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Dirac Monopole

Observations and Experiments

The GUT Version

Search in 2022

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 q

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.
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## Figure 001 Maxwell's Equations, Transformation [view large image] |

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] |

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^{-2}s^{-1}sr^{-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).
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## 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 :

temperature becomes superfluid acting coherently as a whole. The averaged number density is about 10^{9} cm^{-2}, its distribution mimics the probability density of the synthetic monopole (Figure 03,a).
_{s}, which is related to the vector potential A* with the synthetic magnetic field B* = XA* (Figure 03,b). | |

## Figure 03 Synthetic Monople |

- 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

## Figure 05 Cosmic Evolution |
and f_{ijk} 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(

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

_{}= 0 for the false vacuum before SSB (at top of the Mexican hat in Figure 06c)._{}= v = constant for the true vacuum after SSB (see Figure 06,b)._{}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 n _{m}(actually magnetic monopoles as will be shown later) is estimated to be about one per horizon volume, i.e.,

n_{m}~ 1/(ct_{GUT}) ~ 10^{75}/cm^{3}which is much more than the 500/cm^{3}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

- 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

## Figure 08 GUT SU(5) Symmetry Breaking |

## Figure 09 |
As the mass of the monopole is concentrated in this tiny core of radius r_{c}, it is endowed with interesting structure on many different size scales (see Figure 04). |

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 10^{16} 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
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## 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. |