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

Lemma: Plateau covariance of the coordinate operator

Relational expansion and covariance: Route I › Naturality and plateau covariance · lines 1062–1081 · proof included or linked

X-A-ST@1#lem:plateaucoord, X-A-ST@1#eq:coordinatecov

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For $F \in G_{p}$ ,

D_{ε} [F^{*} q] (F^{*} ψ) = F_{dens}^{*} (D_{ε} [q] ψ) (mod κ^{3}) .

Proof. Since $F^{*} σ = σ$ ,

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

The right side vanishes on $U$ because $σ = 1$ , and outside $U$ because $F = id$ there. Apply (eq:naturality).

Displayed formulas in this passage

X-A-ST@1#eq:coordinatecov · line 1065
$D_{ε} [F^{*} q] (F^{*} ψ) = F_{dens}^{*} (D_{ε} [q] ψ) (mod κ^{3}) .$
unlabeled · line 1074
$F^{*} A_{ε} [q] - A_{ε} [F^{*} q] = (1 - σ) (F^{*} η^{- 1} - η^{- 1}) .$

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