Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Source convention sheet

Source convention sheet · lines 198–292

VII-D@1#sc:placement, VII-D@1#sc:covariance, VII-D@1#sc:K, VII-D@1#sc:spectral

{Insertion and placement (eq:B, eq:C, eq:Beps in VII-C).}

η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, □ E_{0} j = j, (E_{0} j) ∣_{t = 0} = 0, \partial_{t} (E_{0} j) ∣_{t = 0} = 0, B_{f} = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x = ⟨ b_{f}, ϕ \otimes h ⟩, W_{g} = \int g h_{00}^{2}, ρ \in C_{c}^{\infty} (R^{4}), ρ \geq 0, ρ real, even and Euclidean-radial, \int ρ = 1, S_{ε} = ρ_{ε} *, ρ_{ε} (x) = ε^{- 4} ρ (x / ε), B_{f, ε} = B_{f} [S_{ε} h, S_{ε} ϕ], b_{f, ε} = (ρ_{ε} \otimes ρ_{ε}) * b_{f} .

{Even covariance (eq:Hhh, eq:Hscalar, eq:positive).}

H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0, Δ_{m}^{+} (x - y) = \int \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}, C_{x}^{μ} H_{hh ρ σ} = δ^{μ}_{(ρ} \partial_{σ)}^{x} Δ_{0}^{+}, C_{x}^{μ} C_{y}^{ν} H_{hh} = - \frac{1}{2} η^{μν} □_{x} Δ_{0}^{+} = 0, H_{w w} = 0.

{Zero-Cauchy mode and mixed covariance (eq:K).}

E_{0} (e^{- i P t}) = - \frac{i t}{2 P} e^{- i P t} + \frac{i sin ( P t )}{2 P ^{2}}, P = ∣ p ∣, K (x, y) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 P} + \frac{i}{8 P ^{2}}) e^{- i P (t - s)} - \frac{i}{8 P ^{2}} e^{i P (t + s)}] .

The factors $P^{- 1}$ and $P^{- 2}$ are locally integrable in three momentum dimensions. No graviton positivity is assumed.

{Profiles and spectral identity (eq:F, eq:spectral, lem:spectral).}

ϑ (u) = ρ (- u, u n), M_{ε} (r) = ϑ (ε r^{2} + m^{2} /2), K_{ε} = K with the multiplier ϑ (εP), F_{f, ε} (y, q) = \int f (x) K_{ε} (x, y) e^{- i ω_{m} (q) t + i q \cdot x} d^{4} x, ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) = \frac{1}{( 2 π ) ^{3}} \int g (y) d^{4} y \int ω_{m} (q) M_{ε} (∣ q ∣)^{2} F_{f, ε} (y, q) F_{f^{'}, ε} (y, q) d^{3} q (f, f^{'}, g real) .

There is one graph type. Its factor $2$ counts the assignments of the two middle legs. The middle observable is not regularized.

Displayed formulas in this passage

VII-D@1#sc:placement · line 208
$η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ, (C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, □ E_{0} j = j, (E_{0} j) ∣_{t = 0} = 0, \partial_{t} (E_{0} j) ∣_{t = 0} = 0, B_{f} = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x = ⟨ b_{f}, ϕ \otimes h ⟩, W_{g} = \int g h_{00}^{2}, ρ \in C_{c}^{\infty} (R^{4}), ρ \geq 0, ρ real, even and Euclidean-radial, \int ρ = 1, S_{ε} = ρ_{ε} *, ρ_{ε} (x) = ε^{- 4} ρ (x / ε), B_{f, ε} = B_{f} [S_{ε} h, S_{ε} ϕ], b_{f, ε} = (ρ_{ε} \otimes ρ_{ε}) * b_{f} .$
VII-D@1#sc:covariance · line 233
$H_{hh α β ρ σ} = P_{α β ρ σ} Δ_{0}^{+}, P_{α β ρ σ} = η_{α (ρ} η_{σ) β} - \frac{1}{2} η_{α β} η_{ρ σ}, H_{ϕϕ} = Δ_{m}^{+}, H_{h ϕ} = H_{ϕ h} = 0, Δ_{m}^{+} (x - y) = \int \frac{d ^{3} q}{( 2 π ) ^{3} 2 ω _{m} ( q )} e^{- i ω_{m} (q) (t - s) + i q \cdot (x - y)}, ω_{m} (q) = ∣ q ∣^{2} + m^{2}, C_{x}^{μ} H_{hh ρ σ} = δ^{μ}_{(ρ} \partial_{σ)}^{x} Δ_{0}^{+}, C_{x}^{μ} C_{y}^{ν} H_{hh} = - \frac{1}{2} η^{μν} □_{x} Δ_{0}^{+} = 0, H_{w w} = 0.$
VII-D@1#sc:K · line 256
$E_{0} (e^{- i P t}) = - \frac{i t}{2 P} e^{- i P t} + \frac{i sin ( P t )}{2 P ^{2}}, P = ∣ p ∣, K (x, y) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot (x - y)} [(- \frac{t}{4 P} + \frac{i}{8 P ^{2}}) e^{- i P (t - s)} - \frac{i}{8 P ^{2}} e^{i P (t + s)}] .$
VII-D@1#sc:spectral · line 275
$ϑ (u) = ρ (- u, u n), M_{ε} (r) = ϑ (ε r^{2} + m^{2} /2), K_{ε} = K with the multiplier ϑ (εP), F_{f, ε} (y, q) = \int f (x) K_{ε} (x, y) e^{- i ω_{m} (q) t + i q \cdot x} d^{4} x, ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) = \frac{1}{( 2 π ) ^{3}} \int g (y) d^{4} y \int ω_{m} (q) M_{ε} (∣ q ∣)^{2} F_{f, ε} (y, q) F_{f^{'}, ε} (y, q) d^{3} q (f, f^{'}, g real) .$

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section: Source convention sheet