Manuscript XVI-L@1 · XVI-L.tex · section · 35081099

The printed formulas, quoted

L.1 The complete source as a jet polynomial › The printed formulas, quoted · lines 263–313

XVI-L@1#eq:L-fc, XVI-L@1#eq:L-ell3, XVI-L@1#eq:L-J3, XVI-L@1#eq:L-C, XVI-L@1#eq:L-M2, XVI-L@1#eq:L-JD

XII-C.tex, Section sec:density (lines above eq:finite-curvature): with $τ = tr_{η} γ$ ,

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν}, b_{0} = 1, b_{1} = \frac{1}{2} τ, b_{2} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν} .

XII-C.tex, eq:finite-curvature: for $n = 1, 2, 3$ ,

Γ_{n}^{ρ}_{μν} R_{n, μν} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} γ_{ν λ} + \partial_{ν} γ_{μ λ} - \partial_{λ} γ_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), r_{n} = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .

XII-C.tex, eq:ell3:

ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) .

XII-C.tex, the definitions printed before eq:J3, and eq:J3:

V_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, V_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1}, J_{3}^{μν} = 2 (V_{2}^{μν} + b_{1} V_{1}^{μν} - γ^{μα} V_{1, α}^{ν} - γ^{ν α} V_{1, α}^{μ}) .

XII-C.tex, eq:Ccoefficients (with $C^{μν α β} (q) = - q (q^{μν} q^{α β} - q^{μ (α} q^{β) ν}) = C_{0} + κ C_{1} + κ^{2} C_{2} + O (κ^{3})$ ):

C_{0}^{μν α β} C_{1}^{μν α β} C_{2}^{μν α β} = η^{μν} η^{α β} - η^{μ (α} η^{β) ν}, = b_{1} C_{0}^{μν α β} + u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}, = b_{2} C_{0}^{μν α β} + b_{1} (u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}) + u_{2}^{μν} η^{α β} + η^{μν} u_{2}^{α β} + u_{1}^{μν} u_{1}^{α β} - u_{2}^{μ (α} η^{β) ν} - η^{μ (α} u_{2}^{β) ν} - u_{1}^{μ (α} u_{1}^{β) ν} .

XII-C.tex, eq:M2:

M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

XII-C.tex, Theorem thm:actualsource, eq:actualsource: the selected cubic vertex $V_{3, D} [γ] = \int σ ℓ_{3} [γ] d^{4} x$ has Euler source

J_{D}^{μν} = σ J_{3}^{μν} + D_{σ}^{μν}, D_{σ}^{μν} = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ - 2 M_{2}^{μν ρ} \partial_{ρ} σ .

XII-C.tex, eq:full-trace-sector, and XII-B (quoted in XII-C.tex as eq:input-trace): for $γ = ψ η$ , $C_{1} = C_{2} = 0$ , $Γ_{2} = - ψ Γ_{1}$ , and

D_^{}|_{=}=-[^{}(_)^+(^)^+(^)^

Displayed formulas in this passage

XVI-L@1#eq:L-fc · line 271
$Γ_{n}^{ρ}_{μν} R_{n, μν} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} γ_{ν λ} + \partial_{ν} γ_{μ λ} - \partial_{λ} γ_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), r_{n} = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .$
XVI-L@1#eq:L-ell3 · line 279
$ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) .$
XVI-L@1#eq:L-J3 · line 284
$V_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, V_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1}, J_{3}^{μν} = 2 (V_{2}^{μν} + b_{1} V_{1}^{μν} - γ^{μα} V_{1, α}^{ν} - γ^{ν α} V_{1, α}^{μ}) .$
XVI-L@1#eq:L-C · line 292
$C_{0}^{μν α β} C_{1}^{μν α β} C_{2}^{μν α β} = η^{μν} η^{α β} - η^{μ (α} η^{β) ν}, = b_{1} C_{0}^{μν α β} + u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}, = b_{2} C_{0}^{μν α β} + b_{1} (u_{1}^{μν} η^{α β} + η^{μν} u_{1}^{α β} - u_{1}^{μ (α} η^{β) ν} - η^{μ (α} u_{1}^{β) ν}) + u_{2}^{μν} η^{α β} + η^{μν} u_{2}^{α β} + u_{1}^{μν} u_{1}^{α β} - u_{2}^{μ (α} η^{β) ν} - η^{μ (α} u_{2}^{β) ν} - u_{1}^{μ (α} u_{1}^{β) ν} .$
XVI-L@1#eq:L-M2 · line 301
$M_{2}^{μν ρ} = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$
XVI-L@1#eq:L-JD · line 306
$J_{D}^{μν} = σ J_{3}^{μν} + D_{σ}^{μν}, D_{σ}^{μν} = 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ - 2 M_{2}^{μν ρ} \partial_{ρ} σ .$
unlabeled · line 266
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν}, b_{0} = 1, b_{1} = \frac{1}{2} τ, b_{2} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν} .$

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