2

Question

Suppose you perform a gauge transformation $f(x)$ that is only $n$ times differentiable, for any $n$. Can the discontinuity in the $(n+1)^{th}$ derivative change any observable?

Clarification

I understand that the answer to the question for small values of $n$ is obvious. If $F_{\mu \nu} = \partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}$, then a gauge transformation, which is insufficiently differentiable, would change the field strength, hence would be detectable. But can one show that for any $n$, some observable in some theory gets affected by such a transformation?

My question is very much related to this one, in fact one of the comments there states that a gauge transformation, which is not $C^{\infty}$, would change the fiber bundle of the gauge field, but I don't see how that would change any observable in any physical theory. Further still, if no observables are changed by such discontinuous gauge transformations, should we not allow them? And if they do change the fiber bundle, then would it not be more correct to think of the physics of the connection as an equivalency class of fiber bundles that are connected by these non-smooth gauge transformations?

I know these are several questions, but their answers become irrelevant if the answer of the first question is in the affirmative.

Stratiev
  • 1,759
  • Related: https://physics.stackexchange.com/q/1324/2451 and links therein. – Qmechanic Jun 30 '20 at 15:04
  • 1
    What do you mean by "should we not allow them"? Who forbids them? There's plenty of cases in physics where you could in principle replace "smooth" by "$n$ times differentiable", but most people just don't bother determining the necessary $n$ and settle for smooth, right? – ACuriousMind Jun 30 '20 at 15:38
  • @ACuriousMind It seems so to me too, but that doesn't mean it's not a meaningful question to ask. One can also just settle to look for gauge transformations that are homotopical to the identity, but there seems to be interesting physics that can be derived from not settling. I was thinking of maybe the existence of higher spin operators like $\partial_{i_1 ... i_m} \phi$ and their correlation functions. Perhaps not restricting the gauge transformations would lead to a restriction on the correlation functions themselves? I'm just wondering and that was a well-defined question to ask(I think)... – Stratiev Jun 30 '20 at 16:20

0 Answers0