Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

The crossed array and complete vertex

Component conventions and the retained internal expression › The crossed array and complete vertex · lines 809–895

Em01-C@3#eq:C, Em01-C@3#eq:vertex, Em01-C@3#eq:vertex-test, Em01-C@3#eq:regularization

Fix a symmetric crossed array

C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .

The four lines are fixed distributions; both tensor products exist. The lower component label $I$ denotes the dual input slot contracted with the corresponding slot of $K^{I, J}$ . No additional fiber metric is inserted in that contraction.

The linear submersion is

f (w) = (y - a, y - b) .

For smooth $H$ , define

Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .

The vertex is

V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .

Its arbitrary-test action is

⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .

This fixes the kinetic, trace-kinetic, and mass coefficients. With the adopted polarization

T_{μν} (ϕ, ψ) = \partial_{(μ} ϕ \partial_{ν)} ψ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ψ - m^{2} ϕ ψ),

they are the coefficients obtained by transferring the unused scalar derivative in the mixed Hessian $- \int g h^{μν} T_{μν} (ϕ, ψ) d^{4} y$ . This is the normalization adopted for the retained vertex Em01B. The vertex mass term and the internal $K_{9, mass}$ term are distinct.

Fix an open $U ⋐ W$ , a cutoff $χ \in C_{c}^{\infty} (R^{8})$ equal to one near $f (U)$ , and a profile $ρ \in C_{c}^{\infty} (R^{8})$ with $\int ρ = 1$ . Define

ρ_{δ} (x) = δ^{- 8} ρ (x / δ), C_{δ} = ρ_{δ} * (χ C), t_{δ} = F_{δ} (K_{ret}) := i V (K_{ret}, C_{δ}) ∣_{U} .

No positivity or symmetry of $ρ$ is assumed. Since $C_{δ}$ is smooth, every positive-scale expression exists.

Displayed formulas in this passage

Em01-C@3#eq:C · line 812
$C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .$
Em01-C@3#eq:vertex · line 841
$V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .$
Em01-C@3#eq:vertex-test · line 850
$⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .$
Em01-C@3#eq:regularization · line 883
$ρ_{δ} (x) = δ^{- 8} ρ (x / δ), C_{δ} = ρ_{δ} * (χ C), t_{δ} = F_{δ} (K_{ret}) := i V (K_{ret}, C_{δ}) ∣_{U} .$
unlabeled · line 826
$f (w) = (y - a, y - b) .$
unlabeled · line 830
$Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .$
unlabeled · line 863
$T_{μν} (ϕ, ψ) = \partial_{(μ} ϕ \partial_{ν)} ψ - \frac{1}{2} η_{μν} (\partial^{λ} ϕ \partial_{λ} ψ - m^{2} ϕ ψ),$