# Gauge invariance of the vector potential

• muscaria
In summary, the vector potential can be expressed in the following way: the magnetic field is invariant with regard to this calibration (just basic gauge invariance), and the divergence of the vector potential has no observable consequence.
muscaria
The vector potential can be expressed in the following way:

∇^2 Ay-∂/∂y (∇∙A)=-μJy

(Here only taking y components)

Vector A is not determined uniquely. We may add derivatives of an arbitrary function (gradient) to the components of A, and the magnetic field does not change (curl of gradient is zero). In other words: the magnetic field is invariant with regard to this calibration (just basic gauge invariance)
The arbitrary function is usually chosen so that we obtain the simplest form for the equation of A. The function is chosen so that ∇∙A=0 (lorentz condition) and we get

∇^2 A=-μJ

By doing this, don't we consider the divergence of A to be completely unphysical? The Lorentz condition is formulated purely as a mathematical convenience. Is it accepted mainly because we think that the magnetic field represents the whole physical effect? Shouldn't the divergence of A have a physical impact on the polarisation of the medium/vacuum? Maybe it could be observed in other non-transverse EM waves? Furthermore, the aharonov-bohm effect proves the physical impact of A even in a region with a vanishing B.
I think that there may be different types of waves (longitudinal) but that the classical theory would not account for them because it focuses on E and B as being the fundamental quantities, and disregarding the totallity of the effects of A and φ (electrostatic) potentials. Also for example the rate of change of φ with respect to time.
Has anyone ever come across this? Anyway, any kind of constructive criticism would be greatly appreciated.. Cheers

No, there's no observable consequence to choosing a particular gauge. You started off by (correctly) saying that the gauge doesn't affect the magnetic field. Even in the Ahonarov-Bohm effect, potentials are not measured directly. Instead, you're effectively observing their line integrals around loops. These are not affected by gauge transformations:
$$\oint ( A + \nabla \chi ) \cdot \mathrm{d} l = \oint A \cdot \mathrm{d} l .$$

I'm not sure what you mean with regards to longitudinal waves. These can exist in certain modifications to Maxwell's theory, but there is no experimental evidence for them. Gauge concepts certainly do change if you want to move away from Maxwell's equations.

Could you tell me which kind of modifications of the Maxwell/Heavyside equations please, do they include divergence of A, or send me a link to something, would be great!

I think i wasn't really clear enough in what i was saying, hopefully this might make sense.
The vector potential is defined purely with respect to B, or as a concequence of the effect of B..When doing this we say A has a purely rotational aspect and disregard a possible effect of the divergence of A on the medium.
Wouldn't the divergence of A have an impact on the medium and therefore shouldn't be reduced to zero because "it may be wrongly defined" (gauge invariance on A works mathematically because of how it's defined)
A fair amount of Nicola Tesla's papers describe how to generate longitudinal waves (would be interesting to build!).
He obtained them by parametrizing the frequency of changing scalar potential (which is not very considered in classical theory).
Another recent experiment by Wesley and monstein seems to be confirming their existence..although i haven't looked into that enough yet.

## What is gauge invariance of the vector potential?

Gauge invariance of the vector potential is a fundamental principle in electromagnetism that states the choice of gauge or mathematical representation of the vector potential does not change the physical phenomena being observed. This means that the equations and solutions for electromagnetic fields remain the same regardless of the specific mathematical representation used.

## Why is gauge invariance important?

Gauge invariance is important because it allows for a more flexible and intuitive approach to solving problems in electromagnetism. It also ensures that the physical predictions and measurements of electromagnetic phenomena are independent of the choice of gauge, making the theory more robust and reliable.

## How does gauge invariance affect the vector potential?

Gauge invariance affects the vector potential by allowing for different mathematical representations of the same physical phenomena. This means that the vector potential can have different values in different gauges, but the resulting electromagnetic fields will be the same. Gauge transformations can also be used to simplify the vector potential and make calculations easier.

## Can gauge invariance be violated?

No, gauge invariance is a fundamental principle in electromagnetism and cannot be violated. However, certain approximations or simplifications in calculations may temporarily seem to violate gauge invariance, but this is due to the limitations of those methods and not a true violation of the principle.

## How is gauge invariance related to electric charge conservation?

Gauge invariance is related to electric charge conservation through the concept of local gauge invariance. This states that the laws of electromagnetism must remain the same at every point in space and time. This leads to the conservation of electric charge, as the gauge transformations must also conserve electric charge in order to maintain gauge invariance.

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