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

Classical coefficient and represented generator

Model 1: the fixed TT plane and removable edge › Classical coefficient and represented generator · lines 547–623

IV-B@1#eq:scalar-response, IV-B@1#eq:oneparticle, IV-B@1#eq:A1

Work first on the declared completion $M$ , with

P = - \partial_{t}^{2} + Δ - m^{2}, Ω_{m} = - Δ + m^{2}, m > 0.

Fix smooth real spatial TT waves $e, r$ with Schwartz Cauchy data, spanning a nondegenerate real symplectic plane. They have no temporal components. For a free scalar $φ$ , define

S_{t_{0}} (a) φ (t) = \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} (- a^{ij} (s) \partial_{i} \partial_{j} φ (s)) d s, a = e, r .

This solves $P S_{t_{0}} (a) φ = - L_{a} φ$ , with zero correction data, where $L_{a} φ = - a^{ij} \partial_{i} \partial_{j} φ$ .

For real $f \in C_{c}^{\infty} ((t_{0}, t_{1}) \times R^{3})$ , write

ℓ_{a, f} (φ) = \int f S_{t_{0}} (a) φ d^{4} x .

Set

v_{f} (s) = \int_{s}^{t_{1}} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t) d t, b_{a, f} = - \partial_{i} \partial_{j} (a^{ij} v_{f}) .

Integration by parts gives

ℓ_{a, f} (φ) = \int_{t_{0}}^{t_{1}} \int b_{a, f} (s, x) φ (s, x) d x d s .

Finite propagation makes the relevant spatial support compact on the finite interval.

Let $u_{0}, u_{e}, u_{r} \in L^{2} (R^{3})$ be the one-particle vectors representing $\int f φ$ , $ℓ_{e, f}$ , and $ℓ_{r, f}$ , respectively. Explicitly, if a real linear Cauchy functional is $\int (α φ_{0} + β ϖ_{0}) d x$ , use

u = \frac{α}{2 ω} + i \frac{ω}{2} β, ω (p) = ∣ p ∣^{2} + m^{2} .

For the response functional,

α = \int_{t_{0}}^{t_{1}} cos ((s - t_{0}) Ω_{m}) b_{a, f} (s) d s, β = \int_{t_{0}}^{t_{1}} \frac{sin (( s - t _{0} ) Ω _{m} )}{Ω _{m}} b_{a, f} (s) d s .

All these vectors are Schwartz.

H_{1} = L^{2} (R) \otimes Γ_{s} (L^{2} (R^{3})), Q = \frac{b + b ^{*}}{2}, Π_{0} = - \frac{i ( b - b ^{*} )}{2},

define

A^{(1)} (λ) = I \otimes Φ (u_{0}) + λ (Q \otimes Φ (u_{e}) + Π_{0} \otimes Φ (u_{r})), Φ (u) = a (u) + a^{*} (u) .

The real $λ$ defines a numerical model. It is not evaluation of a formal nilpotent parameter at a nonzero real number.

Displayed formulas in this passage

IV-B@1#eq:scalar-response · line 557
$S_{t_{0}} (a) φ (t) = \int_{t_{0}}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} (- a^{ij} (s) \partial_{i} \partial_{j} φ (s)) d s, a = e, r .$
IV-B@1#eq:oneparticle · line 590
$u = \frac{α}{2 ω} + i \frac{ω}{2} β, ω (p) = ∣ p ∣^{2} + m^{2} .$
IV-B@1#eq:A1 · line 613
$A^{(1)} (λ) = I \otimes Φ (u_{0}) + λ (Q \otimes Φ (u_{e}) + Π_{0} \otimes Φ (u_{r})), Φ (u) = a (u) + a^{*} (u) .$
unlabeled · line 550
$P = - \partial_{t}^{2} + Δ - m^{2}, Ω_{m} = - Δ + m^{2}, m > 0.$
unlabeled · line 569
$ℓ_{a, f} (φ) = \int f S_{t_{0}} (a) φ d^{4} x .$
unlabeled · line 573
$v_{f} (s) = \int_{s}^{t_{1}} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t) d t, b_{a, f} = - \partial_{i} \partial_{j} (a^{ij} v_{f}) .$
unlabeled · line 579
$ℓ_{a, f} (φ) = \int_{t_{0}}^{t_{1}} \int b_{a, f} (s, x) φ (s, x) d x d s .$
unlabeled · line 597
$α = \int_{t_{0}}^{t_{1}} cos ((s - t_{0}) Ω_{m}) b_{a, f} (s) d s, β = \int_{t_{0}}^{t_{1}} \frac{sin (( s - t _{0} ) Ω _{m} )}{Ω _{m}} b_{a, f} (s) d s .$
unlabeled · line 606
$H_{1} = L^{2} (R) \otimes Γ_{s} (L^{2} (R^{3})), Q = \frac{b + b ^{*}}{2}, Π_{0} = - \frac{i ( b - b ^{*} )}{2},$

Nearby in this section

proposition: Proposition: Self-adjointness and active factors