Manuscript X-E@1 · X-E.tex · definition · 293fae6a

Definition: Retarded metric response and the tree

E4: one tree › Field content and the one-tree observable · lines 1188–1215

X-E@1#def:tree, X-E@1#eq:k1, X-E@1#eq:tree, X-E@1#eq:one-tree

The first-order retarded response of the metric to the switched scalar source is

k_{1} [ϕ] := Δ_{R}^{gr} (- \frac{δ V _{1, g}}{δ h} [ϕ]) = Δ_{R}^{gr} (\frac{1}{2} g Θ) = - 2 P E_{0}^{R} (g Θ),

so that $P_{gr} k_{1} + δ V_{1, g} / δ h = 0$ with retarded support. The tree is the insertion of $k_{1}$ into the affine slot of Lemma lem:affine:

T_{tree} := κ^{2} : A_{1} [k_{1} [ϕ]] := κ^{2} : ϕ (d_{1} [k_{1} [ϕ]]) :,

totally Wick ordered. The one-tree completed observable is

Φ^{tree} (f) = ϕ (f) + κ A_{1} [h] + κ^{2} (Q [h, h] + A_{1} [k_{cl}] + T_{tree}), Q [h, h] := : ϕ (Q [h, h]) :,

with $h = h_{cl} + h$ in $A_{1}$ and $Q$ .

Displayed formulas in this passage

X-E@1#eq:k1 · line 1192
$k_{1} [ϕ] := Δ_{R}^{gr} (- \frac{δ V _{1, g}}{δ h} [ϕ]) = Δ_{R}^{gr} (\frac{1}{2} g Θ) = - 2 P E_{0}^{R} (g Θ),$
X-E@1#eq:tree · line 1201
$T_{tree} := κ^{2} : A_{1} [k_{1} [ϕ]] := κ^{2} : ϕ (d_{1} [k_{1} [ϕ]]) :,$
X-E@1#eq:one-tree · line 1207
$Φ^{tree} (f) = ϕ (f) + κ A_{1} [h] + κ^{2} (Q [h, h] + A_{1} [k_{cl}] + T_{tree}), Q [h, h] := : ϕ (Q [h, h]) :,$