Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

Geometry, carrier, and product data

J0: the pair and extension contract › Geometry, carrier, and product data · lines 417–455

Fix

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .

Fix the inherited smooth ramp $σ_{ε}$ , zero for $t \leq 0$ and one for $t \geq ε$ , the compact scalar switch $g$ , the observation $f$ , and the frozen translation-invariant Hadamard matrix $H$ .

The carrier remains

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

The action Green blocks are

G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R}, H_{T} = H + i G_{act}^{A} .

The metric–scalar blocks vanish. The inverse coefficient is $- 1$ , not the differently normalized X-track coefficient $- 4$ .

We denote by $F_{μ}$ the scalar time-ordered numerical line of mass $μ$ , with the fixed nonzero species fiber factors separated when convenient. This notation does not change the frozen Hadamard data.

Displayed formulas in this passage

unlabeled · line 420
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ_{x}, P_{m} = □ + m^{2},$
unlabeled · line 425
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, 0 < ε < T .$
unlabeled · line 434
$E_{H} = (F_{mc, pol} [[ℏ, κ]], ⋆_{H}) / N_{sol} [[ℏ, κ]], F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .$
unlabeled · line 442
$G_{act}^{R}_{γ γ} = - P E_{0}^{R}, G_{act}^{R}_{ϕϕ} = - E_{m}^{R}, H_{T} = H + i G_{act}^{A} .$