# What's the physical meaning of the gauge invariant quantity $\partial_\mu \varphi(x) - A_\mu(x)$?

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

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

• 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? – ACuriousMind Sep 29 '18 at 13:30
• @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. – jak Sep 29 '18 at 13:41

## 2 Answers

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.

• This is indeed the correct answer, and an explicit demonstration of the statement can be found here en.wikipedia.org/wiki/… – jak Oct 2 '18 at 8:36

In a superfluid, this would be the fluid velocity.