Manuscript XIII-C@1 · XIII-C.tex · section · e6556893

C1: complete incidence inventory for the selected pair

C1: complete incidence inventory for the selected pair · lines 493–533

XIII-C@1#eq:vertex

The interaction is the functional

V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, T_{0}^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .

Its kinetic monomials have one undifferentiated metric slot and two scalar slots with one derivative each. Its mass monomials have no field derivatives. The vertex is not an Euler operator.

The observable is

O = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ) .

Its formula and admission are inherited from XII-E XIIE. For inventory purposes only, abbreviate

X = X_{1} [γ], p = p_{1} [γ; ϕ], \int_{f} A = \int f A d^{4} x .

Put

U_{σ}_{ε} [γ] q = \partial_{α} (σ_{ε} a_{1}^{α β} [γ] \partial_{β} q),

V_{2, g} [γ, 0] ϕ = \partial_{μ} (g a_{2}^{μν} [γ] \partial_{ν} ϕ) + m^{2} g b_{2} [γ] ϕ .

Here $a_{1}, b_{1}, a_{2}, b_{2}$ have their inherited density-expansion definitions.

Displayed formulas in this passage

XIII-C@1#eq:vertex · line 497
$V_{g} = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x, T_{0}^{μν} = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .$
unlabeled · line 510
$O = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} + A_{1} [f, g] (r_{g} [ϕ]; ϕ) .$
unlabeled · line 516
$X = X_{1} [γ], p = p_{1} [γ; ϕ], \int_{f} A = \int f A d^{4} x .$
unlabeled · line 521
$U_{σ}_{ε} [γ] q = \partial_{α} (σ_{ε} a_{1}^{α β} [γ] \partial_{β} q),$
unlabeled · line 526
$V_{2, g} [γ, 0] ϕ = \partial_{μ} (g a_{2}^{μν} [γ] \partial_{ν} ϕ) + m^{2} g b_{2} [γ] ϕ .$

C1: complete incidence inventory for the selected pair

Displayed formulas in this passage

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