Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

Four legs and the distinction between coefficient norms

Model 2: dense TT completion and fixed-number spectral removal › Four legs and the distinction between coefficient norms · lines 807–860

IV-B@1#eq:fourlegs

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On the flat completion define

h_{g} = {z \in L^{2} (R^{3}; Sym^{2} C^{3}) : p^{i} z_{ij} = 0, z_{i}^{i} = 0 a.e.}, h_{φ} = L^{2} (R^{3}),

and

H_{2} = Γ_{s} (h_{g}) \otimes Γ_{s} (h_{φ}) .

Let $P_{TT} (p)$ be the orthogonal TT projection. Use modes

u_{p} (t, x) = \frac{e ^{- i ∣ p ∣ (t - t_{0}) + i p \cdot x}}{( 2 π ) ^{3/2} 2∣ p ∣}, v_{q} (t, x) = \frac{e ^{- i E_{q} (t - t_{0}) + i q \cdot x}}{( 2 π ) ^{3/2} 2 E _{q}}, E_{q} = ∣ q ∣^{2} + m^{2},

with a factor $2$ in the TT field, as in IV CheungIV.

For $u^{+} = u, u^{-} = u$ , and similarly for $v$ , define

H_{ij}^{ϵδ} (p, q) = 2 P_{TT}_{ij, ℓ m} (p) q_{ℓ} q_{m} \int_{t_{0}}^{t_{1}} \int v_{f} (t, x) u_{p}^{ϵ} (t, x) v_{q}^{δ} (t, x) d x d t, ϵ, δ = \pm .

The four operators are annihilation–annihilation, annihilation–creation, creation–annihilation, and creation–creation, with the reality relations supplied by real $f$ .

Put

L = H^{-+} : h_{φ} \to h_{g}, P_{2} = H^{--} \in h_{g} \otimes h_{φ} .

The tensor $P_{2}$ is not the particle-number projection.

Take smooth real even masks $0 \leq a_{R}, b_{R} \leq 1$ , equal to one for $R^{- 1} \leq ∣ p ∣ \leq R$ and $∣ q ∣ \leq R$ , respectively, and supported in the corresponding doubled regions. Set

H_{R}^{ϵδ} (p, q) = a_{R} (p) b_{R} (q) H^{ϵδ} (p, q) .

Displayed formulas in this passage

IV-B@1#eq:fourlegs · line 833
$H_{ij}^{ϵδ} (p, q) = 2 P_{TT}_{ij, ℓ m} (p) q_{ℓ} q_{m} \int_{t_{0}}^{t_{1}} \int v_{f} (t, x) u_{p}^{ϵ} (t, x) v_{q}^{δ} (t, x) d x d t, ϵ, δ = \pm .$
unlabeled · line 810
$h_{g} = {z \in L^{2} (R^{3}; Sym^{2} C^{3}) : p^{i} z_{ij} = 0, z_{i}^{i} = 0 a.e.}, h_{φ} = L^{2} (R^{3}),$
unlabeled · line 818
$H_{2} = Γ_{s} (h_{g}) \otimes Γ_{s} (h_{φ}) .$
unlabeled · line 822
$u_{p} (t, x) = \frac{e ^{- i ∣ p ∣ (t - t_{0}) + i p \cdot x}}{( 2 π ) ^{3/2} 2∣ p ∣}, v_{q} (t, x) = \frac{e ^{- i E_{q} (t - t_{0}) + i q \cdot x}}{( 2 π ) ^{3/2} 2 E _{q}}, E_{q} = ∣ q ∣^{2} + m^{2},$
unlabeled · line 847
$L = H^{-+} : h_{φ} \to h_{g}, P_{2} = H^{--} \in h_{g} \otimes h_{φ} .$
unlabeled · line 856
$H_{R}^{ϵδ} (p, q) = a_{R} (p) b_{R} (q) H^{ϵδ} (p, q) .$

Four legs and the distinction between coefficient norms

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