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I understand the Yang-Mill perspective of $U(1)$-gauge theory. In that, you can easily write down the field of a Dirac magnetic monopole. What interests me is the fact that it's so hard to find (if exists) in our real world.

Why is that the case? Are there other theories implying something like "our universe does not like non-trivial topology (chern class) and thus eagers to cancel the non-triviality by putting two opposites together"?

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    $\begingroup$ They don't exist. $\endgroup$
    – my2cts
    Mar 13, 2020 at 14:01
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    $\begingroup$ They are presumably too massive for current experiments. $\endgroup$
    – Qmechanic ♦
    Mar 13, 2020 at 15:22
  • $\begingroup$ ... contradicting answers. Would you mind providing more details? $\endgroup$
    – Student
    Mar 13, 2020 at 16:09
  • $\begingroup$ Monopoles would not be hard to find if they existed. They would be easily detectable. Thus the failure to detect any suggests they do not exist. Can you clarify whether you are asking why monopoles are hard to find (they aren't) or why they don't exist? $\endgroup$
    – John Rennie
    Mar 13, 2020 at 17:12
  • $\begingroup$ Ok.. provided that they don't exist. Is there any reason for why they don't exist? $\endgroup$
    – Student
    Mar 13, 2020 at 17:38

1 Answer 1

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it's so hard to find [ magnetic monopole] (...) in our real world

Not very hard to explain. Particle spin induces it's magnetic dipole moment in current way : $$ {\vec {\mu }}_{\text{S}}\ =\gamma {\vec {S}} $$ where $\gamma$ is gyromagnetic ratio and $\vec{S}$ is particle spin angular momentum. So magnetic dipole of particle is intrinsic property of it's spin. And alignment of spins of all particles in a body produces permanent magnetic field, which is also a magnetic dipole.

Now when you think about spin is has something to do about rotation along some axis. (Not very good analogy, cause in QM view it's hardly can be said that particle has some fixed rotation axis, however we just don't have a better analogy). So if you have a rotation axis, then you have automatically a pair of poles - North and South. That's intrinsic property of a rotating body. You can't rotate along some axis which would have only 1 pole, thus magnetic monopoles does not exists.

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  • $\begingroup$ Ah! I understand the second paragraph. But not sure about the first. In particular, why should a magnetic pole spin? Can't its spin be zero, thus getting rid of the 2-poles-issue? $\endgroup$
    – Student
    Mar 13, 2020 at 14:50
  • $\begingroup$ It can be seen from the formula that if spin angular momentum is zero, then magnetic moment is zero too. Probably you haven't noticed causality. The point is that particle spin generate particles magnetic moment, ie. makes it magnetic dipole. No spin = No magnetic field at all. Is it clear now ? $\endgroup$
    – Agnius Vasiliauskas
    Mar 13, 2020 at 18:35
  • $\begingroup$ Why the down-vote ? Answer is technically correct and comprehensive. I see no reasons for down-vote. And no alternative explanations either. $\endgroup$
    – Agnius Vasiliauskas
    Mar 13, 2020 at 18:37
  • $\begingroup$ why should a magnetic pole spin? Spin is particle property, not a pole (if any). Particle spinning produces magnetic dipole moment, ie. it makes particle like a "tiny magnet with north and south poles" - rough analogy. If you switch particle spin off,- then you switch magnetic moment off too, thus - both magnetic poles too. You can't control just one magnetic pole of particle,- that's how spin works. $\endgroup$
    – Agnius Vasiliauskas
    Mar 13, 2020 at 20:20

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