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Given a general gauge theory, can you fix the gauge so that field at a single point vanishes?

An equivalent question would be, given a gauge group $G$, can you pick a $g \in G$ so that if $L_\mu$ are the components of a gauge field in the Lie Algebra of $G$, that $L'_\mu = g L_\mu g^{-1} - (\partial_\mu g) g^{-1} = 0$ for all $\mu$ at a single point?

This is possible in general relativity, using Riemann normal coordinates to make the Christoffel symbols zero. Is it possible for any gauge group?

EDIT: I think this is actually very simple. Let $p$ be the point we want to make the gauge vanish if we pick $g(p) = 1$ and extend the function so that $\partial_\mu g = L_\mu$, which is possible via the exponential map since $L_\mu$ are in the Lie Algebra of the group. So, $g = exp[x^\mu L_\mu]$ if p is the origin in your coordinate system.

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  • $\begingroup$ See Weinberg's book, the discussion below eq. 15.1.17 $\endgroup$ Jul 8, 2017 at 22:41

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