any monopole; in fact the flux could be even lower all the way down to zero. The data are in unit of cm-2s-1sr-1 (sr = steradian, is the SI unit of solid angle). The flux upper limit is in the range of 10-15 - 10-16 with this unit. Such flux level is much lower than that for the cosmic ray bombarding the Earth regularly in the range of 1 - 10. The MACRO experiment comprised three different types of detector : liquid scintillator, limited stream tubes, and NTDs (Nuclear Track Detector for detecting cosmic ray tracks inside solid material).

#### Figure 02 Monopole Flux Upper Limit [view large image]

Other similar searches are the OHYA experiment (using array of NTDs) located inside a mine in Japan; while SLIM is a high-altitude experiment. See "Magnetic Monopole Searches" for a summary.

The latest claim of a discovery is from a 2014 paper about "Observation of Dirac Monopoles in a Synthetic Magnetic Field". Actually, every part of the monopole is synthetic in the sense that each one is represented by an artificial object. They are identified in the followings :
• Probability Density ; it is represented by a collection of rubidium atoms cooled to less than 10-7 K. The system at such cold
temperature becomes superfluid acting coherently as a whole. The averaged number density is about 109 cm-2, its distribution mimics the probability density of the synthetic monopole (Figure 03,a).
• Magnetic Field - The system is induced to rotate around the z-axis with azimuthal velocity vs, which is related to the vector potential A* with the synthetic magnetic field B* = XA* (Figure 03,b).
• #### Figure 03 Synthetic Monople

• Nodal Line - it is where the wave function vanishes and represented by a vorticity as shown in Figure 03.
• Magnetic Monopole - it is located at the end of the nodal line from which all the field lines are emanated.

• Another kind of synthetic monopoles is the "Spin Ice" constructed from special crystal structure. Meanwhile, other attempts in liquid crystals, skyrmion lattices and metallic ferromagnets also do not really find the real thing. Looking for them in old rocks doesn't help either.

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[2022 Update]

### The GUT Version

Monopoles are now perceived in a new guise according to the Grand Unified Theories (GUT). It is realized that if GUT were correct, monopoles must have created about 10-36 seconds after the Big Bang, when the forces differentiated. These monopoles, would be very massive - about 1015 times heavier than ordinary particles - and would therefore be impossible to make in the lab. However, the number expected to have survived from the early universe seemed embarrassingly large: there would have been enough to short out the galactic magnetic field; even worse, their total mass would far exceed that of everything else in the universe (far too much, even, for the dark matter). For GUT physics, monopoles are extremely interesting objects: they have an onion-like structure, which contains the whole world of GUT (Figure 04):
• Near the center ( about 10-29 cm ) there is a GUT symmetric vacuum.
• At about 10-16 cm, its content is the electroweak gauge fields of the standard model.
• At 10-15 cm, it is made up of photons and gluons.
• At the edge to the distance of 10-13 cm, there are fermion-antifermion pairs.
• Far beyond nuclear distances it behaves as a magnetically-charged pole of the Dirac type.

• In effect, the sequence of events during the earliest moment of the universe had been fossilized inside the magnetic monopole.

#### Figure 04 Monopole Structure in GUT

In this theory, the monopole is portrayed as a soliton which is stable - cannot decay to lower mass particles. In effect, it closes the loophole for explaining its absence in nature, and prompts the invention of "Cosmic Inflation" to explain away the problem by dilution. GUT also fails in its prediction of "proton decay".

Here's a mathematical description :
According to GUT, there is an additional Spontaneous Symmetry Breakdown (SSB) ~ 10-34 sec after Big Bang (BB), see Figure 05 and Quantum Field History à la Gauge Theory. The Lagrangian density for the Gauage Boson + Higgs (the scalar field in the old day) sectors after SSB is in the form :

#### Figure 05 Cosmic Evolution [view large image]

and fijk is the Levi-Civita tensor, = 0 for equal pairing of i,j,k; +1 for ijk; -1 for jik etc.

