Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

Finite-width free products and ordering

Finite-width free products and ordering · lines 1040–1065

Use the even field pair $(h, ϕ)$ and the fixed Hermitian two-point matrix

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

with

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

The scalar mass may be any $m \geq 0$ . No positivity is assumed for the graviton block.

For finite polynomials define

F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .

The contraction series terminates for each polynomial pair.

Displayed formulas in this passage

unlabeled · line 1045
$H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = 0,$
unlabeled · line 1051
$P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ} .$
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$F ⋆_{H} G = m exp (ℏ Γ_{H}) (F \otimes G) .$