Gauge choice for a magnetic vector potential

[Lodeg]

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?

 

[Lodeg]

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]

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
$$\frac{1}{c} \partial_t \Phi + \vec{\nabla} \cdot \vec{A}=0 \qquad \text{(Lorenz gauge)},$$
$$\vec{\nabla} \cdot \vec{A}=0 \qquad \text{(Coulomb gauge)}.$$

 

[Lodeg]

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

 

[Lodeg]

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

 

[vanhees71]

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

 

[Lodeg]

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

https://en.wikipedia.org/wiki/Gauge_fixing

 

[Lodeg]

Thank you very much