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The homogenous Maxwell equations

$$ \partial_\mu \tilde{F}^{\mu \nu} =0 $$

follow "trivially" from the definition of the dual field strength tensor $\tilde{F}^{\mu \nu} = \epsilon^{\mu \nu \sigma \rho} F_{\sigma \rho}. $

They express a conserved current equation for each component $\nu$. For example, for $\nu=1$, we have

$$ \partial_\mu \tilde{F}^{\mu 1} =0 \quad \leftrightarrow \quad \partial_\mu j^\mu =0 ,$$

where I have defined $j^\mu \equiv \tilde{F}^{\mu 1} $.

When we have a conserved current, we have a conserved charge thanks to Gauss' theorem:

$$ Q \equiv \int_V dx^3 j^0 \quad \text{ with} \quad \partial_0 Q =0 .$$

What is this conserved charge?

It doesn't follow from Noether's theorem and is always conserved irrespectively of if the equations of motion are fulfilled or not, i.e. even "off-shell".

In addition, we get one such conserved charge for each component $\nu$ as mentioned above, so there are, in total, 4 conserved charges here.


In terms of the gauge potential $A_\mu$, the conserved charge reads:

$$ Q= \int_V dx^3 j^0 = \int_V dx^3 \tilde{F}^{0 1} = \int_V dx^3 \epsilon^{01 \sigma \rho} F_{\sigma \rho} = \int_V dx^3 \epsilon^{01 \sigma \rho} ( \partial_\sigma A_\rho - \partial_\rho A_\sigma ).$$


The usual electrical charge associated with the electromagnetic field is

$$ Q= \int_V dx^3 j^0 \quad \text{ where } \quad j^\mu = \partial_\nu F^{\nu\mu}.$$ $$ \rightarrow Q= \int_V dx^3 \partial_\nu F^{\nu0} = \int_V dx^3 \partial_nu ( \partial_\nu A_0 - \partial_0 A_\nu ). $$

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  • 2
    $\begingroup$ It is a U(1) “1-form” global symmetry (not a set of 4 ordinary symmetries). The charged operators are ‘t Hooft loops, which you can think of as inserting the world line of a probe magnetic monopole. The charge on a 2 sphere linking the line measures the magnetic flux. See this brief answer I wrote the other day, or the paper generalized global symmetries by Seiberg et al from a couple of years ago $\endgroup$ – Elliot Schneider Mar 14 '18 at 14:40
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Recall that for the ordinary field strength tensor, the conserved charge is the electric charge, and it is computed by integrating the electric flux through a surface at infinity. Taking the dual swaps electric and magnetic fields, so you end up with a surface integral of the magnetic field, i.e. the magnetic charge.

In the usual case the magnetic charge is always zero. If you add weird topology you can get nonzero magnetic charge but you've shown the topology prevents also prevents this charge from changing.

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