Manuscript XI-E@1 · XI-E.tex · section · 46e55514

The word before the solution map

Route 2: the complete word and the affine recomputation › The word before the solution map · lines 996–1028

XI-E@1#eq:word, XI-E@1#eq:O2

For a $κ$ -independent scalar define

F_{f} = \int f ϕ d^{4} x,

O_{f} O_{2, f} = F_{f} + κ B_{f} + κ^{2} O_{2, f}, = \int f [(X_{1} [γ] \cdot \partial X_{1} [γ] - X_{2} [γ, 0]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [γ] X_{1}^{ν} [γ] \partial_{μ} \partial_{ν} ϕ] d^{4} x .

This is the coordinate expansion of $ϕ \circ X^{- 1}$ . The scalar responses $ρ_{1}, ρ_{2}$ and the graviton response $- T$ are solution-map outputs, not terms added to (eq:word). No admission of (eq:O2) is needed for the first-order identity below.

For this coefficient calculation let

d_{1} F_{f} = \frac{1}{2} \int f c \cdot \partial ϕ d^{4} x .

This is notation for the proposed first scalar co-transformation. We do not thereby define $d_{1}$ on every generator or construct $s$ .

Displayed formulas in this passage

XI-E@1#eq:word · line 1002
$O_{f} O_{2, f} = F_{f} + κ B_{f} + κ^{2} O_{2, f}, = \int f [(X_{1} [γ] \cdot \partial X_{1} [γ] - X_{2} [γ, 0]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [γ] X_{1}^{ν} [γ] \partial_{μ} \partial_{ν} ϕ] d^{4} x .$
XI-E@1#eq:O2 · line 1002
$O_{f} O_{2, f} = F_{f} + κ B_{f} + κ^{2} O_{2, f}, = \int f [(X_{1} [γ] \cdot \partial X_{1} [γ] - X_{2} [γ, 0]) \cdot \partial ϕ + \frac{1}{2} X_{1}^{μ} [γ] X_{1}^{ν} [γ] \partial_{μ} \partial_{ν} ϕ] d^{4} x .$
unlabeled · line 999
$F_{f} = \int f ϕ d^{4} x,$
unlabeled · line 1022
$d_{1} F_{f} = \frac{1}{2} \int f c \cdot \partial ϕ d^{4} x .$

The word before the solution map

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