Magnetic Monopole: Unsolved Puzzle

arrangement of spin in spin ice

We may should not bring journalism style into science publication please. I admit that it would be  more attractive  but also bad… 😦 and sometime more than bad…it can be mislead!

Magnetic monopole; one of physics holy grail was reported detected for the first time in Science daily and Physorg. I was remember my first reaction when I read that news in 2009. I am a little bit half believe actually 😛

The real paper, as usual, pretty different 😀

Actually, it is not really monopole but closely like quasi particle. Quasi particle can indeed has unique properties such as spin that does not satisfy Bose-Einstein nor Fermi-Dirac statistic.

Well, the question come up. If magnetic monopole really exist, does it violence the validity of one of Maxwell equation (divH=0)? But even it was quasi particle, its also confusing. It is difficult enough to understand how the dipole-dipole as a “building blocks” can produce a resultant effect as a monopole? Though, it will be easier to understand the resultant effect as a stronger dipole, quadrapole, or neutral, depending on the relative orientation between them.

Paul Dirac in 1934 has been postulated the possibility of magnetic monopole in one of his classic paper. Dirac analyzed the possibility of magnetic monopole existance in the context of Abelian gauge theory (Maxwell electrodynamics). He conclude that the existence  of magnetic monopole can explain why electric charge is quantized.

Maxwell equation showed that the existence of magnetic monopole cam naively imply inconsistency to the equation it self. However, Dirac showed that the inconsistency can be avoided if we assume an existence of an axis singularity which pulled from the center of monopole to the infinity. This singularity line is known as Dirac String. Then, the physical implication  is that the magnetic monopole potential can not be defined for the entire space, but there are areas where there is an overlap between the two different magnetic potential due to the gauge transformation.

Forty years after, in 1974, Gerardus t’Hooft from Utrecht and Alexander Polyakov from Landau Institute, Moscow (now living in Princeton) postulated the naturally existence of magnetic monopole in the framework of non-abelian gauge theory in which the properties of its gauge transformation group is compact (as in the old theory of  Georgi-Glashow SO (3) for electroweak), in contrast to the existence of magnetic monopole in the abelian gauge theory that demands the existence of Dirac strings.

In non-abelian gauge theory, magnetic monopole appears as a topological soliton and its magnetic charge is describe by the “topological charge”, eternal because of the topological condition and independent from the motion equation; contrast with the Noether charge in which its immortality is guaranteed by the motion equation.

The existence of quasi particle in spin ice material is a consequence from the crystal structure material that was being studied (spin ice material is a material that has a crystal structure similar to ice) which allows unusual excitation configuration. The theory itself  has been exist for a long in physics community (see the paper here).

So, what is the implication of the Dirac postulate toward div H=0? For example, whether the monopole existance will imply the modification of div H=0 to contain monopole inside the equation? If yes, in what condition the equation will reduce to div H=0?

According to Maxwell, div H=0 implies the absence of magnetic charge. And thus, the magnetic induction H can be expressed as H=curl A, for vector quantity A, commonly called the vector potential. If we postulate the existence of magnetic charge, div H=m, then H no longer can be expressed as a curl of the potential vector potential A, and we must construct another potential vector to be consistent with the equation of div H=m.

This is not a problem in classical electrodynamics, because the phi potential scalar and A vector potential was introduced as a mathematical tool to formulate the electric and magnetic field. Physics quantity E and H (electric and magnetic field) can be measured. We can redefine, or even dispose  A quantity if he is no longer consistent with the formulation of E and H. But in quantum  mechanics, this is become a problem, because the A potential vector is a physical observable! (See the Aharonov-Bohm phenomenon). Therefore, we have to keep use the identity: H=curl A to be consistent with the quantum formulation.

If so, how can it be consistent with the magnetic charge formulation? By calculus vector we can clearly see that div curl A=0 contradict to div H=m.

Dirac suggested a genius trick to solve this problem. According to Dirac (and has been re-formulated by Wu and Yang), as a vector, A continues to exist but not defined for the whole space (global). There are areas where the A vector potential is not defined (‘ill defined”), so the integral of div H for whole space M, the total magnetic charge.  Thus, to cover the entire space wee need another potential vector A so there are areas where overlaps between two potential vectors happen. Because these vectors must be physically the same, they will only distinguishable by a gauge transformation, the form of the integer multiples. From this integer multiplies we can obtain (by Hamiltonian) the electric charge quantization.

So, if magnetic monopole does exist, why it can not easily detected? Meanwhile, as comparison, the monopole charge is very abundant. t’Hooft showed that the monopole mass is inversely proportional to the coupling constant (fine constant structure). Quantitavely, the mass of monopole=137*Mw, where Mw is the mass of vector boson (one of topological soliton properties which are non-pertubatif phenomenon, where the mass/energy is always varies inversely with the related coupling constant theory, so the information can not be extracted through  perturbatif method such as Feynman diagram).

This is show that the existence of magnetic monopole is very massive (heavy) and not easy to found.

In other side, monopole as a topological solution (in the context of non-abelian group theory) only exist if the gauge group are compact also. For example, the theory of Georgi-Glashow SO (3) for electroweak. This theory allows the existence of monopole. Unfortunatelly, these theory are outdated in the competition with Weinberg-Salam electroweak theory that has been prove true by the discovery of neutral currents in CERN at 70s. However, the Weinberg-Salam theory is based on non compact gauge group, SU(2)xU(1). Consequently, magnetic monopole is not exist in Weinberg-Salam electroweak theory. If Grand Unified Theory (which is still currently constructed) someday has been establish, and based on the compact group, then the GUT monopole must be exist, where the mass is very massive: 137 times super-heavy vector boson mass!


10 thoughts on “Magnetic Monopole: Unsolved Puzzle

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