Manuscript XVI-I@1 · XVI-I.tex · section · a7046a8d

A declared density representative

The cubic interaction and its exact jet Hessian › A declared density representative · lines 824–885

XVI-I@1#eq:ell3

Retain the supplied cubic density representative

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

with the following atoms:

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - h^{μν}, u_{2}^{μν} = h^{μ}_{ρ} h^{ρ ν},

b_{1} = \frac{1}{2} tr_{η} h, b_{2} = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν},

Γ_{n}^{ρ}_{μν} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} h_{ν λ} + \partial_{ν} h_{μ λ} - \partial_{λ} h_{μν}), n = 1, 2, 3,

R_{n, 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}),

and

r_{n} = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .

Only coefficients appearing in these finite expressions are used. This is the unintegrated representative supplied in the original XVI-I through XII-C, not an Euler-class replacement.

Each monomial has three field-jet factors and total field derivative order two. The density has jet order at most two. The distinction between total derivative order and the maximum jet order of an individual factor will be used below.

Choose

χ_{I} \in C_{c}^{\infty} (M; R), V (h) = \int_{M} χ_{I} (w) ℓ_{3} (h) (w) d^{4} w .

The preparation correction belongs to the insertion kernel. It is not added again as an interaction vertex.

Displayed formulas in this passage

XVI-I@1#eq:ell3 · line 827
$ℓ_{3} (h) = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}),$
unlabeled · line 832
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - h^{μν}, u_{2}^{μν} = h^{μ}_{ρ} h^{ρ ν},$
unlabeled · line 839
$b_{1} = \frac{1}{2} tr_{η} h, b_{2} = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{μν} h^{μν},$
unlabeled · line 844
$Γ_{n}^{ρ}_{μν} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} h_{ν λ} + \partial_{ν} h_{μ λ} - \partial_{λ} h_{μν}), n = 1, 2, 3,$
unlabeled · line 852
$R_{n, μν} = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}),$
unlabeled · line 865
$r_{n} = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .$
unlabeled · line 878
$χ_{I} \in C_{c}^{\infty} (M; R), V (h) = \int_{M} χ_{I} (w) ℓ_{3} (h) (w) d^{4} w .$