Actually, a gauge theory does not, by itself, possess any monopole configuration. It is produced only via the introduction of scalar field. Therefore, the concept can be introduced just by the breaking of its symmetrical phase to the true vacuum phase. That is, the vacuum state from a false one in the potential U() at = 0 to the true one at = v (see Figure 06,c). The Lagrangian density for the scalar field alone after SSB in spherical coordinates (taking only the radial component r) and at stationary state (for which = 0) is :

#### Figure 06 [view large image] GUT SSB and Soliton

1. = 0 for the false vacuum before SSB (at top of the Mexican hat in Figure 06c).
2. = v = constant for the true vacuum after SSB (see Figure 06,b).
3. v, as r . This is the essence of GUT magnetic monopole, its property is further elucidated in the followings :

• This is known as hedgehog configuration (see Figure 06a) because the magnitude of the field increases from zero at the origin up to v at infinity as prescribed by the hyperbolic function tanh(x) in Eq.(4) which is often denoted as f(r).

• This configuration cannot be turned into the uniform vacuum state (Figure 06,b), so it is topologically stable. In order for the field to be continuous, it has to be localized as lump of energy (in other words, a particle, a.k.a. soliton).

• The energy density of the scalar field is given by :

• Occurrance of SSB is similar to phase transition (for example from water to ice with decreasing temperature).
The misalignment of individual crystal's axis of symmetry creates topological defect at the boundary called domain wall (in 2-dimension). The soliton itself creates zero-dimensional defect. The number of such defects nm (actually magnetic monopoles as will be shown later) is estimated to be about one per horizon volume, i.e.,
nm ~ 1/(ctGUT) ~ 1075/cm3 which is much more than the 500/cm3 for the number density of photons as measured from CMB (see "Early Universe"). Moreover, these huge amount of solitons (magnetic monopoles) cannot decay to particles of lower energy because it is topologically stable - cannot transform to other topological shape (see examples in Figure 07).

#### Figure 07 Topology

For full-fledged GUT after SU(5) has broken to SU(3) X [SU(2)XU(1)] + (Scalar Field), the Lagrangian density is prescribed by Eq.(1) (see also Figure 08), the solutions (in Nature Units for which c = = 1, gm = erg cm-1) are in the forms :

#### Figure 08 GUT SU(5) Symmetry Breaking [view large image]

Since the W gauge field rapidly approaches its asymptotic value outside a core with radius of order rc; it is excited only inside the core with energy Ecore. The size rc is chosen to minimize the sum of Ecore and the energy stored in the magnetic field outside the core Eout, i.e., via dE/drc = 0, where E = Ecore + Eout (see Figure 06,a).

#### Figure 09 Phase Transitions

As the mass of the monopole is concentrated in this tiny core of radius rc, it is endowed with interesting structure on many different size scales (see Figure 04).

This concludes GUT's take on SSB (see further details in "MAGNETIC MONOPOLES", 1984). Unfortunately, the theory is not supported by observational evidence. This is just one of the instances of no correspondence between mathematics and reality.

Similar process in the Electro-weak SSB generated very different end products at 10-12 sec after BB. It breaks the left-right symmetry of the leptons and endows mass to the gauge bosons but does not produce things like soliton nor magnetic monopole (see "SSB+ESB"). It could be the different in energy scale, e.g., the end products are different from gas to liguid, and from liquid to solid (see Figure 09).

Another attempt to search for the monopole is also based on flimsy theoretical ground. This time, it is on an analogy of strong electric field creating electric charge (see "Search for single magnetic charges in the largest of fields"). The search takes advantage of the enormous magnetic field induced by collisions between lead ions accelerated by the Large Hadron Collider (LHC) at CERN. This magnetic field can reach strengths of 1016 tesla, making it the largest magnetic field measured on Earth (see Figure 10). It did not find a statistically significant signal of a magnetic charge trapped in their detector, and therefore ruled out the existence of monopoles with masses up to 75 GeV. Nevertheless, the result is considered a great success

#### Figure 10 One More Search for Monopole [view large image]

in the context of particle-physics research, because it opens up a new avenue for studies of magnetic monopoles. It also maps out the limits within which we can further hunt for magnetic monopoles.

[End of 2022 Update]