How does Gauge fixing fix anything?

[carllacan]

I've looked everywhere and I haven't found an explanation of why is it useful to introduce gauge conditions. I've also searched in this forum, and none of the existing threads I've read answered my question. I apologize if there is and I have failed to find it.

My problem is that, as I see it, gauge fixing fixes nothing. We can still perform transformations that will not affect the field. Let me expose my reasoning:

The electromagnetic field potential is undefined up to the derivative of a function because if we transform A^μ like A^μ → A'^μ = A^μ + ∂^μ Λ the field is the same. In order to fix this we impose ∂_μ A ^μ = 0.

Now A needs to satisfy this condition, and so does A', so we have
∂_μ A' ^μ = 0
∂_μ A ^μ + ∂_μ ∂^μ Λ = 0
∂_μ ∂^μ Λ = 0

and now we have restricted the kinds of Λ we can use for the gauge transformations, but we still haven't ruled out all of them, so we can impose Coulomb's conditions ∂ⁱ A _i = 0 and A₀ = 0.

Now A and A' need to satisfy these conditions, so we have

∂_i A' ⁱ = 0
∂_i A ⁱ + ∂_i ∂ⁱ Λ = 0
∂_i ∂ⁱ Λ = 0

So now we have two conditions on the functions Λ
∂_i ∂ⁱ Λ = 0
Λ₀ = 0

and we have restricted the transforms we can do, but we haven't ruled out all of them, since we can still transform using a Λ that satisfies those conditions. Shouldn't the gauge fixing rule out every possible function Λ, so there is one and only one potential for every field?

 

[Dale]

The Lorenz gauge is only a partial gauge fixing
https://en.m.wikipedia.org/wiki/Lorenz_gauge_condition

 

[carllacan]

The Lorenz gauge is only a partial gauge fixing
https://en.m.wikipedia.org/wiki/Lorenz_gauge_condition

Ok, but what about the Coulomb gauge?

 

[Dale]

I'm honestly not sure. I have never actually used the Coulomb gauge.

 

[carllacan]

I'm honestly not sure. I have never actually used the Coulomb gauge.

Oh, ok.

But, Coulomb gauge aside, is it not a problem that the Lorentz condition only fixes the problem partially? What is the point of using it then?

 

[Dale]

After a brief look it looks like the Coulomb gauge is not a partial gauge fixing, but I am not confident in that.

 

[Dale]

What is the point of using it then?

The point is that it is a relativistically covariant gauge condition. If it is satisfied in one frame then it is satisfied in all frames. The same is not true of the Coulomb gauge, which is why I never use it.

Many times covariance is more important than uniqueness.

 

[haushofer]

Shouldn't the gauge fixing rule out every possible function Λ, so there is one and only one potential for every field?

No, not necessarily. There are many cases in which you are left with some remaining gauge degrees of freedom, like in General Relativity and String Theory. Like Dale says, it depends on what you want. Covariant gauge choices are nice because your calculations remain explicitly covariant. But your calculation shouldn't depend on your gauge choice.

If you are a bit familiar with string theory, one case in which this is clear is in analyzing the equations of motion. You can do this by imposing non-covariant gauge conditions (the so-called "static gauge"), but at the end of your analysis you have to check again that everything is still covariant (you have to check again if the spacetime coordinates satisfy the Lorentz algebra). And that is a very tedious calculation, I can tell you. This turns out to be the case only if (among other things) a certain amount of spacetime dimensions is assumed. In a covariant analysis you don't have to do this, and there the same amount of spacetime dimensions is again needed but for a different reason. Same result, but different calculations.

Fixing a gauge is similar to fixing a coordinate system in classical mechanics. Often, there you want to fix coordinates up to a Galilei transformation. This leaves you with the class of inertial observers. Only in particular calculations you want to uniquely fix the coordinates, such that you stick to one observer at one place at one instant in time.