Gauge Theory Explained for EE Students

[tim9000]

Is anyone able to explain in basic terms a thick EE student could understand, the significance and a bit about what Guage Fixing, or maybe gauge theory in general is? (or especially for FE simulation)
making heads or tails of
https://en.wikipedia.org/wiki/Gauge_fixing

is not easy. A nice simple electrical or magnetic example would be helpful.

Cheers

 

[mfb]

The easiest example is from electrostatics. What is the potential of a point charge (charge q at the origin)?

$\frac{q}{4\pi \epsilon_0 r} + 1 V$?

$\frac{q}{4\pi \epsilon_0 r} + 0 V$?

$\frac{q}{4\pi \epsilon_0 r} - 1 V$?

$\frac{q}{4\pi \epsilon_0 r} - 2.4642 V$?
...?

All those lead to the same physics, as they all lead to the same electric field (the constant doesn't change the derivative). We are free to choose one to make life easier. The usual choice is the potential with the easiest description:

$\frac{q}{4\pi \epsilon_0 r} + 0 V$ = $\frac{q}{4\pi \epsilon_0 r}$

This is just a single global constant you can choose. If you go to electrodynamics, you get more freedom in the choice of the potentials. There is no "correct" choice, but some can make your life easier, some make it harder.

 

[tim9000]

The easiest example is from electrostatics. What is the potential of a point charge (charge q at the origin)?

$\frac{q}{4\pi \epsilon_0 r} + 1 V$?

$\frac{q}{4\pi \epsilon_0 r} + 0 V$?

$\frac{q}{4\pi \epsilon_0 r} - 1 V$?

$\frac{q}{4\pi \epsilon_0 r} - 2.4642 V$?
...?

All those lead to the same physics, as they all lead to the same electric field (the constant doesn't change the derivative). We are free to choose one to make life easier. The usual choice is the potential with the easiest description:

$\frac{q}{4\pi \epsilon_0 r} + 0 V$ = $\frac{q}{4\pi \epsilon_0 r}$

This is just a single global constant you can choose. If you go to electrodynamics, you get more freedom in the choice of the potentials. There is no "correct" choice, but some can make your life easier, some make it harder.

Ok so the constant doesn't change the equation of the field, just maybe like the point in the field we're looking at. So I see you're saying we may as well choose the constant that is zero to give us an easy potential. But isn't there something to do with the scaling variable?

Thanks

 

[mfb]

Which scaling variable?
In electrostatics (no magnetic field) this is all the freedom you have - you can add a global constant to the potential without changing the field. In electrodynamics, you have more freedom to change the potentials without changing the fields.

 

[tim9000]

Which scaling variable?
In electrostatics (no magnetic field) this is all the freedom you have - you can add a global constant to the potential without changing the field. In electrodynamics, you have more freedom to change the potentials without changing the fields.

Oh, ok. I'm not really sure, I just thought I saw some scaling factor represented by 'Ψ'.
What about magnetostatics etc.?
Thanks

 

[mfb]

There you have more freedom, you can add a time-constant field to A which has zero curl.

 

[tim9000]

There you have more freedom, you can add a time-constant field to A which has zero curl.

Interesting, that's not the 'Ψ' I was talking about perhaps is it?

So what does the time-constant field do? I take it something about it being a path invarient field (curl = 0 is important)
I've always struggled to understand the significance of A, I remember hearing it's a mathematical/theoretical construct, but they found some proof that is real in quantum mechanics? Maybe I'm getting too far off track. lol

 

[mfb]

So what does the time-constant field do?

Nothing in terms of physics, that's the point.

I've always struggled to understand the significance of A, I remember hearing it's a mathematical/theoretical construct, but they found some proof that is real in quantum mechanics?

