Can anybody explain the physical difference between Dirac monopole and Polyakov monopole?

First, let me write down what I know briefly.

Dirac monopole

  1. It comes from the symmetry of Maxwell equation. By assuming that magnetic field for a point source magnetic charge $g$.

\begin{align} B(r,t) = \frac{g}{4\pi r^2} \frac{\vec{r}}{r} \end{align} Since the divergence of $B$ gives non-vanishing value due to delta function $\nabla \cdot \nabla(\frac{1}{r})=\delta(r)$. Thus we introduce the so-called Dirac String, ($i.e$, add some solenoid field)

  1. Dirac string is non-obeservable due to Dirac's charge quantization

Polyakov-'t Hooft monopole.

  1. It comes from soliton Dynamics. $i.e$ $SO(3)$ model

  2. We can compute the mass (Energy)

  3. For large distance Polyakov-'t Hooft monopole behaves like Dirac monopole

You can comment anything including above things.
This question arise from the comment of my previous question [Compact QED and Non-compact QED - Polyakov textbook ] by Stephen Powell

  • $\begingroup$ I'm not sure what your question is. The Dirac monopole is a solution for the pure gauge theory, the Polyakov-'t Hooft monopole arises in the presence of a Higgs-like symmetry breaking. So they aren't the same because they aren't solutions for the same theory. What exactly is your question about that? (Also, I don't think the "Polyakov" monopole the answer by StephenPowell mentions are the "Polyakov-'t Hooft monopoles" one usually speaks of in the continuum.) $\endgroup$
    – ACuriousMind
    Commented Aug 27, 2015 at 13:42
  • $\begingroup$ @ACuriousMind, Oh... Polyakov monopole and Polyakov-'t Hooft monopoles are different?...Maybe i will modify the question after finding some proper references. $\endgroup$
    – phy_math
    Commented Aug 27, 2015 at 13:54
  • $\begingroup$ @ACuriousMind, What i found on the google, was note.pdf which cover the dirac monopole and Polyakov-'tHooft monopole $\endgroup$
    – phy_math
    Commented Aug 27, 2015 at 13:56
  • 1
    $\begingroup$ @ACuriousMind. Can you recommend some materials dealing with Polyakov monopole and Polyakov-'t Hooft monopole? $\endgroup$
    – phy_math
    Commented Aug 27, 2015 at 13:59

1 Answer 1

  1. A (generalized) 't Hooft-Polyakov monopole and

  2. a Dirac monopole with a Dirac string attached

are two types of magnetic monopoles, which differ in several ways, as OP and user ACuriousMind correctly state.

  1. On one hand, a (generalized) 't Hooft-Polyakov monopole is a regular, soliton-like, finite-energy solution to the classical Euler-Lagrange field equations of some GUT (with an action principle that extends the standard model). Its existence is unavoidable if a certain topological condition is satisfied in the GUT.

  2. On the other hand, while Dirac monopoles were mostly conceived by Dirac as a theoretical laboratory to study charge quantization, the modern interpretation is that a Dirac monopole is an effective description far away from the monopole that fails near the finite core region of the monopole. Moreover a Dirac monopole requires a non-standard action principle, cf. e.g. this Phys.SE post and links therein.

For further differences and details, see Ref. 1 and the linked Wikipedia pages.


  1. F.A. Bais, To be or not to be? Magnetic monopoles in non-abelian gauge theories, arXiv:hep-th/0407197. (Hat tip: Hunter.)
  • $\begingroup$ Thanks for kind answer. Can you give me some example(or comment) of topological condition related with 't Hooft-Polyakov monopole? $\endgroup$
    – phy_math
    Commented Aug 30, 2015 at 23:33

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