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

Mixed Wick product and retarded coefficient

The one-oscillator mixed insertion › Mixed Wick product and retarded coefficient · lines 2590–2619

Product-vacuum normal ordering gives

: h^{μν} T_{μν} (ϕ) := h^{μν} : T_{μν} (ϕ) : .

There are no oscillator–scalar contractions and no pair of metric factors at this insertion. For real compact $g$ ,

V (g) = Q \otimes V_{e} (g) + Π \otimes V_{r} (g), V_{a} (g) = \frac{1}{2} \int g a^{μν} : T_{μν} (ϕ) : .

The vertex preserves $D_{mix}$ , by Lemma lem:vertexdomain and finite oscillator number shifts. It is symmetric there. No essential self-adjointness claim is made for this mixed cubic vertex.

In its product with one scalar insertion the normal uncontracted block has one oscillator and three scalar factors. It cancels in the retarded subtraction. The surviving block has one oscillator and one scalar. All numerical coefficients are the same ones as in Section sec:external; multiplication by matrix elements of $e, r$ is smooth and does not raise relative scaling degree.

Displayed formulas in this passage

unlabeled · line 2593
$: h^{μν} T_{μν} (ϕ) := h^{μν} : T_{μν} (ϕ) : .$
unlabeled · line 2599
$V (g) = Q \otimes V_{e} (g) + Π \otimes V_{r} (g), V_{a} (g) = \frac{1}{2} \int g a^{μν} : T_{μν} (ϕ) : .$

Mixed Wick product and retarded coefficient

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