Manuscript VII-C@1 · VII-C.tex · section · 8b2ade29

The exact regularized coefficient

The mandatory mixed triple › The exact regularized coefficient · lines 1024–1081

VII-C@1#eq:Gdef, VII-C@1#eq:M, VII-C@1#eq:F

Let $f, f^{'}, g$ be real compact smooth functions, with $f, f^{'}$ supported in $M_{+}$ , and set

W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .

Define

G_{ε} (f, g, f^{'}) = ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) .

This coefficient is unambiguously defined in the old polynomial domain.

There is exactly one fully contracted graph type: the scalar legs of the two insertions are joined, and the two graviton legs are joined to the two legs of $W_{g}$ . The two assignments of the $W_{g}$ legs give a factor of $2$ . No $w w$ contraction is present in this graph.

Let

μ (u) = ρ (- u, u n), n \in S^{2} .

Euclidean radiality makes this independent of $n$ . It is real, smooth, rapidly decreasing, and $μ (0) = 1$ . For the scalar field define

M_{ε} (r) = ρ (ε (- ω_{m} (r), r n)) = μ (ε r^{2} + \frac{m ^{2}}{2}) .

Here and below $ω_{m} (r) = r^{2} + m^{2}$ .

Let $K_{ε}$ be the right-hand side of (eq:K) with the additional multiplier $μ (ε ∣ p ∣)$ in the momentum integral. This is exactly the covariance between $w [S_{ε} h] (x)$ and $h_{00} (y)$ .

For $q \in R^{3}$ , put

F_{f, ε} (y, q) = \int f (x) K_{ε} (x, y) e^{- i ω_{m} (q) t + i q \cdot x} d^{4} x .

Displayed formulas in this passage

VII-C@1#eq:Gdef · line 1032
$G_{ε} (f, g, f^{'}) = ev_{0} [ℏ^{3}] (B_{f, ε} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}, ε}) .$
VII-C@1#eq:F · line 1074
$F_{f, ε} (y, q) = \int f (x) K_{ε} (x, y) e^{- i ω_{m} (q) t + i q \cdot x} d^{4} x .$
unlabeled · line 1028
$W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .$
unlabeled · line 1051
$μ (u) = ρ (- u, u n), n \in S^{2} .$
VII-C@1#eq:M · line 1058
$M_{ε} (r) = ρ (ε (- ω_{m} (r), r n)) = μ (ε r^{2} + \frac{m ^{2}}{2}) .$

Nearby in this section

lemma: Lemma: Spectral form of the triple