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

Lemma: Spectral form of the triple

The mandatory mixed triple › The exact regularized coefficient · lines 1082–1121 · proof included or linked

VII-C@1#lem:spectral, VII-C@1#eq:spectral

For every $ε > 0$ ,

G_{ε} (f, g, f^{'}) = \frac{1}{( 2 π ) ^{3}} \int g (y) d^{4} y \int_{R^{3}} ω_{m} (q) M_{ε} (∣ q ∣)^{2} F_{f, ε} (y, q) F_{f^{'}, ε} (y, q) d^{3} q .

For $f^{'} = f$ and $g \geq 0$ , this expression is nonnegative.

Proof. By (eq:CH), a contraction with $h_{00}$ forces the insertion vector index to be $0$ . Thus the scalar contraction is

\partial_{t} \partial_{t^{'}} (S_{ε} \otimes S_{ε}) Δ_{m}^{+} (x - x^{'}) = \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) (t - t^{'}) + i q \cdot (x - x^{'})} .

The factor $2$ from the two $W_{g}$ -leg assignments cancels the factor $1/2$ in this spectral density. The two minus signs in the definitions of the insertions cancel as well. Hermiticity gives the complex conjugate factor in (eq:spectral).

For fixed $ε > 0$ , the mollifier multipliers and the compact smooth smearings make the resulting integrals convergent. The positivity assertion uses only the positive scalar spectral measure, not positivity of the graviton covariance.

Displayed formulas in this passage

VII-C@1#eq:spectral · line 1085
$G_{ε} (f, g, f^{'}) = \frac{1}{( 2 π ) ^{3}} \int g (y) d^{4} y \int_{R^{3}} ω_{m} (q) M_{ε} (∣ q ∣)^{2} F_{f, ε} (y, q) F_{f^{'}, ε} (y, q) d^{3} q .$
unlabeled · line 1103
$\partial_{t} \partial_{t^{'}} (S_{ε} \otimes S_{ε}) Δ_{m}^{+} (x - x^{'}) = \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) (t - t^{'}) + i q \cdot (x - x^{'})} .$

Nearby in this section

section: The exact regularized coefficient