Manuscript IX-C@1 · IX-C.tex · section · 03ce8e6b

Covariance and selected coefficients

Inherited data and the separation from V › Covariance and selected coefficients · lines 367–423

IX-C@1#eq:K, IX-C@1#eq:Q

The Wick convention is

F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .

The fixed covariance blocks are

H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,

with

P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ} .

We inherit

H_{w w} = 0, w = L h,

and the mixed component

K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .

These are fixed input identities, not new calculations IXA (lem:tensor, eq:K) IXB (eq:tensor, eq:K).

IX-B fixes

Q_{f, g} = - 2 \int f (x) g (y) K (x, y) \partial_{t} ϕ (x) h_{00} (y) d^{4} x d^{4} y .

Its existence and admission are inherited from IX-A; its sign and multiplicity are fixed in IX-B IXA (prop:knownoutputs, item 3) IXB (prop:partial, eq:Qpartial).

Displayed formulas in this passage

IX-C@1#eq:K · line 393
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
IX-C@1#eq:Q · line 413
$Q_{f, g} = - 2 \int f (x) g (y) K (x, y) \partial_{t} ϕ (x) h_{00} (y) d^{4} x d^{4} y .$
unlabeled · line 370
$F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .$
unlabeled · line 376
$H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,$
unlabeled · line 383
$P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ} .$
unlabeled · line 389
$H_{w w} = 0, w = L h,$