Manuscript IX-A-ST@1 · IX-A-ST.tex · section · 6e99d1b2

A local comparison covariance, not a conical quantum state

Presentation independence and selected existing coefficients › A local comparison covariance, not a conical quantum state · lines 1207–1242

For the selected coefficient comparisons only, fix in the auxiliary developed coordinates

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

H_{ϕϕ} (x, y) = Δ_{m}^{+} (x - y), m \geq 0, H_{h ϕ} = 0,

where

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

The graviton block is not asserted positive. This is neither a global cone Hadamard state nor a represented interacting gravitational field.

On its existing bulk domain, retain the usual contraction formula

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

For polynomials the sum is finite. The distinction between this old domain and a quantum extension is standard in the functional formulation FR.

Displayed formulas in this passage

unlabeled · line 1211
$H_{hh, α β ρ σ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ},$
unlabeled · line 1218
$H_{ϕϕ} (x, y) = Δ_{m}^{+} (x - y), m \geq 0, H_{h ϕ} = 0,$
unlabeled · line 1223
$Δ_{m}^{+} (t, x) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) t + i p \cdot x} .$
unlabeled · line 1233
$F ⋆_{H} G = r \geq 0 \sum \frac{ℏ ^{r}}{r !} ⟨ F^{(r)}, H^{\otimes r} G^{(r)} ⟩ .$