Manuscript X-B-ST@1 · X-B-ST.tex · section · d03c150c

A sourced sign check

The mixed term, second-order boundary, and diagnostics › A sourced sign check · lines 1557–1591

X-B-ST@1#eq:sourcedsign

For $m = 0$ , $h = ψ η$ , one has

a_{1} [h] = ψ η^{- 1}, a_{2} [h] = 0, L_{1, g} [h] u = \partial_{μ} (g ψ \partial^{μ} u) .

On a time-dependent region of the retained causal compact, let $g = g_{0} (t)$ and $ψ = ψ (t)$ , with spatial cutoffs outside that compact. Define

Ψ (t) = \int_{0}^{t} σ ψ d s, Γ (t) = \int_{0}^{t} g_{0} ψ d s .

For the free solution $u = t$ ,

w^{0} = - Ψ, ρ_{1, g} [h; t] = - Γ.

Therefore

ev_{t} A_{1} [h] = \int f (Ψ - Γ) d^{4} x, ev_{t} M [h] = - \int f Ψ g_{0} ψ d^{4} x .

Here $ev_{t}$ means classical evaluation of a linear class, justified precisely in Section sec:normalization. It is not a quantum expectation value or a state on the Wick algebra.

Stage B-ii: four scope closures inside the same file B-ii: four scope closures

Displayed formulas in this passage

X-B-ST@1#eq:sourcedsign · line 1576
$ev_{t} A_{1} [h] = \int f (Ψ - Γ) d^{4} x, ev_{t} M [h] = - \int f Ψ g_{0} ψ d^{4} x .$
unlabeled · line 1560
$a_{1} [h] = ψ η^{- 1}, a_{2} [h] = 0, L_{1, g} [h] u = \partial_{μ} (g ψ \partial^{μ} u) .$
unlabeled · line 1567
$Ψ (t) = \int_{0}^{t} σ ψ d s, Γ (t) = \int_{0}^{t} g_{0} ψ d s .$
unlabeled · line 1572
$w^{0} = - Ψ, ρ_{1, g} [h; t] = - Γ.$