# Gauge choice for a magnetic vector potential

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## Main Question or Discussion Point

How do we verify whether a condition on the magnetic vector potential A constitutes a possible gauge choice ?
Specifically, could a relation in the form A x F(r,t) be a gauge , where F is an arbitrary vector field?

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How do we verify whether a condition on the magnetic vector potential A constitutes a possible gauge choice ?
Specifically, could a relation in the form: A x F(r,t) = 0, be a gauge , where F is an arbitrary vector field?

vanhees71
Gold Member
2019 Award
This looks a bit too much constrained, because it practically says that $\vec{A}=\lambda \vec{F}$, but you can impose only one "scalar" condition like
$$\frac{1}{c} \partial_t \Phi + \vec{\nabla} \cdot \vec{A}=0 \qquad \text{(Lorenz gauge)},$$
$$\vec{\nabla} \cdot \vec{A}=0 \qquad \text{(Coulomb gauge)}.$$

In that case, would a condition like ∫∫ A x F(r,t) dS = 0 , be acceptable as a guage?

More generally, could a condition like ∫∫ L(A) x F(r,t) dS = 0 be a gauge, where L is a linear operator?

vanhees71
Gold Member
2019 Award
I've never seen gauge-constraints involving integrals. This looks very complicated. What do you think it may be good for?

This allows simplifying some expressions.

Are there any criteria to judge the validity of such a condition?

Are there any reference that mention different gauge conditions other than Coulomb and Lorenz conditions?

vanhees71
Well, if it helps you with a concrete example, the only validation is to check that the final solutions for the physical fields, $\vec{E}$ and $\vec{B}$, really solve the problem. In the literature there are also more gauge fixing conditions, particularly in QFT. Some simple ones are temporal gauge, $A^0=0$, or axial gauge $A^3=0$, and also the $R_{\xi}$ gauges, based on the action principle rather than a specific constraint for the gauge fields. There are also some more less common ones. See