Manuscript III@2 · III.tex · proposition · 82dad3e5

Proposition: Coefficient gauge action

The coupled first-order jet and retarded harmonic section › The changed scalar equations · lines 1069–1095 · proof included or linked

III@2#prop:jet-action

For field-independent $ξ$ , define

T_{ξ} (h, ϕ_{0}, ϕ_{1}) = (h + K ξ, ϕ_{0}, ϕ_{1} + X_{ξ} ϕ_{0}) .

This preserves (eq:jet-scalar) and $G^{(1)} (h) = 0$ , and

T_{ξ} T_{ζ} = T_{ξ + ζ}, T_{ξ}^{- 1} = T_{- ξ} .

Proof. The transformed forced equation has left-hand side

P (ϕ_{1} + X_{ξ} ϕ_{0}) + L_{h + K ξ} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + P X_{ξ} ϕ_{0} - [P, X_{ξ}] ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + X_{ξ} P ϕ_{0} = 0.

The metric equation is preserved because $G^{(1)} K = 0$ . The group identities are immediate.

Displayed formulas in this passage

unlabeled · line 1072
$T_{ξ} (h, ϕ_{0}, ϕ_{1}) = (h + K ξ, ϕ_{0}, ϕ_{1} + X_{ξ} ϕ_{0}) .$
unlabeled · line 1077
$T_{ξ} T_{ζ} = T_{ξ + ζ}, T_{ξ}^{- 1} = T_{- ξ} .$
unlabeled · line 1085
$P (ϕ_{1} + X_{ξ} ϕ_{0}) + L_{h + K ξ} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + P X_{ξ} ϕ_{0} - [P, X_{ξ}] ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + X_{ξ} P ϕ_{0} = 0.$