Manuscript IX-B-ST@1 · IX-B-ST.tex · assumption · f6af1523

Assumption: Locked tensor and old-domain data

Chart formulae and the exact dependency ledger › Covariance and mixed kernel · lines 499–537

IX-B-ST@1#ass:tensor, IX-B-ST@1#eq:K, IX-B-ST@1#eq:oldproduct

Retain the VII-C tensor identities and old Wick-symbol convention CheungVIIC ({:tensor}, {:K}):

C_{x}^{μ} H_{hh, ρ σ} = δ^{μ}_{(ρ} \partial_{σ)}^{x} Δ_{0}^{+}, H_{w w} = 0.

The nonzero mixed component is

K (x, y) = H_{w h}^{0}_{00} (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)}], P = ∣ p ∣.

The reverse component is $K (x, y)$ . The covariance selects the $0$ -component at a contraction with $h_{00}$ .

On the old bulk polynomial domain, retain

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

Self-contractions within a Wick symbol are excluded. The usual old-domain operations used at positive cutoff are defined. The unchanged mixed-kernel approximants converge in the fixed-support normal stages of VII-C.

Displayed formulas in this passage

IX-B-ST@1#eq:K · line 509
$K (x, y) = H_{w h}^{0}_{00} (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 P} + \frac{i}{8 P ^{2}}) e^{- i P (t - s)} - \frac{i}{8 P ^{2}} e^{i P (t + s)}], P = ∣ p ∣.$
IX-B-ST@1#eq:oldproduct · line 527
$F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .$
unlabeled · line 503
$C_{x}^{μ} H_{hh, ρ σ} = δ^{μ}_{(ρ} \partial_{σ)}^{x} Δ_{0}^{+}, H_{w w} = 0.$

Assumption: Locked tensor and old-domain data

Displayed formulas in this passage

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