Generalization of relation between charge and magnetic moment

Question

In classical theory, for electrons or protons, charge and angular momentum combine to give the magnetic moment.

Does a similar consequence hold for the generalization of charge to other forces, like the strong force or weak force?

Answer 1

I'm not sure what kind of answer you want, but let me try:

The generalization of electromagnetic forces is what is generally known as a gauge theory, which possesses a symmetry group called the gauge group $\mathcal{G}$. Electromagnetism (EM) is the simplest of gauge theories since it has the simplest gauge group that yields non-trivial physics. In EM, one usually differentiates between the electrical field $E$ and the magnetic field $B$, but they are really not that different, as they both arise through the vector potential $A_\mu$ by calculating the field strength $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. The electric field is then $E_i = F_{0i}$ and the magnetic field is $B^i = \epsilon^{ijk}F_{jk}$, where the roman indices indicate that we only sum over spatial components and Einstein summation is used.

For general gauge theories, this keeps working, it is just that you have one component $A_\mu^a$ of some total vector potential as a Lie-algebra valued function, and to each of these a field strength $F^a_{\mu\nu}$. E.g. in the strong interaction, $\mathcal{G} = \mathrm{SU}(3)$, and there are eight gluons, to each belonging one $A^a_\mu$, and thus one chromoelectric field $E^a_i$ and one chromomagnetic field $(B^i)^a$, so if you want to define a "chromomagnetic moment", you could proceed just as in EM and define it separately for each gluon.

The problem is that such a definition is essentially nonsense, since there are no macroscopic gluon fields. All interaction takes place on a microscopic scale that is inherently quantum, there is no scale at which you could treat the interaction classically. EM is, due to the lack of self-interaction of the photons, the only (interesting) gauge theory which permits a treatment of it in the classical limit that coincides with our naive notions of field lines and electric/magnetic fields. All other gauge theories arise from non-abelian gauge groups, their force carriers interact among themselves, and thus look very different from electromagnetism (e.g. the strong force shows confinement, and no detectable charge relating to it at scales above the subatomic).

In short, I think one can talk about electric/magnetic fields for general gauge theories, it is just not very illuminating, to say the least. The intricacies of gauge theories cannot be understood if one wishes to relate them to electromagnetism, and you should not try to rephrase them in language specific to EM at any cost.