Manuscript V@1 · V.tex · theorem · e7ddf37d

Theorem: Mixed first retarded coefficient

The one-oscillator mixed insertion › Mixed Wick product and retarded coefficient · lines 2620–2651 · proof included or linked

V@1#thm:mixed, V@1#eq:mixed

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For real compact $g, f$ , put

q_{a} = Q_{g a} u_{f}, v_{a} = K q_{a}, a = e, r, u_{f} = - G_{A} f .

Then on $D_{mix}$ ,

R_{g, f} := R_{1, 1} (V (g), ϕ (f)) = Q \otimes Φ (v_{e}) + Π \otimes Φ (v_{r}) .

The unsmeared coefficient is

ϕ_{g}^{(1)} = - Q \otimes G_{R} Q_{g e} ϕ - Π \otimes G_{R} Q_{g r} ϕ .

The normalization has no extra diagonal parameter for this insertion pair.

Proof. Apply the contraction calculation of Theorem thm:external separately to $e$ and $r$ , retaining the commuting oscillator factors. The same normal-block cancellation and the same unique scalar contraction extensions give the result.

Displayed formulas in this passage

V@1#eq:mixed · line 2628
$R_{g, f} := R_{1, 1} (V (g), ϕ (f)) = Q \otimes Φ (v_{e}) + Π \otimes Φ (v_{r}) .$
unlabeled · line 2623
$q_{a} = Q_{g a} u_{f}, v_{a} = K q_{a}, a = e, r, u_{f} = - G_{A} f .$
unlabeled · line 2636
$ϕ_{g}^{(1)} = - Q \otimes G_{R} Q_{g e} ϕ - Π \otimes G_{R} Q_{g r} ϕ .$

Theorem: Mixed first retarded coefficient

Displayed formulas in this passage

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