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

Density and scalar vertex

Density and scalar vertex · lines 936–973

At flat background use

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

Then

\partial^{μ} T_{μν} [ϕ] = (□_{η} + m^{2}) ϕ \partial_{ν} ϕ .

The mass sign therefore agrees with $P_{m}$ .

Its polarization is

T_{μν} (u, v) = \partial_{(μ} u \partial_{ν)} v - \frac{1}{2} η_{μν} (\partial^{ρ} u \partial_{ρ} v - m^{2} uv) .

Define

a_{λ}^{μν} (γ) = - det (η + λγ) (η + λγ)^{- 1 μν}, b_{λ} (γ) = - det (η + λγ) .

Displayed formulas in this passage

unlabeled · line 940
$T_{μν} [ϕ] = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{μν} (\partial^{ρ} ϕ \partial_{ρ} ϕ - m^{2} ϕ^{2}) .$
unlabeled · line 948
$\partial^{μ} T_{μν} [ϕ] = (□_{η} + m^{2}) ϕ \partial_{ν} ϕ .$
unlabeled · line 956
$T_{μν} (u, v) = \partial_{(μ} u \partial_{ν)} v - \frac{1}{2} η_{μν} (\partial^{ρ} u \partial_{ρ} v - m^{2} uv) .$
unlabeled · line 965
$a_{λ}^{μν} (γ) = - det (η + λγ) (η + λγ)^{- 1 μν}, b_{λ} (γ) = - det (η + λγ) .$