Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Switching and an off-shell-label witness

The external first retarded coefficient › Switching and an off-shell-label witness · lines 1130–1157

V-B@1#eq:switch

On a free scalar,

Q_{g h} ϕ = g L_{h} ϕ - (\partial_{μ} g) (r h)^{μν} \partial_{ν} ϕ .

Thus replacing $L_{g h}$ by $g L_{h}$ is generally incorrect.

For a pure gauge,

Q_{g K ξ} ϕ = - g P (X_{ξ} ϕ) - (\partial_{μ} g) (r K ξ)^{μν} \partial_{ν} ϕ .

Consequently,

δ_{ξ} ϕ_{g}^{(1)} = g X_{ξ} ϕ + G_{m, R} [(\partial_{μ} g) (r K ξ)^{μν} \partial_{ν} ϕ - [P, g] X_{ξ} ϕ] .

This formula retains the trace-reversed switched tensor. Constancy of $g$ near $supp f$ need not eliminate terms in its causal past.

Displayed formulas in this passage

V-B@1#eq:switch · line 1133
$Q_{g h} ϕ = g L_{h} ϕ - (\partial_{μ} g) (r h)^{μν} \partial_{ν} ϕ .$
unlabeled · line 1141
$Q_{g K ξ} ϕ = - g P (X_{ξ} ϕ) - (\partial_{μ} g) (r K ξ)^{μν} \partial_{ν} ϕ .$
unlabeled · line 1147
$δ_{ξ} ϕ_{g}^{(1)} = g X_{ξ} ϕ + G_{m, R} [(\partial_{μ} g) (r K ξ)^{μν} \partial_{ν} ϕ - [P, g] X_{ξ} ϕ] .$