The discussion centers around the concept of gauge transformations, particularly in the context of electromagnetism (E&M). Participants explore the definition, implications, and applications of gauge transformations, seeking clarity and illustrative examples.
Participants generally agree on the basic definition of gauge transformations but express differing views on their applications and benefits outside of electromagnetism. The discussion remains unresolved regarding the generalization of gauge transformations to other types of potentials.
Participants note that while gauge transformations are well understood in E&M, their application in other fields remains less clear, and there is a lack of concrete examples outside of electromagnetics.
Your absolutely correct, I should have been more specific.fatra2 said: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
How does it make calculations easier?Nick R said: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...
Thank you.Born2bwire said: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 said: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?