Manuscript VII-A@1 · VII-A.tex · lemma · 11da3fe6

Lemma: Plateau covariance

Plateau covariance and the second-order intertwiner › Covariance of the switched density operator · lines 1130–1176 · proof included or linked

VII-A@1#lem:plateaucov, VII-A@1#eq:plateaucov

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For $F \in G_{U}$ and every smooth scalar $ψ$ ,

D_{g} [F^{*} g] (F^{*} ψ) = F_{dens}^{*} (D_{g} [g] ψ) .

The identity also holds coefficientwise for the formal transformations (eq:formalF).

Proof. Regard $A_{g}$ as a contravariant tensor density and $ρ_{g}$ as a scalar density. Their unswitched definitions are natural under pullback.

By (eq:Ag),

A_{g} [g] = (1 - g) A_{η} + g A_{g} .

Since $F^{*} g = g$ ,

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

The difference between this expression and $A_{g} [F^{*} g]$ is

(1 - g) (F^{*} A_{η} - A_{η}) .

It vanishes: on $U$ the factor $1 - g$ is zero, while outside $U$ the diffeomorphism and its derivative are the identity. Hence

F^{*} A_{g} [g] = A_{g} [F^{*} g] .

The same argument applies to $ρ_{g} [g]$ .

The divergence of a vector density and the pairing with a scalar derivative are natural. Applying these operations proves (eq:plateaucov). Its formal coefficient identities follow by expansion.

Displayed formulas in this passage

VII-A@1#eq:plateaucov · line 1133
$D_{g} [F^{*} g] (F^{*} ψ) = F_{dens}^{*} (D_{g} [g] ψ) .$
unlabeled · line 1155
$F^{*} A_{g} [g] = (1 - g) F^{*} A_{η} + g A_{F^{*} g} .$
unlabeled · line 1149
$A_{g} [g] = (1 - g) A_{η} + g A_{g} .$
unlabeled · line 1162
$(1 - g) (F^{*} A_{η} - A_{η}) .$
unlabeled · line 1167
$F^{*} A_{g} [g] = A_{g} [F^{*} g] .$

Nearby in this section

remark: Remark: Why switch derivatives have not disappeared