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A famous locally gauge invariant quantity is

$$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \, , $$ which is interpreted as the measurable electric and magnetic field strengths.

Now, another locally gauge invariant quantity is

$$ C_\mu = \partial_\mu \varphi(x) - A_\mu(x) \, , $$ where $ \varphi(x)$ is the phase of the wave function $\Psi(x) = A e^{i \varphi(x)}$, which, for example, describes a single electron.

What's the standard interpretation of this gauge invariant quantity $C_\mu$?


A local gauge transformation is

$$ \Psi(x) \to e^{i \epsilon(x)} \Psi(x) $$ $$ A_\mu(x) \to A_\mu(x) + \partial_\mu \epsilon(x) $$

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  • $\begingroup$ 1. Your question is tagged quantum-electrodynamics, but in QED there is a fermion field, and no single wavefunction. Are you doing more traditional QM where a single particle is coupled to the EM field instead? 2. $\varphi(x)$ is meaningless because absolute phases are meaningless, so why would you expect $C_\mu(x)$ to have any significance? $\endgroup$ – ACuriousMind Sep 29 '18 at 13:30
  • $\begingroup$ @ACuriousMind 1.) yes I'm considering QM and changed the tag accordingly. 2.) This quantity is defined, for example, by Maldacena in arxiv.org/abs/1410.6753 Eq. 6.4 and he calls it the “gauge invariant gradient of the field ϕ”. At first I thought he means the covariant derivative, but this is clearly not the case, so I was wondering about the physical meaning of the quantity and how it shows up in a more conventional context. $\endgroup$ – jak Sep 29 '18 at 13:41
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Up to a constant, this is the kinetic momentum, the quantum generalization of $mv$. You know that in a magnetic field the canonical momentum picks up an extra $eA$, and here we’ve just subtracted it back out again. The kinetic momentum measures how much “oomph” is in the particles if they hit you.

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  • $\begingroup$ This is indeed the correct answer, and an explicit demonstration of the statement can be found here en.wikipedia.org/wiki/… $\endgroup$ – jak Oct 2 '18 at 8:36
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In a superfluid, this would be the fluid velocity.

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