You can prove effects of electromagnetism exist in a region where there is no magnetic field. The introduction of A solves that apparent puzzle: A cannot be zero in those regions, and while you cannot measure A, you can measure the curl of A integrated over some region. And that again refers to the magnetic field (it has to exist somewhere, but it doesn't have to be where your particles are). That leads to the Aharonov-Bohm-effect.
Gauge invariance can change A, but not the curl of A, so the quantum mechanical measurement doesn't add anything new in terms of gauge invariance.

 

[Jeff Rosenbury]

I think the word "gauge" can be loosely translated as "phase". It is phase for the U(1) group and mathematical extensions of the concept for other groups. Electromagnetism is a (the?) U(1) group. Thus phase (i.e. complex numbers) is important to electrical engineers. Strong force engineers (if any existed) would be more concerned with octonions which are like phase on steroids.

My understanding is that the rules (Maxwell's equations) fall out of the math for a U(1) group. But there are other solutions which also work like mfb points out. Adding one volt to everything doesn't really change how the equations work. We choose to use 0V as the base because it is convenient. But we could define everything with ground at 10,000V. Then a 9V battery would give 10,009 volts output. Of course that's silly. Instead we choose ground to be 0V.

As I understand it, any choice of reference that doesn't affect the underlying algebra is a degree of freedom. This might include a scalar base value as mfb illustrated, or a scaling variable such as unit choice.

Other choices do affect the algebra. The U(2) group uses quaternions. These are non-commutative, or more specifically anti-commutative. This means a⋅b = - b⋅a rather than the more common a ⋅ b = b ⋅ a seen in real and complex numbers. (The U(3) group uses octonions which do something similar with the associative principle.)

I'm not sure why SU(2) and SU(3) are special.

You might get a better, but perhaps less understandable answer on the Quatum Physics forum. This really isn't my field.

 

[tim9000]

Nothing in terms of physics, that's the point.

Ah

You can prove effects of electromagnetism exist in a region where there is no magnetic field. The introduction of A solves that apparent puzzle: A cannot be zero in those regions, and while you cannot measure A, you can measure the curl of A integrated over some region. And that again refers to the magnetic field (it has to exist somewhere, but it doesn't have to be where your particles are). That leads to the Aharonov-Bohm-effect.
Gauge invariance can change A, but not the curl of A, so the quantum mechanical measurement doesn't add anything new in terms of gauge invariance.

I'd never heard of the 'Aharonov-Bohm-effect' until you just mentioned it, that's quite a bit to get my head around. Could you talk a bit more about how 'Gauge invariance can change A, but not the curl of A', or even just the Aharonov-Bohm-effect in general?

So is it similar to if I said I had two electromagnetic coils wound in such a way that their MMFs along any possible path would cancel out and the magnetic flux would be zero (along any reluctance path) yet there would still be an A?

Thanks

 

[tim9000]

I think the word "gauge" can be loosely translated as "phase". It is phase for the U(1) group and mathematical extensions of the concept for other groups. Electromagnetism is a (the?) U(1) group. Thus phase (i.e. complex numbers) is important to electrical engineers. Strong force engineers (if any existed) would be more concerned with octonions which are like phase on steroids.

My understanding is that the rules (Maxwell's equations) fall out of the math for a U(1) group. But there are other solutions which also work like mfb points out. Adding one volt to everything doesn't really change how the equations work. We choose to use 0V as the base because it is convenient. But we could define everything with ground at 10,000V. Then a 9V battery would give 10,009 volts output. Of course that's silly. Instead we choose ground to be 0V.

As I understand it, any choice of reference that doesn't affect the underlying algebra is a degree of freedom. This might include a scalar base value as mfb illustrated, or a scaling variable such as unit choice.

Other choices do affect the algebra. The U(2) group uses quaternions. These are non-commutative, or more specifically anti-commutative. This means a⋅b = - b⋅a rather than the more common a ⋅ b = b ⋅ a seen in real and complex numbers. (The U(3) group uses octonions which do something similar with the associative principle.)

I'm not sure why SU(2) and SU(3) are special.

