Manuscript X-A-ST@1 · X-A-ST.tex · lemma · d9ccbde1

Lemma: Plateau covariance of the scalar Euler operator

Relational expansion and covariance: Route I › Naturality and plateau covariance · lines 1083–1105 · proof included or linked

X-A-ST@1#lem:plateaueuler, X-A-ST@1#eq:scalarcov

For $F \in G_{p}$ ,

E_{g} [F^{*} q] (F^{*} ψ) L_{g} [F^{*} q, F^{*} ψ] = F_{dens}^{*} (E_{g} [q] ψ), = F_{dens}^{*} L_{g} [q, ψ] (mod κ^{3}) .

Proof. The same calculation with $g$ gives

F^{*} A_{g} [q] - A_{g} [F^{*} q] = (1 - g) (F^{*} η^{- 1} - η^{- 1}) = 0.

For the volume coefficient the corresponding difference is $(1 - g) (F_{dens}^{*} 1 - 1) = 0$ . Naturality proves both identities.

Displayed formulas in this passage

X-A-ST@1#eq:scalarcov · line 1086
$E_{g} [F^{*} q] (F^{*} ψ) L_{g} [F^{*} q, F^{*} ψ] = F_{dens}^{*} (E_{g} [q] ψ), = F_{dens}^{*} L_{g} [q, ψ] (mod κ^{3}) .$
unlabeled · line 1098
$F^{*} A_{g} [q] - A_{g} [F^{*} q] = (1 - g) (F^{*} η^{- 1} - η^{- 1}) = 0.$