Manuscript XII-E@1 · XII-E.tex · section · 9471ea56

The two legs of the first-order completion

E2: the sheet, the two legs, and the scalar completion › The two legs of the first-order completion · lines 493–517

XII-E@1#eq:legs, XII-E@1#eq:Aone, XII-E@1#eq:alphabeta

The first-order completion of XII-D, copied with its switch derivatives retained, is

X_{1}^{μ} [r] V_{1, g} [r] ψ A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, ϕ_{1} [r; ψ] = - E_{m}^{R} (V_{1, g} [r] ψ), = \int f (ϕ_{1} [r; ϕ] - X_{1}^{μ} [r] \partial_{μ} ϕ) d^{4} x .

It is linear in $r$ and in $ϕ$ . Its first root and transposed root are XII-D's

A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} z, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]),

with $α_{f, g} [ϕ]$ smooth, compact, supported in $C_{K} = J^{-} (K) \cap {t \geq 0}$ and flat at $t = 0$ .

Displayed formulas in this passage

XII-E@1#eq:legs · line 497
$X_{1}^{μ} [r] V_{1, g} [r] ψ A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, ϕ_{1} [r; ψ] = - E_{m}^{R} (V_{1, g} [r] ψ), = \int f (ϕ_{1} [r; ϕ] - X_{1}^{μ} [r] \partial_{μ} ϕ) d^{4} x .$
XII-E@1#eq:Aone · line 497
$X_{1}^{μ} [r] V_{1, g} [r] ψ A_{1} [f, g] (r; ϕ) = - E_{0}^{R} \partial_{λ} (σ_{ε} a_{1}^{λ μ} [r]), = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ, ϕ_{1} [r; ψ] = - E_{m}^{R} (V_{1, g} [r] ψ), = \int f (ϕ_{1} [r; ϕ] - X_{1}^{μ} [r] \partial_{μ} ϕ) d^{4} x .$
XII-E@1#eq:alphabeta · line 509
$A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} z, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]),$