Manuscript XIII-B@1 · XIII-B.tex · section · b8a5c0f5

Product data and action inverses

Prerequisite sheet and convention audit › Product data and action inverses · lines 441–480

XIII-B@1#eq:action-inverse, XIII-B@1#eq:roots

The inherited carrier is

E_{H} = (F_{mc, pol} [[ℏ, κ]], ⋆_{H}) / N_{sol} [[ℏ, κ]], F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .

The relevant action inverses are

G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R}, (G_{act}^{A})^{T} = G_{act}^{R}

in the even sector. Although $E_{m}^{R}$ inverts the scalar equation, the scalar action Hessian is $- P_{m}$ . This sign enters the binary response.

The causal part satisfies

H - H^{T} = i Δ, Δ = G_{act}^{R} - G_{act}^{A}

on the even blocks used here. Admissible differences $d$ are smooth graded-symmetric shell bisolutions, not necessarily compactly supported. XIII-A's rank-one example consequently remains within the admissible class XIIA.

For later comparison, the inherited roots are

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

They are not newly selected product data.

Displayed formulas in this passage

XIII-B@1#eq:action-inverse · line 452
$G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R}, (G_{act}^{A})^{T} = G_{act}^{R}$
XIII-B@1#eq:roots · line 473
$A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} x, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .$
unlabeled · line 444
$E_{H} = (F_{mc, pol} [[ℏ, κ]], ⋆_{H}) / N_{sol} [[ℏ, κ]], F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .$
unlabeled · line 464
$H - H^{T} = i Δ, Δ = G_{act}^{R} - G_{act}^{A}$