You might get a better, but perhaps less understandable answer on the Quatum Physics forum. This really isn't my field.

I'm deeply interested in the content of your post, however could you dumb it down a bit for me? How are you defining the word 'phase'? (like a function at a frequency?)
Is an "octonions" to do with spatial (or other) dimensions?

I take your point from your ponient battery example but what was the U(1) and U(2) you were referring to?

Cheers

 

[Jeff Rosenbury]

I'm deeply interested in the content of your post, however could you dumb it down a bit for me? How are you defining the word 'phase'? (like a function at a frequency?)
Is an "octonions" to do with spatial (or other) dimensions?

I take your point from your ponient battery example but what was the U(1) and U(2) you were referring to?

Cheers

Phase as in the phasor representation of a complex number. I'm kind of leaving "phase" undefined. If I carefully define it, it doesn't fit for the word "gauge". But for electrical engineers who use complex math regularly, a gauge field can be thought of as field of complex numbers.

U(1) is a unitary group. Roughly that means a matrix of ones (units). U(1) extends the real numbers by one "dimension" giving phase. U(2) gives extends to quarternions.

 

[tim9000]

Phase as in the phasor representation of a complex number. I'm kind of leaving "phase" undefined. If I carefully define it, it doesn't fit for the word "gauge". But for electrical engineers who use complex math regularly, a gauge field can be thought of as field of complex numbers.

U(1) is a unitary group. Roughly that means a matrix of ones (units). U(1) extends the real numbers by one "dimension" giving phase. U(2) gives extends to quarternions.

Fascinating. Thank you for giving me a few wikipedia pages to read through

 

[mfb]

I'd never heard of the 'Aharonov-Bohm-effect' until you just mentioned it, that's quite a bit to get my head around.

It's the typical experiment for the effect you mentioned.

Could you talk a bit more about how 'Gauge invariance can change A, but not the curl of A'

Similar to the electrostatic case: you can change the potential, but you cannot change its spatial derivatives.

 

[tim9000]

It's the typical experiment for the effect you mentioned.Similar to the electrostatic case: you can change the potential, but you cannot change its spatial derivatives.

hmm, So changing the amount it moves, rather than the way it moves.

What about 'So is it similar to if I said I had two electromagnetic coils wound in such a way that their MMFs along any possible path would cancel out and the magnetic flux would be zero (along any reluctance path) yet there would still be an A?'

Thank you

 

[mfb]

So changing the amount it moves, rather than the way it moves

I don't understand that part.

What about 'So is it similar to if I said I had two electromagnetic coils wound in such a way that their MMFs along any possible path would cancel out and the magnetic flux would be zero (along any reluctance path) yet there would still be an A?'

No.

 

[tim9000]

Well when you say spatial derivatives I assume you mean you cannot change like the first and second derivatives for x, y, z, but you can move like the parallel plates appart further so the particle has further to travel?

No.

No in the sense that if you have a net MMF of zero along all paths (like a counter wound toroid) there will still be flux? (somehow?) or in someother way about magnetic vector potential? ( a quantity I only sort of understand)

Thanks!

 

[mfb]

Well when you say spatial derivatives I assume you mean you cannot change like the first and second derivatives for x, y, z, but you can move like the parallel plates appart further so the particle has further to travel?

We do not change the physical setup at all.
All we change is the description by changing the potential without changing the fields. The electric field is the spatial derivative of the potential.

No in the sense that if you have a net MMF of zero along all paths (like a counter wound toroid) there will still be flux? (somehow?) or in someother way about magnetic vector potential? ( a quantity I only sort of understand)

No in the sense that your setup would not do anything you describe and has no similarity to the topics discussed before.

 

[tim9000]

We do not change the physical setup at all.
All we change is the description by changing the potential without changing the fields. The electric field is the spatial derivative of the potential.
No in the sense that your setup would not do anything you describe and has no similarity to the topics discussed before.

Alright