## Magnetic Monopole

### Contents

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

### 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 01). 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 as listed below :
• 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.

### 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 [view large image]

• 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.

### 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.