Manuscript IV@1 · IV.tex · section · 351cb2d7

One-particle spaces and normalization

Model 2: dense vacuum TT form and the actual cutoffs › One-particle spaces and normalization · lines 1461–1524

Define

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

The measure on $h_{g}$ is $d p$ , not $d p / (2∣ p ∣)$ . Frequency factors are in the 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} .

The TT field and scalar field are represented by

k_{ij} = 2 \int (u_{p} a_{g, ij} (p) + u ˉ_{p} a_{g, ij}^{†} (p)) d p, ϕ = \int (v_{q} a_{ϕ} (q) + v ˉ_{q} a_{ϕ}^{†} (q)) d q

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

The tensor CCR contain $Π_{ij, l m} (p)$ .

The factor two in the gravitational field fixes the classical symplectic convention

σ_{g} (k, k^{'}) = \frac{1}{4} \int (k_{ij} k^{'} ˙_{ij} - k ˙_{ij} k^{'}_{ij}) d x .

For real TT data $(q, p)$ , the coherent label is

z (ℓ) = \frac{1}{2} (\frac{∣ ℓ ∣}{2} q (ℓ) + \frac{i}{2∣ ℓ ∣} p (ℓ)),

and Fourier inversion gives

σ_{g} (k_{z}, k_{z^{'}}) = 2 Im ⟨ z, z^{'} ⟩ .

This fixes Model 2's normalization independently of Model 1.

A dense test space is

d_{g} = {z \in C_{c}^{\infty} (R^{3} ∖ {0}; Sym^{2} C^{3}) : Π z = z}, d_{ϕ} = C_{c}^{\infty} (R^{3}) .

Density follows by cutting off away from zero, smoothing on annuli, and applying the smooth projection $Π$ . Let $D_{2}$ be the algebraic finite-particle space generated by these labels.

Displayed formulas in this passage

unlabeled · line 1464
$h_{g} = {z \in L^{2} (R^{3}, d p; Sym^{2} C^{3}) : p_{i} z_{ij} = 0, z_{ii} = 0 a.e.}, h_{ϕ} = L^{2} (R^{3}, d q) .$
unlabeled · line 1473
$u_{p} (t, x) = \frac{e ^{- i ∣ p ∣ (t - t_{0}) + i p \cdot x}}{( 2 π ) ^{3/2} 2∣ p ∣}, v_{q} (t, x) = \frac{e ^{- i E_{q} (t - t_{0}) + i q \cdot x}}{( 2 π ) ^{3/2} 2 E _{q}}, E_{q} = ∣ q ∣^{2} + m^{2} .$
unlabeled · line 1482
$k_{ij} = 2 \int (u_{p} a_{g, ij} (p) + u ˉ_{p} a_{g, ij}^{†} (p)) d p, ϕ = \int (v_{q} a_{ϕ} (q) + v ˉ_{q} a_{ϕ}^{†} (q)) d q$
unlabeled · line 1490
$H_{2} = Γ_{s} (h_{g}) \otimes Γ_{s} (h_{ϕ}) .$
unlabeled · line 1497
$σ_{g} (k, k^{'}) = \frac{1}{4} \int (k_{ij} k^{'} ˙_{ij} - k ˙_{ij} k^{'}_{ij}) d x .$
unlabeled · line 1502
$z (ℓ) = \frac{1}{2} (\frac{∣ ℓ ∣}{2} q (ℓ) + \frac{i}{2∣ ℓ ∣} p (ℓ)),$
unlabeled · line 1508
$σ_{g} (k_{z}, k_{z^{'}}) = 2 Im ⟨ z, z^{'} ⟩ .$
unlabeled · line 1514
$d_{g} = {z \in C_{c}^{\infty} (R^{3} ∖ {0}; Sym^{2} C^{3}) : Π z = z}, d_{ϕ} = C_{c}^{\infty} (R^{3}) .$