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

Marked observable

The independently defined marked insertion › Marked observable · lines 1843–1884

Em01-F@3#eq:insertion, Em01-F@3#eq:observable

The tensor $G$ is an external compact smooth probe, not a second quantum species. For compact smooth $h, G, ψ$ , define

B_{f, g} (h, G, ψ) = \int [f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ})] X_{h}^{ν} \partial_{μ} ψ d^{4} b + \int f (b) [R * \partial_{α} (a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ})] (b) \partial_{μ} ψ (b) d^{4} b + \int f (b) [R * \partial_{α} D_{dens}^{α μ} (G, h)] (b) \partial_{μ} ψ (b) d^{4} b - \int f (b) X_{r [h, G]}^{μ} (b) \partial_{μ} ψ (b) d^{4} b + \int g (b) [a_{1}^{μν} [r [h, G]] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r [h, G]] F ψ] (b) d^{4} b .

Set

O_{G}^{ret} [γ, ϕ] = λ^{2} B_{f, g} (γ, G, ϕ) .

The ordering of $h, G$ in the first line is part of the definition. The insertion is not assumed symmetric in these roles or identified with an unmarked metric Hessian.

Displayed formulas in this passage

Em01-F@3#eq:insertion · line 1847
$B_{f, g} (h, G, ψ) = \int [f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ})] X_{h}^{ν} \partial_{μ} ψ d^{4} b + \int f (b) [R * \partial_{α} (a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ})] (b) \partial_{μ} ψ (b) d^{4} b + \int f (b) [R * \partial_{α} D_{dens}^{α μ} (G, h)] (b) \partial_{μ} ψ (b) d^{4} b - \int f (b) X_{r [h, G]}^{μ} (b) \partial_{μ} ψ (b) d^{4} b + \int g (b) [a_{1}^{μν} [r [h, G]] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r [h, G]] F ψ] (b) d^{4} b .$
Em01-F@3#eq:observable · line 1875
$O_{G}^{ret} [γ, ϕ] = λ^{2} B_{f, g} (γ, G, ϕ) .$

Marked observable

Displayed formulas in this passage

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