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In scalar QED, applying Noether's theorem for internal global symmetries results in a Noether current that is dependent on the gauge because of the presence of the covariant derivative.

$$j_\mu=-i(\varphi^*\partial_\mu\varphi-\varphi\partial_\mu\varphi^*)-2eA_\mu\varphi^*\varphi.$$

When integrating the current to get the conserved charge, the term dependent on the gauge doesn't obviously seem to cancel out.

My question is how do you interpret this conserved charge that depends on the gauge field? For example, when you take a free complex scalar field, the conserved charge is very simply interpreted as the difference between the number of particles and anti-particles. However, I can't find a similar simple interpretation when it is coupled to a gauge field.

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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/137061/2451 and links therein. $\endgroup$
    – Qmechanic
    May 13, 2019 at 10:27
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    $\begingroup$ The current must be gauge invariant so it must depend on the gauge field. $\endgroup$
    – Prahar
    Jul 30, 2021 at 9:01

1 Answer 1

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The total current is gauge invariant, because the first term is also not gauge invariant, because of the partial derivative and not covariant derivative. In fact the current is just like for free charged scalar just replacing partial derivative by covariant derivative.

As intuition to the appearance of the second term, now the matter current does change, and only combination of some current attributed to the field, as happens for momentum and energy for example

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