• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,862 answers , 20,637 comments
1,470 users with positive rep
502 active unimported users
More ...

Do we have a quantum field theory of monopoles?

+ 2 like - 0 dislike

Recently, I read a review on magnetic monopole published in late 1970s, wherein some conjectures of properties possibly possessed by a longingly desired quantum field theory of monopoles are stated.

My question is what our contemporary understanding of the quantum field theory of monopoles is. Do we have a fully developed one? Any useful ref. is also helpful.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user huotuichang
asked Nov 16, 2013 in Mathematics by sfman (270 points) [ no revision ]

2 Answers

+ 2 like - 1 dislike

This is almost, but not quite, a duplicate of What tree-level Feynman diagrams are added to QED if magnetic monopoles exist?.

In principle quantum electrodynamics includes magnetic monopoles as well as electrons, so yes we do have a theory to describe them. However we expect monopoles to be many orders of magnitude heavier than electrons, and that causes problems trying to describe both with a perturbative calculation.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user John Rennie
answered Nov 16, 2013 by John Rennie (470 points) [ no revision ]
+ 1 like - 0 dislike

This answer is based on David Tong's lectures on solitons - Chapter 2 - Monopoles.

The general answer to the question is that it is known how to construct a quantum mechanical theory of magnetic monopoles acting as individual particles among themselves and also perturbatively in the background of the standard model fields.

t' Hooft - Polyakov monopoles appear as solitons in non-Abelian gauge theories, i.e. as stable static solutions of the classical Yang-Mills-Higgs equations. These solutions depend on some free parameters called moduli. For exmple the center of mass vector of the monopole is a modulus, since monopoles centered around any point in space are solutions since the basic theory is translation invariant. The full moduli space in this case is:

$\mathcal{M_1} = \mathbb{R}^3 \times S^1$.

The first factor is the monopole center of mass, the second factor $S^1$ will provide after quantization an electric charge to the monopole by means of its winding number.

A two monopole solution will have apart of its geometric coordinates an and charge another compact manifold giving it more internal dynamics. This part is called the Atiyah-Hitchin manifold after Atiyah and Hitchin who were the first to investigate the monopole moduli spaces and compute many of their characteristics:

$\mathcal{M_2} = \mathbb{R}^3 \times \frac{S^1 \times \mathcal{M_{AH}}}{\mathbb{Z}_2}$.

The knowledge about the arbitrary Atiyah-Hitchin manifolds is not complete. We can compute its metric and its symplectic structure. It is known thta they are HyperKaehler, which suggests that they can be quantized in a supersymmetric theory. Also, some topological invariants are also known.

These moduli spaces can be quantized (i.e., associated with Hilbert spaces on which the relevant operators can act), and the resulting theory will be a quantum mechanical theory of the monopoles. For example the for the charge 2 monopole one can in principle find the solutions representing the scattering of the two monopoles. It should be emphasized that this is a quantum mechanical theory and not a quantum field theory.

One way to understand that is to let the moduli vary very slowly (although strictly speaking the solutions are only for constant moduli). Then the resulting solutions will correspond to the classical scattering of the monopoles.

Basically, one can find the interaction of the monopoles with the usual fields of the theory by expanding the Yang-Mills theory around the monopole solution, then quantize the moduli space. In particular, the Dirac equation in the monopole background has zero modes which can be viewed as particles in the infrared limit.

This post imported from StackExchange Physics at 2014-08-22 05:08 (UCT), posted by SE-user David Bar Moshe
answered Nov 20, 2013 by David Bar Moshe (3,875 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights