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

Representation and the external coefficient

The external-metric first retarded product › Representation and the external coefficient · lines 2339–2368

V@1#eq:scalarbound

The off-shell retarded-product framework is described in DF; Figure fig:external-contraction depicts the single insertion pair calculated here.

If $Π_{\leq M}$ projects onto scalar sectors at most $M$ ,

∥ ϕ (q_{g, h, f}) Π_{\leq M} ∥ \leq (M + M + 1) ∥ K q_{g, h, f} ∥,

and

∣ ⟨ Ψ, ϕ (q_{g, h, f}) X ⟩ ∣ \leq 2 ∥ K q_{g, h, f} ∥ ∥ Ψ ∥ (N_{ϕ} + 1)^{1/2} X .

These follow from the creation and annihilation estimates.

For temporally ordered supports, $T_{2} = ϕ (f) V (g; h)$ when the vertex is earlier, so $R_{1, 1} = i [ϕ (f), V (g; h)]$ . For the opposite ordering $T_{2} = V (g; h) ϕ (f)$ , so the retarded coefficient is zero. It also vanishes at spacelike separation. It is $T_{2}$ , not the retarded coefficient without its subtraction, that equals an ordered product.

Displayed formulas in this passage

V@1#eq:scalarbound · line 2347
$∥ ϕ (q_{g, h, f}) Π_{\leq M} ∥ \leq (M + M + 1) ∥ K q_{g, h, f} ∥,$
unlabeled · line 2353
$∣ ⟨ Ψ, ϕ (q_{g, h, f}) X ⟩ ∣ \leq 2 ∥ K q_{g, h, f} ∥ ∥ Ψ ∥ (N_{ϕ} + 1)^{1/2} X .$

Representation and the external coefficient

Displayed formulas in this passage

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