Manuscript XI-D@1 · XI-D.tex · section · 994a5c1b

Classical scalar completion

The inherited held-fixed-metric defect › Classical scalar completion · lines 952–977

XI-D@1#eq:forward

Now set $H = h + γ$ and

A_{2}^{cand} [h, k, γ; ϕ] = A_{2}^{sc} [h + γ, k; ϕ] - T (f, g) [ϕ] .

Under the admitted classical transformation

h^{'} = h + K ξ, k^{'} = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ} K ξ,

hold $γ$ fixed. Full transformation of $H$ would require the additional second coefficient $L_{ξ} γ$ . X-A's complete classical intertwiner and Lemma lem:affine therefore give

A_{2}^{cand} [h^{'}, k^{'}, γ; ϕ] - A_{2}^{cand} [h, k, γ; ϕ] = - A_{1} [L_{ξ} γ; ϕ] .

The tree has no external $h, k, γ$ argument and contributes no variation under this operation. The backward difference has the opposite sign.

Displayed formulas in this passage

XI-D@1#eq:forward · line 968
$A_{2}^{cand} [h^{'}, k^{'}, γ; ϕ] - A_{2}^{cand} [h, k, γ; ϕ] = - A_{1} [L_{ξ} γ; ϕ] .$
unlabeled · line 955
$A_{2}^{cand} [h, k, γ; ϕ] = A_{2}^{sc} [h + γ, k; ϕ] - T (f, g) [ϕ] .$
unlabeled · line 960
$h^{'} = h + K ξ, k^{'} = k + L_{ξ} h + K ζ + \frac{1}{2} L_{ξ} K ξ,$

Classical scalar completion

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