Manuscript XII-D@1 · XII-D.tex · section · 308a2080

The first root and its transpose

D2: both completion legs and the compact root › The first root and its transpose · lines 653–693

XII-D@1#eq:alpha-pair, XII-D@1#eq:alpha, XII-D@1#eq:scalar-support, XII-D@1#eq:beta

Put $v_{f} = E_{m}^{A} f$ . Compact integration by parts in (eq:Aone) gives

A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ]^{ab} (z) r_{ab} (z) d^{4} z,

where

α_{co}^{ab} (z) α_{sc}^{ab} (z) α_{f, g} = - σ_{ε} (z) a^{λ μ, ab} \partial_{λ} E_{0}^{A} (f \partial_{μ} ϕ) (z), = g (z) a^{μν, ab} \partial_{μ} v_{f} (z) \partial_{ν} ϕ (z) - \frac{m ^{2}}{2} g (z) η^{ab} v_{f} (z) ϕ (z), = α_{co} + α_{sc} .

Both legs are present.

If $K = supp f$ , define the corrected scalar support set

O_{f, g} = K \cup (supp g \cap J^{-} (K)) .

The map $ϕ \mapsto α_{f, g} [ϕ]$ depends only on the scalar near $O_{f, g}$ . Its range is smooth, compact, and supported in $C_{K}$ .

Transpose the metric inverse with its prescribed sign:

β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .

The minus sign follows from $G_{gr}^{R} = - P E_{0}^{R}$ and (eq:alpha-pair); it is not an independent convention.

Displayed formulas in this passage

XII-D@1#eq:alpha-pair · line 657
$A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ]^{ab} (z) r_{ab} (z) d^{4} z,$
XII-D@1#eq:alpha · line 662
$α_{co}^{ab} (z) α_{sc}^{ab} (z) α_{f, g} = - σ_{ε} (z) a^{λ μ, ab} \partial_{λ} E_{0}^{A} (f \partial_{μ} ϕ) (z), = g (z) a^{μν, ab} \partial_{μ} v_{f} (z) \partial_{ν} ϕ (z) - \frac{m ^{2}}{2} g (z) η^{ab} v_{f} (z) ϕ (z), = α_{co} + α_{sc} .$
XII-D@1#eq:scalar-support · line 676
$O_{f, g} = K \cup (supp g \cap J^{-} (K)) .$
XII-D@1#eq:beta · line 686
$β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .$