For a $U(1)$-gauge theory, we can fix $A_0 = 0$ by choosing a temporal gauge. Can we do the same for all of the gauge components of the $SU(2)$ gauge field, i.e., $W^a_0 = 0$ for $a \in \{1,2,3\}$? Does this uniquely determine a gauge?
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1The question https://physics.stackexchange.com/q/33133 isn't a duplicate, but its answers contain the answer to at least the second part of your question, and implicitly to the first part, too. – Chiral Anomaly Sep 03 '19 at 13:03