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

Covariance and mixed kernel

Chart formulae and the exact dependency ledger › Covariance and mixed kernel · lines 473–498

IX-B-ST@1#eq:covariance

The supplied comparison covariance has blocks

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

where

P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, m \geq 0,

and

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

No positivity of the metric block is assumed.

Displayed formulas in this passage

IX-B-ST@1#eq:covariance · line 476
$H_{hh, α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,$
unlabeled · line 484
$P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, m \geq 0,$
unlabeled · line 491
$Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} q}{( 2 π ) ^{3} 2 ω _{m} ( q )} e^{- i ω_{m} (q) (t - s) + i q \cdot (x - y)}, ω_{m} (q) = ∣ q ∣^{2} + m^{2} .$