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

The necessary reciprocity identity

Source switching and an exact-upgrade obstruction › The necessary reciprocity identity · lines 1114–1131

X-G@1#eq:skew

For a fixed background $w$ define

N_{w} = σ_{ε} Q_{w} .

Since $Q_{w}$ is formally self-adjoint,

N_{w}^{*} = Q_{w} σ_{ε}, N_{w} - N_{w}^{*} = [σ_{ε}, Q_{w}] .

Thus source multiplication generally destroys the symmetry of the action Hessian.

The following on-shell test excludes an equation-zero representative repair.

Displayed formulas in this passage

X-G@1#eq:skew · line 1121
$N_{w}^{*} = Q_{w} σ_{ε}, N_{w} - N_{w}^{*} = [σ_{ε}, Q_{w}] .$
unlabeled · line 1117
$N_{w} = σ_{ε} Q_{w} .$