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

Covariance and old product

Shared analytical inputs and notation › Covariance and old product · lines 481–512

The comparison covariance is

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

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

Use

Δ_{m}^{+} (z) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) z^{0} + i p \cdot z}, ω_{m} (p) = ∣ p ∣^{2} + m^{2},

and

S_{μν} (x, x^{'}) = \partial_{x^{μ}} \partial_{x^{' ν}} Δ_{m}^{+} (x - x^{'}) .

These are supplied chart formulae, not covariances derived from the conical metric.

Displayed formulas in this passage

unlabeled · line 484
$H_{hh, α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0,$
unlabeled · line 490
$P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, m \geq 0.$
unlabeled · line 497
$Δ_{m}^{+} (z) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) z^{0} + i p \cdot z}, ω_{m} (p) = ∣ p ∣^{2} + m^{2},$
unlabeled · line 505
$S_{μν} (x, x^{'}) = \partial_{x^{μ}} \partial_{x^{' ν}} Δ_{m}^{+} (x - x^{'}) .$