Manuscript X-F-ST@1 · X-F-ST.tex · theorem · e71f03ef

Theorem: Background-gauge commutator

F2: background gauge, kinetic correction, and retarded jet law › The commutator identity · lines 914–939 · proof included or linked

X-F-ST@1#thm:key, X-F-ST@1#eq:key

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For every smooth $ξ, r$ ,

F^{(1)} [K ξ] r Q^{(1)} [K ξ] r = L_{ξ}^{d} (F^{(0)} r) - F^{(0)} (L_{ξ}^{c} r), = L_{ξ}^{d} (P_{gr} r) - P_{gr} (L_{ξ}^{c} r) .

Proof. The background gauge-fixing density is natural under simultaneous pullback of $r, q ˉ$ . Differentiating its invariant compact-variation pairing gives

F [F_{s}^{*} r; F_{s}^{*} η] = F_{s}^{*} F [r; η] .

Its derivative at zero is the first identity. Combine it with (eq:cov2) and (eq:P). Equivalently, differentiate $Q [F_{s}^{*} η] (F_{s}^{*} r) = F_{s}^{*} (Q [η] r)$ .

Displayed formulas in this passage

X-F-ST@1#eq:key · line 917
$F^{(1)} [K ξ] r Q^{(1)} [K ξ] r = L_{ξ}^{d} (F^{(0)} r) - F^{(0)} (L_{ξ}^{c} r), = L_{ξ}^{d} (P_{gr} r) - P_{gr} (L_{ξ}^{c} r) .$
unlabeled · line 932
$F [F_{s}^{*} r; F_{s}^{*} η] = F_{s}^{*} F [r; η] .$

Nearby in this section

section: The commutator identity