Manuscript X-G@1 · X-G.tex · section · 01566e14

The mixed operator retained from X-F

Frozen conventions and a support correction › The mixed operator retained from X-F · lines 445–473

X-G@1#eq:commutator, X-G@1#eq:mixed

For the background de Donder gauge,

Q [η + κ w] = P_{gr} + κ Q_{w} + O (κ^{2}), Q_{w} := Q^{(1)} [w] .

The unswitched $Q_{w}$ is the second variation of a quadratic background action and is formally self-adjoint.

Naturality supplies

Q_{K ξ} u = L_{ξ} (P_{gr} u) - P_{gr} (L_{ξ} u) .

This local identity holds for all smooth $ξ$ , not only $ξ \in g_{U}$ .

X-F's mixed source prescription is

k_{mix} [w; u] = - Δ_{R}^{gr} (σ_{ε} Q_{w} u) .

It is (eq:mixed), rather than a silently density-switched replacement, that is tested in Section sec:obstruction.

Displayed formulas in this passage

X-G@1#eq:commutator · line 457
$Q_{K ξ} u = L_{ξ} (P_{gr} u) - P_{gr} (L_{ξ} u) .$
X-G@1#eq:mixed · line 466
$k_{mix} [w; u] = - Δ_{R}^{gr} (σ_{ε} Q_{w} u) .$
unlabeled · line 448
$Q [η + κ w] = P_{gr} + κ Q_{w} + O (κ^{2}), Q_{w} := Q^{(1)} [w] .$