Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

The fixed free matrix

Problem, fixed model, and precise scope › The fixed free matrix · lines 241–287

VII-C@1#eq:Hhh, VII-C@1#eq:P, VII-C@1#eq:Hscalar, VII-C@1#eq:positive, VII-C@1#eq:star

The active even blocks are

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} = H_{ϕ h} = 0,

where $m \geq 0$ is fixed 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)}, ω_{m} (p) = ∣ p ∣^{2} + m^{2} .

No positivity is assumed for the graviton block.

The remaining gauge-fixed multiplet components are spectators in every calculation below. An enlargement of the full multiplet algebra must, in particular, admit the displayed even-sector products. Therefore an obstruction in this sector is sufficient for the stated conclusion. No inference about physical observables in gauge cohomology is made; that is a separate problem in the perturbative gravitational framework BFR.

We use the Wick-symbol convention

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

For finite polynomials this sum is finite. A local expression such as $\int g h_{00}^{2}$ denotes its Wick symbol; no self-contraction is included inside that expression.

Displayed formulas in this passage

VII-C@1#eq:Hhh · line 244
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = H_{ϕ h} = 0,$
VII-C@1#eq:P · line 244
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = H_{ϕ h} = 0,$
VII-C@1#eq:Hscalar · line 244
$H_{hh α β ρ σ} (x, y) P_{α β ρ σ} H_{ϕϕ} (x, y) = P_{α β ρ σ} Δ_{0}^{+} (x - y), = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, = Δ_{m}^{+} (x - y), H_{h ϕ} = H_{ϕ h} = 0,$
VII-C@1#eq:positive · line 257
$Δ_{m}^{+} (x - y) = \int_{R^{3}} \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{m} ( p )} e^{- i ω_{m} (p) (t - s) + i p \cdot (x - y)}, ω_{m} (p) = ∣ p ∣^{2} + m^{2} .$
VII-C@1#eq:star · line 278
$F ⋆_{H} G = r \geq 0 \sum \frac{ℏ ^{r}}{r !} ⟨ F^{(r)}, H^{\otimes r} G^{(r)} ⟩ .$