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

Proposition: Kinetic operator and its support

F2: background gauge, kinetic correction, and retarded jet law › Background de Donder gauge · lines 876–910 · proof included or linked

X-F-ST@1#prop:Q, X-F-ST@1#eq:Q1

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Define

Q [q ˉ] r = \frac{1}{2} D E [q ˉ] r + F [r; q ˉ] .

It is the fluctuation-quadratic action Hessian, and

Q [η + κh] = P_{gr} + κ Q^{(1)} [h] + O (κ^{2}), Q^{(1)} [h] r = G_{2} [h, r] + F^{(1)} [h] r .

The operator $Q^{(1)} [h]$ is formally self-adjoint and of order at most two in $r$ . Its coefficients are linear in the two-jet of $h$ , and

supp (Q^{(1)} [h] r) \subset supp h .

Dependence on a background coefficient $κ^{2} k$ first changes $Q$ at order $κ^{2}$ , hence affects the observable beyond the order considered here.

Proof. Differentiate (eq:Eexp) in a fluctuation direction. The term linear in $r$ at order $κ$ is $G_{2} [h, r]$ . Expansion of (eq:backgroundgauge) gives (eq:C1)–(eq:gf1), hence the other term. Both arise as second variations of local quadratic actions, proving formal self-adjointness. Their explicit local differential formulas prove order, coefficient dependence, and support.

Displayed formulas in this passage

X-F-ST@1#eq:Q1 · line 883
$Q [η + κh] = P_{gr} + κ Q^{(1)} [h] + O (κ^{2}), Q^{(1)} [h] r = G_{2} [h, r] + F^{(1)} [h] r .$
unlabeled · line 879
$Q [q ˉ] r = \frac{1}{2} D E [q ˉ] r + F [r; q ˉ] .$
unlabeled · line 892
$supp (Q^{(1)} [h] r) \subset supp h .$

Nearby in this section

section: Background de Donder gauge