# What is the action for an electromagnetic field if including magnetic charge

Recently, I try to write an action of an electromagnetic field with magnetic charge and quantize it. But it seems not as easy as it seems to be. Does anyone know anything or think of anything like this?

• With magnetic monopole charges, $A_\mu$ is no longer well-defined. One has to use $F_{\mu\nu}$ itself - or something similar. Then the two sets of equations of motion can't be simultaneously derived from an action. It's plausible that there exists no action at all although it's not a proven fact. The quantum theory still exists. We deal with various cases of well-defined QFTs that don't have an action, at least a nicely manifestly Lorentz-invariant one, e.g. the (2,0) theory in 6 dimensions. – Luboš Motl Apr 3 '13 at 6:35
• Related: physics.stackexchange.com/q/27755/2451, physics.stackexchange.com/q/22747/2451 and links therein. – Qmechanic Apr 3 '13 at 9:30
• Or introduce a second potential, like here – Retarded Potential Apr 3 '13 at 22:00

## 1 Answer

In the usual formulation, where we have Maxwell's equations $$\partial_{\nu}F^{\mu\nu}=j^\mu$$ $$\partial_{\nu}*F^{\mu\nu}=0$$ a scenario with magnetic charge is included by making the potential pure gauge at infinity, but with a non zero winding number. (Dirac monopole) So this modelling of magnetic charge is a global property, rather than a local one.

If you now try, instead, to do it locally by modifying the second equation above to have a magnetic current on the RHS, then the equation, which normally expresses the fact that $F$ is a closed two form, now no longer has this interpretation. If $F$ is not closed, then it's no longer the derivative of a potential, so the usual incorporation of EM into electrodyamics using the minimal coupling prescription $$\partial_{\mu} \rightarrow \partial_{\mu}-ieA_{\mu}$$ doesn't work. For this reason I'd say it's not possible to do it using the normal gauge theory approach.

Edit: after reading Lubos' comment, I should maybe add the caveat "not possible to do it using the conventional U(1) QED gauge theory in four dimensions"

Edit2: there are apparently also approaches involving two potentials, discussed here and here, but I'm not sure that the proliferation of degrees of freedom is a good thing!

• I understand that it will break gauge invariance. But the classical equation still exists, is there a way to quantize it? – Xiao-Qi Sun Apr 3 '13 at 10:52
• If I have a system represented by the equations $\partial_{\nu}F^{\mu\nu}=j^{\mu}$;$\partial_{\nu}*F^{\mu\nu}=i^{\mu}$ where $j^{\mu}$ is the electric source current and $i^{\mu}$ is the magnetic source current, then to quantize, I'd first need to identify the dynamical variables. In conventional theory this is $A_{\mu}$, but here I can't do this (because F isn't closed), so I don't know how to proceed. – twistor59 Apr 3 '13 at 11:26