When we introduce minimal coupling for the Dirac spinor field, we introduce terms into the Lagrangian, by the substitution i∂∂xμ↦i∂∂xμ+eAμ, so that the Lagrangian is invariant under arbitrary changes of phase at every point of space-time, ψ(x)↦e−ieG(x)ψ(x),Aμ↦Aμ+∂μG(x),
I expect that in fussing about this I'm missing something very obvious. Am I? What is it?
EDIT: So it appears from the Answers, for which Thanks, that the anti-commutator given is valid for both space-like and time-like separation for the free Dirac field, and it's valid for space-like separation in QED, but it's nae valid for time-like separation in QED. Put somewhat loosely, this is OK in the canonical formalism because this anti-commutator is valid for the phase space. I'd rather like to know what a valid expression for the anti-commutator is at time-like separation in QED, but since that would, in a sense, be a solution of the theory I guess I'll have to whistle for it. I'll be grateful for Comments if this is an obviously obtuse reading of the Answers, though I ask that you consider that I would like a manifestly covariant version (so to speak, perhaps obscurely) of the canonical formalism before leaping in.
Answer
Because (iγμ∂∂xμ+m)ξξ′iΔ(x−x′) is a symbolic expression for a given analytical right-hand side. It is written so for convenience (not yet calculated) but it is a specific expression like δ(x−x′) or δ(x−x′)′. It should not acquire any "gauge extension" by definition. This expression does not contain a "particle momentum".
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