Gauge choice for a magnetic vector potential

• Lodeg
In summary, the condition A x F(r,t) = 0 may be a gauge choice if it solves the physical field equations.
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?

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?

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?

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?

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

Thank you very much

1. What is a magnetic vector potential?

A magnetic vector potential is a mathematical construct used in electromagnetism to describe the magnetic field in a given region. It is a vector quantity that is related to the magnetic field through the equation B = ∇ x A, where B is the magnetic field and A is the magnetic vector potential.

2. Why is gauge choice important for a magnetic vector potential?

Gauge choice refers to the freedom to choose a particular mathematical form for the magnetic vector potential, as long as it satisfies certain conditions. The choice of gauge can affect the physical interpretation of the magnetic vector potential and the resulting magnetic field. Therefore, it is important to carefully consider the gauge choice when working with magnetic vector potentials.

3. What are the different gauge choices for a magnetic vector potential?

There are several gauge choices that are commonly used in electromagnetism, including the Coulomb gauge, the Lorenz gauge, and the transverse gauge. Each of these gauge choices has its own advantages and disadvantages and is suitable for different situations. It is important to understand the differences between these gauges in order to choose the most appropriate one for a given problem.

4. How does gauge choice affect the physical interpretation of the magnetic vector potential?

The gauge choice can affect the physical interpretation of the magnetic vector potential in several ways. For example, in the Coulomb gauge, the magnetic vector potential is related to the electric potential, while in the Lorenz gauge, it is related to the electric and magnetic fields. Additionally, different gauges can lead to different mathematical expressions for the same physical situation, making it important to carefully consider the physical interpretation of the chosen gauge.

5. Can gauge transformations be used to change the magnetic vector potential?

Yes, gauge transformations can be used to change the mathematical form of the magnetic vector potential without affecting the physical situation. This is because gauge transformations do not change the physical fields, but only the mathematical representation of those fields. However, it is important to note that not all gauge transformations are physically meaningful, and some may lead to unphysical solutions.

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