Manuscript V-B@1 · V-B-v1.tex · theorem · 3fadc1ad

Theorem: External first retarded coefficient

The external first retarded coefficient › Subcritical coefficients and representation · lines 1079–1123 · proof included or linked

V-B@1#thm:external, V-B@1#eq:external

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For the stated normalization,

R_{1, 1} (V (g; h), ϕ (f)) = ϕ (q_{g, h, f}), q_{g, h, f} = Q_{g h} u_{f}, u_{f} = - G_{m, A} f .

The test $q_{g, h, f}$ is real, smooth, and compact. The unsmeared coefficient is

ϕ_{g, h}^{(1)} = - G_{m, R} Q_{g h} ϕ, P ϕ_{g, h}^{(1)} = - Q_{g h} ϕ = - L_{g h} ϕ .

For spatial-TT $h$ ,

q_{g, h, f} = - \partial_{i} (g h^{ij} \partial_{j} u_{f}) .

Proof. The completely normal-ordered cubic block is identical in the time-ordered and ordinary products and cancels in their difference. Contracting the scalar insertion against either differentiated stress factor, and against the trace and mass terms, gives the Euler operator $Q_{g h}$ .

Equation (eq:Feynman), followed by the prefactor $i$ , gives $- G_{m, R} Q_{g h} ϕ$ . Smearing with $f$ , transposing the Green operator, and using formal self-adjointness yields

- \int (G_{m, A} f) Q_{g h} ϕ = ϕ (Q_{g h} u_{f}) .

The coefficients of $Q_{g h}$ are compact, so the resulting test is compact smooth. The remaining assertions follow by applying $P$ and using the free equation.

Displayed formulas in this passage

V-B@1#eq:external · line 1082
$R_{1, 1} (V (g; h), ϕ (f)) = ϕ (q_{g, h, f}), q_{g, h, f} = Q_{g h} u_{f}, u_{f} = - G_{m, A} f .$
unlabeled · line 1091
$ϕ_{g, h}^{(1)} = - G_{m, R} Q_{g h} ϕ, P ϕ_{g, h}^{(1)} = - Q_{g h} ϕ = - L_{g h} ϕ .$
unlabeled · line 1099
$q_{g, h, f} = - \partial_{i} (g h^{ij} \partial_{j} u_{f}) .$
unlabeled · line 1115
$- \int (G_{m, A} f) Q_{g h} ϕ = ϕ (Q_{g h} u_{f}) .$

Theorem: External first retarded coefficient

Displayed formulas in this passage

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