Gauge Fixing in Yang-Mills gauge theory

tom.stoer

I don't see where in that paper you are referring to.. They seem to be pretty focused on just axial and weyl gauges
It's about the general method they are using.

They fix the temporal gauge.
They construct a unitary operator fixing a second, residual gauge symmetry respecting the temporal gauge.
They show that in the physical Hilbert space both gauge conditions are satisfied (they use implicitly Dirac's constraint quantization for first-class constraints)

Their method applies to all gauges which can be written as "temporal gauge + physical gauge".

So the idea is quite general, even if the gauge-fixing operators are different. Unfortunately they do not discuss Gribov ambiguities.

tom.stoer

What is the reason to focus on the Lorentz gauge?

Perturbatively it's fine, and non-perturbatively it's useless, so don't care about its existence in these cases.

michael879

I thought it would help me, but it turns out the temporal gauge and the $\hat{r}\cdot\vec{A}=0$ gauge are all I need for what I'm doing. So thanks for the help! :D

*edit* I am trying to solve the classical "Maxwell's equations" of an SU(N) gauge theory. I needed to fix a gauge in order to simplify the problem, and eliminating a component of A does that

tom.stoer

Hi, when constructing a classical solution all the Gribov copies do is providing other solutions satisfying the same gauge condition. But there is no standard way to construct Gribov copies so from your point of view they are just new solutions.

And classically they are completely uncritical; they are problematic only when quantizing the theory.

Classical solutions of Yang-Mills equations are instantons, merons as tunneling events between Gribov vacua, Gribov vacua as non-trivial Gribov copies of the trivial vacuum, the Wu-Yang monopole to name a few.

In the post above I described a two-stage gauge fixing. The reason for its existence us simple. Consider the temporal gauge

$A_0=0$

and a time-independent SU(N)-valued gauge transformation U.

Then

$U^\dagger (A_0 - i\partial_0)U = 0$

so a time-independent transformation U but with non-trivial spatial dependency respects the temporal gauge. That means one can start from

$A_i=0$

and construct pure-gauge fields

$A_i^\prime = -i U^\dagger \partial_i U$

b/c the vacuum is a solution of the e.o.m. and b/c the e.o.m. respect the symmetry U, this is a way to construct other solutions of the e.o.m.

Gribov copies are (afaik) related to topologically non-trivial gauge transformations U.

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