# Invariance of maxwell's equations under Gauge transformation

• jacobrhcp
Also I'm not sure what you're trying to say.In summary, the original problem is to show that the source-free Maxwell equations are invariant under a local gauge transformation. This means that the field strength tensor defined in terms of the potentials A and phi is unchanged when the potentials are transformed according to a gauge transformation. This is because the potentials are not unique and can be changed by adding constants without affecting the fields. To prove this, one must show that substituting the gauge transformation into the definition of the field strength tensor results in the same tensor. This can be done by considering the connection between the potentials and how the field strength tensor is defined in terms of them. Additional resources, such as the chapters on relativity and gauge freedom in
jacobrhcp
[SOLVED] invariance of maxwell's equations under Gauge transformation

## Homework Statement

Show that the source-free Maxwell equations $$\partial_{\mu} F^{\mu\nu}=0$$ are left invariant under the local gauge transformation

$$A_{\mu}(x^{\nu})\rightarrow A'_{\mu}(x^{\nu})=A_{\mu}(x^{\nu})+\frac{\partial}{\partial x^{\mu}}\varphi(x^{\nu})$$

for an arbitrary scalar function $$\varphi(x^{\nu})$$

## Homework Equations

The definition of the field strength tensor: $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

## The Attempt at a Solution

This is a course I don't quite have the foreknowledge of. There is no book, so some things I can't quite figure out. The teacher gave the solution on the blackboard already, so that's not the real problem. I wrote it down real good and neat, but I don't get it at all.

I just don't know what a Gauge transformation is, or what the definition of maxwell's equation in that tensor is, I've only seen it as four equations just yet (both integral and differential form). If someone could just help me understand the problem, or give me some useful links to understand it, I'd appreciate it a lot... or mainly what the bigger part of those symbols mean.

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You could just show that field strength tensor itself is invariant under the gauge transformation. If you substitute the gauge transformation into the definition of the field strength tensor this is just a matter of mixed partial derivatives being equal. I.e. d^2f/dxdy=d^f/dydx. Isn't it?

Read the chapter on relativity in Griffith's electrodynamics book. And the chapter about gauge freedom too.

Short digest:

Remember that in a any given situation involving electromagnetic fields, the potentials phi and A are not unique. All that matters is that when differentiated according to maxwell's equations, they give the right fields E and B. For instance, adding a constant c1 to phi and a constant vector c2 to A does not change the derivatives, and so phi' = phi + c1 and A' = A + c2 are again valid potentials for the physical situation. Choosing a value for phi and A is called choosing a gauge, and a switch from one gauge to another, such as going from phi and A to phi' and A' above is called a... gauge transformation.

As it turns out, for any function f(x,t), the gauge transformation phi' = phi - df/dt and A' = A + grad(f) leaves the fields unchanged! (but you got to change phi and A simultaneously!)

This is just what the question is asking you to show.

Dick: that's right.

Quasar: thanks, and in my house there are like 5 copies of Griffiths and I my teacher pointed out a specific chapter in Peter Jackson's classical electrodynamics I should read to catch up. So I'm confident it should work out :)

Because I am stupid and i don't understand can anybody explain me better how it turns out that if i substitute Aμ' =Αμ +θμF ,F scalar fuction, i get the same Fμν ??

Well, you need to show that

$$F'_{\mu\nu} \left(A'_{\lambda}\left(A_{\sigma}(x)\right)\right) = F_{\mu\nu} \left(A_{\sigma}\right)$$

knowing the connection between the potentials A'(A) and how F is defined in terms of A.

 The LaTex is apparently not working properly.

[Later Edit] See post #7.

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You're missing a \ before the first right.

## 1. What is the significance of the invariance of Maxwell's equations under Gauge transformation?

The invariance of Maxwell's equations under Gauge transformation means that the fundamental laws of electromagnetism remain unchanged regardless of the choice of gauge or reference frame. This allows for a more universal and consistent understanding of electromagnetic phenomena.

## 2. How does Gauge transformation affect the equations of electromagnetism?

Gauge transformation does not change the actual equations of electromagnetism, but rather changes the way in which they are expressed. It allows for different mathematical representations of the same physical phenomenon, making it easier for different observers to understand and analyze the same system.

## 3. What is the role of the gauge field in Maxwell's equations?

The gauge field is a mathematical construct that represents the electromagnetic potential in the equations of electromagnetism. It is responsible for maintaining the invariance of Maxwell's equations under Gauge transformation.

## 4. Can you provide an example of how Gauge transformation is used in practical applications?

One example of how Gauge transformation is used in practical applications is in quantum field theory. Here, Gauge invariance is a fundamental principle that is used to describe the behavior of particles and their interactions with electromagnetic fields.

## 5. Are there any limitations to the invariance of Maxwell's equations under Gauge transformation?

While Gauge transformation is a powerful concept that allows for a better understanding of electromagnetism, it does have its limitations. For instance, it cannot account for certain phenomena such as the Aharonov-Bohm effect, which requires a modified version of Maxwell's equations.

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