Manuscript Em01-F@3 · Em01-F-v3.tex · corollary · 225e59af

Corollary: Derived matter vertex

Density and scalar vertex · lines 1069–1097 · proof included or linked

Em01-F@3#cor:vertexnormalization, Em01-F@3#eq:mattervertex

The first matter-interaction coefficient is $λ V_{g}$ , where

V_{g} [γ, ϕ] = - \frac{1}{2} \int g γ_{μν} T^{μν} [ϕ] d^{4} y .

Its mixed derivative is

V_{g}^{''} [h, ψ] = - \int g h_{μν} T^{μν} (ϕ, ψ) d^{4} y .

Proof. The first density coefficient is

\frac{1}{2} g (a_{1}^{μν} [γ] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [γ] ϕ^{2}) = - \frac{1}{2} g γ_{μν} T^{μν} [ϕ] .

The scalar derivative doubles the symmetric polarization.

Displayed formulas in this passage

Em01-F@3#eq:mattervertex · line 1072
$V_{g} [γ, ϕ] = - \frac{1}{2} \int g γ_{μν} T^{μν} [ϕ] d^{4} y .$
unlabeled · line 1079
$V_{g}^{''} [h, ψ] = - \int g h_{μν} T^{μν} (ϕ, ψ) d^{4} y .$
unlabeled · line 1088
$\frac{1}{2} g (a_{1}^{μν} [γ] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [γ] ϕ^{2}) = - \frac{1}{2} g γ_{μν} T^{μν} [ϕ] .$