1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I How does Gauge fixing fix anything?

  1. Jun 19, 2016 #1
    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?
     
    Last edited: Jun 19, 2016
  2. jcsd
  3. Jun 19, 2016 #2

    Dale

    Staff: Mentor

  4. Jun 19, 2016 #3
  5. Jun 19, 2016 #4

    Dale

    Staff: Mentor

    I'm honestly not sure. I have never actually used the Coulomb gauge.
     
  6. Jun 19, 2016 #5
    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?
     
  7. Jun 19, 2016 #6

    Dale

    Staff: Mentor

    After a brief look it looks like the Coulomb gauge is not a partial gauge fixing, but I am not confident in that.
     
  8. Jun 19, 2016 #7

    Dale

    Staff: Mentor

    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.
     
  9. Jun 22, 2016 #8

    haushofer

    User Avatar
    Science Advisor

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted