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

Theorem: Second-order plateau-compatible relational scalar

Plateau covariance and the second-order intertwiner › Scalar and coordinate intertwiners · lines 1243–1288 · proof included or linked

VII-A@1#thm:second, VII-A@1#eq:Phiinvariant, VII-A@1#eq:Ofinvariant

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For the switched action (eq:switchedaction), the harmonic Cauchy prescription (eq:harmonic)–(eq:harmonicdata), and the formal gauge transformations admitted by $G_{U}$ ,

Φ_{g} [F_{κ}^{*} g; ϕ_{0}] = Φ_{g} [g; ϕ_{0}] (mod κ^{3}) .

Consequently,

O_{f} [g] (F_{κ}^{*} g, ϕ_{0}) = O_{f} [g] (g, ϕ_{0}) (mod κ^{3}) .

The coefficients are (eq:A1) and (eq:A2), with (eq:phi1), (eq:phi2), (eq:w), and (eq:v).

Proof. Formal inversion of (eq:coordinateintertwiner) gives

X_{F_{κ}^{*} g}^{- 1} = F_{κ}^{- 1} \circ X_{g}^{- 1} (mod κ^{3}) .

Together with (eq:scalarintertwiner),

Φ_{g} [F_{κ}^{*} g; ϕ_{0}] = (ϕ_{g} [g; ϕ_{0}] \circ F_{κ}) \circ (F_{κ}^{- 1} \circ X_{g}^{- 1}) = ϕ_{g} [g; ϕ_{0}] \circ X_{g}^{- 1} (mod κ^{3}) .

This proves (eq:Phiinvariant). Smearing proves (eq:Ofinvariant). Taylor expansion gives the stated coefficients.

Displayed formulas in this passage

VII-A@1#eq:Phiinvariant · line 1248
$Φ_{g} [F_{κ}^{*} g; ϕ_{0}] = Φ_{g} [g; ϕ_{0}] (mod κ^{3}) .$
VII-A@1#eq:Ofinvariant · line 1255
$O_{f} [g] (F_{κ}^{*} g, ϕ_{0}) = O_{f} [g] (g, ϕ_{0}) (mod κ^{3}) .$
unlabeled · line 1275
$Φ_{g} [F_{κ}^{*} g; ϕ_{0}] = (ϕ_{g} [g; ϕ_{0}] \circ F_{κ}) \circ (F_{κ}^{- 1} \circ X_{g}^{- 1}) = ϕ_{g} [g; ϕ_{0}] \circ X_{g}^{- 1} (mod κ^{3}) .$
unlabeled · line 1268
$X_{F_{κ}^{*} g}^{- 1} = F_{κ}^{- 1} \circ X_{g}^{- 1} (mod κ^{3}) .$

Theorem: Second-order plateau-compatible relational scalar

Displayed formulas in this passage

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