Manuscript Em06-A@1 · Em06-A-v1.tex · section · ee9923c6

One-particle map and scalar preparation

The quantum family is constructed, not inferred › One-particle map and scalar preparation · lines 609–638

Em06-A@1#eq:classicalfield, Em06-A@1#eq:classicalpairing

Choose once and for all $z_{cl} \in S (R^{3}; C)$ . The corresponding real smooth free scalar is

ϕ_{cl} (t, x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} \frac{z _{cl} ( p ) e ^{- i ω_{p} t + i p \cdot x} + z _{cl} ( p ) e ^{i ω_{p} t - i p \cdot x}}{2 ω _{p}} d^{3} p .

It satisfies

\int_{M} F ϕ_{cl} d^{4} x = 2 Re ⟨ k_{F}, z_{cl} ⟩ .

Real compact smooth Cauchy data are included in this preparation class. With the unitary spatial Fourier transform, their amplitude is

z_{cl} (p) = \frac{ω _{p}}{2} ϕ_{cl} (0) (p) + \frac{i}{2 ω _{p}} \partial_{t} ϕ_{cl} (0) (p) .

Displayed formulas in this passage

Em06-A@1#eq:classicalfield · line 613
$ϕ_{cl} (t, x) = \frac{1}{( 2 π ) ^{3/2}} \int_{R^{3}} \frac{z _{cl} ( p ) e ^{- i ω_{p} t + i p \cdot x} + z _{cl} ( p ) e ^{i ω_{p} t - i p \cdot x}}{2 ω _{p}} d^{3} p .$
Em06-A@1#eq:classicalpairing · line 625
$\int_{M} F ϕ_{cl} d^{4} x = 2 Re ⟨ k_{F}, z_{cl} ⟩ .$
unlabeled · line 632
$z_{cl} (p) = \frac{ω _{p}}{2} ϕ_{cl} (0) (p) + \frac{i}{2 ω _{p}} \partial_{t} ϕ_{cl} (0) (p) .$