What is a Gauge? Illustrative Explanation

[Winzer]

I looked at the wiki article, and it's not clicking.
If someone could describe it through an illustration that would
be best.

 

[Nick R]

Making an arbitrary (but valid) change in potentials is making a gauge transformation.

For example, in E&M, you have the scalar potential and vector potential that give you the electric and magnetic fields.

You can add the gradient of an arbitrary scalar to the vector potential if you make a corresponding change to the scalar potential, and the changes will not affect E or B at all.

The purpose is to make calculations easier.

Thats how I understand it...

 

[fatra2]

Hi there,

I thought of looking around the web to see what could be this "guage", for which I found many different meaning. I guess this the most stupid question to ask, but could you precise what type of "gauge" you are talking about.

By the way, I suppose that you mean "gAUge" and not "gUAge".

Cheers

 

[Winzer]

Hi there,

I thought of looking around the web to see what could be this "guage", for which I found many different meaning. I guess this the most stupid question to ask, but could you precise what type of "gauge" you are talking about.

By the way, I suppose that you mean "gAUge" and not "gUAge".

Cheers

Your absolutely correct, I should have been more specific.
I am talking about gauge transformations.

Making an arbitrary (but valid) change in potentials is making a gauge transformation.

For example, in E&M, you have the scalar potential and vector potential that give you the electric and magnetic fields.

You can add the gradient of an arbitrary scalar to the vector potential if you make a corresponding change to the scalar potential, and the changes will not affect E or B at all.

The purpose is to make calculations easier.

Thats how I understand it...

How does it make calculations easier?
What else do we gain by doing a gauge transformation?
When are used?

 

[Born2bwire]

In electrodynamics, the underlying properties of the electric and magnetic fields are described by the scalar and vector potentials. While the fields are unique for a given problem, the potentials are not. There is a degree of freedom when choosing our potentials. This causes us to define additional constraints that allow us to find unique solutions to the potentials despite their invariance on the physical solution. This allows consistency in the results and can ease calculations. The shifting of the potentials within the degrees of freedom that still give rise to the same physical fields is a gauge transformation.

One set of constraints is called the Lorenz condition. The Lorenz condition does not fully constrain the potentials but it does give rise to a reduction in the mathematics. This is because it allows us to express the scalar and vector potentials as decoupled inhomogeneous wave equations. By decoupling the potentials, it allows us to use simpler mathematical methods to derive their solutions, like using a dyadic Green's function to relate the sources to the potentials.

 

[Winzer]

In electrodynamics, the underlying properties of the electric and magnetic fields are described by the scalar and vector potentials. While the fields are unique for a given problem, the potentials are not. There is a degree of freedom when choosing our potentials. This causes us to define additional constraints that allow us to find unique solutions to the potentials despite their invariance on the physical solution. This allows consistency in the results and can ease calculations. The shifting of the potentials within the degrees of freedom that still give rise to the same physical fields is a gauge transformation.

One set of constraints is called the Lorenz condition. The Lorenz condition does not fully constrain the potentials but it does give rise to a reduction in the mathematics. This is because it allows us to express the scalar and vector potentials as decoupled inhomogeneous wave equations. By decoupling the potentials, it allows us to use simpler mathematical methods to derive their solutions, like using a dyadic Green's function to relate the sources to the potentials.

Thank you.
Are these transformations mostly applied to E&M? Or can we generalize for other kinds of potentials?
Could anyone give me a simple example calculation example or a link to one?

 

[Winzer]

For others interested, here is a site on gauge transformations:http://everything2.com/title/gauge+transformation

 

[Born2bwire]

Thank you.
Are these transformations mostly applied to E&M? Or can we generalize for other kinds of potentials?
Could anyone give me a simple example calculation example or a link to one?

I have not really seen them outside of electromagnetics. They are much more prominent in quantum field theory but in a way that is still largely connected with electromagnetics. You could of course apply this to any general potential that would allow you to do so. However, I am at a lost of an example outside of electromagnetics where you could do so in a way that gives positive benefits. Heck, I'm sure I have seen an example in some of my mathematics courses, this would seem like something well suited for a lot of abstract problems.