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When reading about the Wu-Yang monopole, it seems to me that the method that Wu & Young used (using patches and gauge transformations to cover $S^2$) is more systematic and general than Dirac's method for the Dirac monopole analysis (and for reaching the Dirac quantisation condition), but it overall seems to me that the two methods describe the same thing: the Dirac monopole and the same quantisation condition.

Is there a difference in the physical results? Also, in what way is the Wu-Young method more general? For example, I know that the gauge group involved here is $U(1)$ (because of the electromagnetic field), so how does the method generalize to other groups? I have seen the derivation of the 't Hooft-Polyakov monopole, which has to do with $SU(2)$, but the method used there has nothing to do with the Wu-Yang method and the end result is a non-singular monopole.

Qmechanic
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TheQuantumMan
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1 Answers1

2

  1. On one hand, Dirac-type and Wu-Yang-type magnetic monopoles usually refer to different mathematical descriptions of the same$^1$ underlying class of physical phenomenon in various space dimensions and with various gauge group:

    • A Dirac-type monopole uses singular Dirac-strings/delta-distributions and globally defined singular gauge fields on $\mathbb{R}^d$.

    • The Wu-Yang bundle construction avoids singularities by considering a punctured space $\mathbb{R}^d\backslash \{0\}$ and locally defined gauge potentials.

  2. On the other hand, a 't Hooft-Polyakov-type monopole is a globally defined, regular, classical field configuration on $\mathbb{R}^d$, cf. my Phys.SE answer here.

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$^1$ However, some authors call the $U(1)$-monopole in 3D for a Dirac monopole and the $SU(2)$-monopole in 5D for a Yang monopole, etc.

Qmechanic
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