Manuscript IX-A@1 · IX-A.tex · section · 55aa6f6d

The fixed free Wick matrix

Fixed spacetime, fields, and kernel carrier › The fixed free Wick matrix · lines 556–593

IX-A@1#eq:Hhh, IX-A@1#eq:P, IX-A@1#eq:Hphi, IX-A@1#eq:oldstar

For later comparison only, fix

H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = 0,

where $m \geq 0$ and

Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) (t - s) + i p \cdot (x - y)} .

No positivity of the graviton block is required.

The old bulk Wick product is

A ⋆_{H} B = r \geq 0 \sum \frac{ℏ ^{r}}{r !} ⟨ A^{(r)}, H^{\otimes r} B^{(r)} ⟩ .

It is retained as an operation on its existing bulk domain. It is not used to define a product on all of $P_{Σ}$ .

Displayed formulas in this passage

IX-A@1#eq:Hhh · line 559
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = 0,$
IX-A@1#eq:P · line 559
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = 0,$
IX-A@1#eq:Hphi · line 559
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = 0,$
IX-A@1#eq:oldstar · line 583
$A ⋆_{H} B = r \geq 0 \sum \frac{ℏ ^{r}}{r !} ⟨ A^{(r)}, H^{\otimes r} B^{(r)} ⟩ .$
unlabeled · line 574
$Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) (t - s) + i p \cdot (x - y)} .$