Gauge Theory and fundamental fields not being able to be measured

[Stratus]

Hello again everybody, i'm back again for another question that i've come across in my studies.

 

According to many books I've read, a feature of gauge theory is that the fundamental field from which the excitations come from can not be measured in any way, but change by a gauge transformation. Whereas the excitation's observable quantities like charge and velocity can be measured, but can not be changed by a gauge transformation.

 

That makes sense to me, but this next part is where I have trouble.

 

I came across this problem in a book, then went to Wikipedia looking for an answer to my question. In Wikipedia's words

"For example, in electromagnetism the electric and magnetic fields, E and B, are observable, while the potentials V ("voltage") and A (the vector potential) are not.^[3] Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B."

Is this saying that it's impossible to measure the potential V, and in that case, how do voltage meters work? I'm not familiar with vector potential yet myself, but I imagine that there would be a way to calculate or measure it as well.

 

As well, I understand a gauge transformation is a "change in a fields configuration". what is referred to here as being configured differently? I don't understand what is being configured differently in one of the fundamental fields?

Thank you

[ajb]

A is the electromagnetic potential, it is a field on space-time. It is not the potential difference.

 

I don't know how much you know about differential forms, but let us consider the de Rham differential d which takes an n-form to an n+1 form. The key property is that dd =0, in fact that is all that we really need here.

 

The electromagnetic field A is a one-form and the field strength is F(A)=dA. Now consider A'= A+df, where f is a zero form. You see that F(A') = d(A+df) = dA =F(A). This is a gauge transformation and you see that the field strength is unchanged.

 

Now the components of F are related to the electric and magnetic fields.

Edited September 20, 2013 by ajb

[Stratus]

A is the electromagnetic potential, it is a field on space-time. It is not the potential difference.

 

I don't know how much you know about differential forms, but let us consider the de Rham differential d which taken an n-form to an n+1 form. The key property is that dd =0, in fact that is all that we really need here.

 

The electromagnetic field A is a one-form and the field strength is F(A)=dA. Now consider A'= A+df, where f is a zero form. You see that F(A') = d(A+df) = dA =F(A). This is a gauge transformation and you see that the field strength is unchanged.

 

Now the components of F are related to the electric and magnetic fields.

hmm i don't think i'm quite there yet, as I somewhat understand the equations, but am not in calculus now. Hopefully I'll be able to re-read this later and it would make more sense (i'm in trig/pre-calc right now). I know a little bit about differential forms, but not quite enough right now.

 

seems like most things in physics are currently over my head, but hopefully it will make more sense as i proceed in my math studies. I always have a habit of trying to skip the basics and understand the complex stuff, one of my character flaws.

 

right now I think i'll just continue trying to understand what I can until I get far enough into math

[ajb]

right now I think i'll just continue trying to understand what I can until I get far enough into math

Differential forms are considered quite advanced mathematics for standard physics, though they are very important.

 

The key here is we have a map d and this map "squares to zero" thus d(A +df) =dA. For EM theory this is what we mean by a gauge transformation. You see you have the freedom of adding a df to every potential A and you do not change the field strength F. This means that physically you cannot define A absolutely, you always have the ambiguity up to the function f.

 

Picking a gauge is really just declaring an A and fixing it. That is you remove the freedom of adding a df. None of the physics should depend on this choice.

Edited September 20, 2013 by ajb

[Stratus]

Differential forms are considered quite advanced mathematics for standard physics, though they are very important.

 

The key here is we have a map d and this map "squares to zero" thus d(A +df) =dA. For EM theory this is what we mean by a gauge transformation. You see you have the freedom of adding a df to every potential A and you do not change the field strength F. This means that physically you cannot define A absolutely, you always have the ambiguity up to the function f.

 

Picking a gauge is really just declaring an A and fixing it. That is you remove the freedom of adding a df. None of the physics should depend on this choice.

Ah, so if I understand what you're saying, that because we can always add another df to A, we can not say that we know exactly what A is. But, because adding another df effectively makes no different, it does not change the field strength F? So, a transformation would really just be having the freedom to add another df to the equation? At least in EM theory?

 

Also, if I can ask, which areas of mathematics are these ideas normally covered? Such as, which areas of mathematics are most commonly used in physics? Ideally particle physics.

Edited September 21, 2013 by Stratus

[ajb]

Ah, so if I understand what you're saying, that because we can always add another df to A, we can not say that we know exactly what A is. But, because adding another df effectively makes no different, it does not change the field strength F? So, a transformation would really just be having the freedom to add another df to the equation? At least in EM theory?

Yes, that is the idea.

 

It is a little more complicated for general gauge theories, but the idea is the same.

 

Also, if I can ask, which areas of mathematics are these ideas normally covered? Such as, which areas of mathematics are most commonly used in physics? Ideally particle physics.

Differential geometry.