# Dirac quantization of non-Abelian monopoles or instantons

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In a 3+1 dimensional $U(1)$ gauge theory the Dirac quantization condition (and its generalization to dyons) states for any two dyons with electric charge $q_{e1}$ and $q_{e2}$ and magnetic charge $q_{m1}$ and $q_{m2}$, respectively, the following condition needs to be satisfied so that the theory is a consistent quantum theory:
$\begin{equation} q_{e1}\cdot q_{m2}-q_{e2}\cdot q_{m1}\in\mathbb{Z} \end{equation}$

in proper units. The simplest way to see this is to require that the Dirac string coming out from the monopole being invisible to local detection.

People have also studied monopoles and instantons of non-Abelian gauge fields. Among the literature I found, in this subject people usually study the classical solution to the field equation with finite action, and a classic example will be the BSTP instanton of $SU(2)$ gauge field. All these are classical considerations.

Quantum mechanically speaking, should there be a generalization of the Dirac quantization condition to the instantons of non-Abelian gauge fields?

The condition that I am imagining is that inserting an instanton in the system will not have any observable effect to local probes far away from the point where this instanton is inserted. More specifically, consider the following thought experiment for an $SU(2)$ gauge theory with a scalar particle in the fundamental representation. Dragging this particle slowly around a closed loop may result in some (dynamical and Aharonov-Bohm) phase (in this case a unitary matrix in general). Now insert an instanton at the spacetime origin, and dragging this particle slowly around the same closed loop again. We can compare the unitary matrices obtained before and after the insertion of instantons. I think for a legal instanton the resulting unitary matrices should be the same. This condition may put some constraints on which instantons are allowed quantum mechanically.

Is this correct? If so, what is the corresponding condition for the $SU(2)$ instanton then? I could not find any paper on this in the literature, and I appreciate if anyone gives an answer or provides relevant references.

recategorized Aug 13, 2016

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A very good review may be found here: http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf.

With a particle one derives the Dirac quantization condition by looking at loops of charged particles around each other.  To define the action you need to complete the loop into a surface over which the field strength can be integrated.  Different choices of surface must give the same answer up to unobservable (i.e., 2\pi) phases.  The strongest condition then comes from looking at the flux through a two sphere surrounding the particle.  So, for particles, we need to look at flux through a two sphere, which amounts to requiring that the topology of the U(1) bundle on that sphere is well defined.

For instantons there is one more transverse dimension so we should look at three sphere and require that the bundle be well defined here as well.  This leads to a quantization condition which is exactly the analog of the Dirac condition.  The reference cited explains this in greater detail. answered Jan 19, 2017 by anonymous

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