# How does Gauge fixing fix anything?

• I
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 Aμ like Aμ → A'μ = Aμ + ∂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 + ∂μμ Λ = 0
μμ 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 + ∂ii Λ = 0
ii Λ = 0

So now we have two conditions on the functions Λ
ii 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?

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Dale
Mentor
2020 Award
I'm honestly not sure. I have never actually used the Coulomb gauge.

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
Mentor
2020 Award
After a brief look it looks like the Coulomb gauge is not a partial gauge fixing, but I am not confident in that.

Dale
Mentor
2020 Award
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