# The residual gauge symmetry of Yang-Mills theory after Wick rotation

I am a bit puzzled by a statement in this question here. In particular, the claim that the residual gauge symmetry in Yang-Mills theory disappears upon Wick rotation to the Euclidean theory.

For concreteness, lets consider a gauge group $$U(1)$$, so our action is simply $$S=-\frac{1}{4}\int d^{4}x F_{\mu\nu}F^{\mu\nu}$$ Where $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ is the field strength tensor, and for now we are in Minkowski space with signature $$(-1,+1,+1,+1)$$.

It is clear that this action has a gauge symmetry, $$A_{\mu}\to A_{\mu}':=A_{\mu}+\partial_{\mu}\Lambda$$, for an arbitrary scalar function $$\Lambda$$. Furthermore, by choosing $$\Lambda$$ such that $$\Box\Lambda=-\partial^{\mu}A_{\mu}$$, we find that we can set $$\partial^{\mu}A_{\mu}'=0$$, this is called the Lorenz gauge.

There is a residual gauge freedom given by choosing $$\Lambda'$$ such that $$\Box\Lambda'=0$$, since under this gauge transformation $$\partial^{\mu}A_{\mu}'\mapsto \partial^{\mu}A_{\mu}'':=\partial^{\mu}A_{\mu}'+\Box\Lambda'=0$$. So to fully fix the gauge, we also have to choose a function satisfying $$\Box\Lambda'=(-\partial_{t}^{2}+\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2})\Lambda'=0$$.

The claim is that after a Wick rotation to Euclidean theory, this residual gauge freedom disappears. Definging $$t=-i\tau$$, the Wick rotated action is: $$S^{E}=\frac{1}{4}\int d^{4}xF_{\mu\nu}F^{\mu\nu}$$ where the signature is now $$(+1,+1,+1,+1)$$.

This action again has a gauge symmetry $$A_{\mu}\to A_{\mu}+\partial_{\mu}\Lambda$$, and we can again choose Lorenz gauge by picking a $$\Lambda$$ such that $$\Box\Lambda=-\partial^{\mu}A_{\mu}$$, where now $$\Box=\partial^{\mu}\partial_{\mu}=\nabla^{2}$$, the Laplacian.

It appears to me that we once again have residual gauge freedom. Namely, that we can transform by a function $$\Lambda$$ such that $$\nabla^{2}\Lambda=0$$, so it really seems like the residual gauge symmetry has not disappeared in the Wick rotated theory!

My guess is that I am either misinterpreting what it means for the residual gauge freedom to disappear, or that I am missing an issue with the existence of solutions to the equations defining $$\Lambda$$ in Euclidean space. I would really appreciate some help resolving this issue.

In chapter 74 of Srednicki's QFT book (which the previous Phys.SE post refers to) it is implicitly assumed that in Euclidean 4D $$x$$-space the gauge field $$A_{\mu}$$ vanishes sufficiently fast as $$|x|\to \infty$$. In particular, large gauge transformations are implicitly excluded. In this case, the only harmonic function $$\Lambda$$ (i.e. such that $$\Box\Lambda=0$$) is the zero-function $$\Lambda\equiv 0$$, cf. the maximum principle, so that the gauge is completely fixed.
• Thank you, this answer is definitely starting to clear things up. However, since (in Lorenz gauge), we have the equation of motion $\Box A_{\mu}=0$ for the gauge fields, would this not also imply that the $A_{\mu}$ themselves are rapidly decaying harmonic functions and thus zero? Nov 29, 2021 at 0:18