Manuscript XI-E@1 · XI-E.tex · section · 46e55514

The local source is explicit

Route 3: the cubic Einstein tree and its computed variation › The local source is explicit · lines 1427–1510

XI-E@1#eq:H2, XI-E@1#eq:u3, XI-E@1#eq:C3

Retain XI-A's curvature convention and

S_{EH} = - \frac{2}{κ ^{2}} \int - q R [q] d^{4} x, q = η + κγ .

Write

- q G^{μν} (q) = κ H_{1}^{μν} [γ] + κ^{2} H_{2}^{μν} [γ] + O (κ^{3}), A γ = 2 H_{1} [γ] .

For completeness, a finite prescription for $H_{2}$ is as follows:

Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}),

Γ_{2}^{ρ}_{μν} = - \frac{1}{2} γ^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}),

R_{1, μν} = \partial_{ρ} Γ_{1}^{ρ}_{μν} - \partial_{ν} Γ_{1}^{ρ}_{μ ρ},

R_{2, μν} = \partial_{ρ} Γ_{2}^{ρ}_{μν} - \partial_{ν} Γ_{2}^{ρ}_{μ ρ} + Γ_{1}^{ρ}_{ρ λ} Γ_{1}^{λ}_{μν} - Γ_{1}^{ρ}_{ν λ} Γ_{1}^{λ}_{μ ρ} .

Put

r_{1} = η^{μν} R_{1, μν}, r_{2} = η^{μν} R_{2, μν} - γ^{μν} R_{1, μν},

C_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, C_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1} .

Then, with $τ = tr_{η} γ$ ,

H_{2}^{μν} = C_{2}^{μν} + \frac{1}{2} τ C_{1}^{μν} - γ^{μα} C_{1, α}^{ν} - γ^{ν α} C_{1, α}^{μ} .

The cubic Einstein vertex has metric Euler derivative

J_{3} [γ] = 2 H_{2} [γ] .

Its actual classical retarded metric response is

u_{3} [γ] = - G_{gr}^{R} J_{3} [γ] = P E_{0}^{R} (2 H_{2} [γ]) .

The sign follows from the response rule and the frozen inverse. Define its affine scalar-completed insertion by

C_{3} (f, g) [γ, ϕ] = A_{1} [u_{3} [γ]; ϕ] .

This has field degree $(2, 1)$ , unlike the scalar-cubic tree $T$ . It is a named classical sourced evaluation, not a quantum retarded contraction constructed by this paper.

Displayed formulas in this passage

XI-E@1#eq:H2 · line 1483
$H_{2}^{μν} = C_{2}^{μν} + \frac{1}{2} τ C_{1}^{μν} - γ^{μα} C_{1, α}^{ν} - γ^{ν α} C_{1, α}^{μ} .$
XI-E@1#eq:u3 · line 1495
$u_{3} [γ] = - G_{gr}^{R} J_{3} [γ] = P E_{0}^{R} (2 H_{2} [γ]) .$
XI-E@1#eq:C3 · line 1502
$C_{3} (f, g) [γ, ϕ] = A_{1} [u_{3} [γ]; ϕ] .$
unlabeled · line 1430
$S_{EH} = - \frac{2}{κ ^{2}} \int - q R [q] d^{4} x, q = η + κγ .$
unlabeled · line 1436
$- q G^{μν} (q) = κ H_{1}^{μν} [γ] + κ^{2} H_{2}^{μν} [γ] + O (κ^{3}), A γ = 2 H_{1} [γ] .$
unlabeled · line 1444
$Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}),$
unlabeled · line 1451
$Γ_{2}^{ρ}_{μν} = - \frac{1}{2} γ^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}),$
unlabeled · line 1458
$R_{1, μν} = \partial_{ρ} Γ_{1}^{ρ}_{μν} - \partial_{ν} Γ_{1}^{ρ}_{μ ρ},$
unlabeled · line 1463
$R_{2, μν} = \partial_{ρ} Γ_{2}^{ρ}_{μν} - \partial_{ν} Γ_{2}^{ρ}_{μ ρ} + Γ_{1}^{ρ}_{ρ λ} Γ_{1}^{λ}_{μν} - Γ_{1}^{ρ}_{ν λ} Γ_{1}^{λ}_{μ ρ} .$
unlabeled · line 1471
$r_{1} = η^{μν} R_{1, μν}, r_{2} = η^{μν} R_{2, μν} - γ^{μν} R_{1, μν},$
unlabeled · line 1476
$C_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, C_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1} .$
unlabeled · line 1491
$J_{3} [γ] = 2 H_{2} [γ] .$