Manuscript XI-C@1 · XI-C.tex · section · acec1422

Vertex and coordinate operator

Copied handoff and frozen conventions › Vertex and coordinate operator · lines 425–465

XI-C@1#eq:vertex, XI-C@1#eq:coordinate

The only contracted vertex is

V_{1, g} [γ, ϕ] = - \frac{1}{2} \int_{M_{T}} g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x,

where

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{α} ϕ \partial^{α} ϕ - m^{2} ϕ^{2}) .

It is a classical local polynomial before any quantum admission.

Write

a_{1}^{α μ} [r] = \frac{1}{2} η^{α μ} tr_{η} r - r^{α μ} .

The coordinate map is the linear operator

L_{ε} r := X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .

The layer formula alone applies only when $D r = 0$ . It cannot be imposed on a sourced response with nonzero divergence.

The cubic Einstein vertex is not contracted here. Its coupling degree permits contributions at the same order as the present tree. Its omission is a scope restriction, not a consequence of counting one metric leg.

Displayed formulas in this passage

XI-C@1#eq:vertex · line 428
$V_{1, g} [γ, ϕ] = - \frac{1}{2} \int_{M_{T}} g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x,$
XI-C@1#eq:coordinate · line 449
$L_{ε} r := X_{1} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α \cdot} [r]) = E_{0}^{R} (σ_{ε} D r - σ_{ε}^{'} a_{1}^{0 \cdot} [r]) .$
unlabeled · line 435
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} (\partial_{α} ϕ \partial^{α} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 444
$a_{1}^{α μ} [r] = \frac{1}{2} η^{α μ} tr_{η} r - r^{α μ} .$