Manuscript IX-C-ST@1 · IX-C-ST.tex · assumption · 892f50c4

Assumption: Locked mixed kernel and old product

Shared analytical inputs and notation › Covariance and old product · lines 513–560

IX-C-ST@1#ass:old, IX-C-ST@1#eq:K

Retain the VII-C tensor identities

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

The nonzero mixed component with $h_{00}$ is

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)}] .

The reverse ordered mixed component is $K (x, y)$ ; the contraction selects vector index $0$ . The displayed infrared powers are locally integrable in three momentum dimensions. High-frequency integrals are distributional.

The full old compact polynomial microcausal algebra $P_{mc}$ is included faithfully in $P_{Σ}$ . Its unchanged coefficients are

C_{j}^{H} : P_{mc} \otimes P_{mc} ⟶ P_{mc}, 0 \leq j \leq 3,

with $C_{0}^{H} (F, G) = F G$ , supplied by

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

The old convention is Hermitian:

C_{j}^{H} (G^{*}, F^{*}) = C_{j}^{H} (F, G)^{*} .

The microcausal wavefront condition excludes joint covectors entirely in either closed causal cone, including zero individual covectors in those cones.

Displayed formulas in this passage

IX-C-ST@1#eq:K · line 522
$K (x, y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4∣ p ∣} + \frac{i}{8∣ p ∣ ^{2}}) e^{- i ∣ p ∣ (t - s)} - \frac{i}{8∣ p ∣ ^{2}} e^{i ∣ p ∣ (t + s)}] .$
unlabeled · line 516
$H_{w w} = 0, C_{x}^{μ} H_{hh, ρ σ} = δ^{μ}_{(ρ} \partial^{x}_{σ)} Δ_{0}^{+} .$
unlabeled · line 545
$C_{j}^{H} : P_{mc} \otimes P_{mc} ⟶ P_{mc}, 0 \leq j \leq 3,$
unlabeled · line 549
$F ⋆_{H} G = j \geq 0 \sum \frac{ℏ ^{j}}{j !} ⟨ F^{(j)}, H^{\otimes j} G^{(j)} ⟩ .$
unlabeled · line 554
$C_{j}^{H} (G^{*}, F^{*}) = C_{j}^{H} (F, G)^{*} .$

Assumption: Locked mixed kernel and old product

Displayed formulas in this passage

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