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

Three scalar one-particle vectors

Model 1: the fixed vacuum TT plane › Three scalar one-particle vectors · lines 914–967

IV@1#eq:vf, IV@1#eq:gplane, IV@1#eq:ug

Take real $f \in C_{c}^{\infty} ((t_{0}, T) \times R^{3})$ , and define

v_{f} (s, x) = \int_{s}^{T} (\frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t, \cdot)) (x) d t .

This sign convention is fixed throughout the paper: $(\partial_{s}^{2} + Ω_{m}^{2}) v_{f} = f$ , with zero terminal data. It differs by a minus sign from the advanced inverse of $P$ .

Exchanging the finite-time integrals gives, for TT $a$ ,

ℓ_{a, f} = - \int_{t_{0}}^{T} v_{f} a^{ij} \partial_{i} \partial_{j} ϕ_{0} d^{4} x .

Consequently set

b_{a, f} = - \partial_{i} \partial_{j} (a^{ij} v_{f}), g_{a, f} = I_{{t \geq t_{0}}} b_{a, f}, g_{0, f} = f .

Then $ℓ_{a, f} = \int g_{a, f} ϕ_{0}$ . The spatial support on the finite slab is compact, all spatial derivatives are bounded, and the only possible spacetime nonsmoothness is the initial-time edge.

For such a real $g$ , define

α_{g} = \int_{t_{0}}^{T} cos ((t - t_{0}) Ω_{m}) g (t) d t, β_{g} = \int_{t_{0}}^{T} \frac{sin (( t - t _{0} ) Ω _{m} )}{Ω _{m}} g (t) d t .

Spatial smoothness and finite propagation imply $α_{g}, β_{g} \in C_{c}^{\infty} (R^{3})$ . If $(φ, ϖ)$ are the scalar data,

ℓ_{g} = \int (α_{g} φ + β_{g} ϖ) d x .

Define

u_{g} (p) = \frac{1}{2 ω _{p}} \int_{t_{0}}^{T} e^{i ω_{p} (t - t_{0})} g (t, p) d t = \frac{α _{g}}{2 ω} + i \frac{ω}{2} β_{g} .

Since $m > 0$ , these vectors are Schwartz. Write $u_{a, f} = u_{g_{a, f}}$ .

Displayed formulas in this passage

IV@1#eq:vf · line 917
$v_{f} (s, x) = \int_{s}^{T} (\frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} f (t, \cdot)) (x) d t .$
IV@1#eq:gplane · line 933
$b_{a, f} = - \partial_{i} \partial_{j} (a^{ij} v_{f}), g_{a, f} = I_{{t \geq t_{0}}} b_{a, f}, g_{0, f} = f .$
IV@1#eq:ug · line 958
$u_{g} (p) = \frac{1}{2 ω _{p}} \int_{t_{0}}^{T} e^{i ω_{p} (t - t_{0})} g (t, p) d t = \frac{α _{g}}{2 ω} + i \frac{ω}{2} β_{g} .$
unlabeled · line 928
$ℓ_{a, f} = - \int_{t_{0}}^{T} v_{f} a^{ij} \partial_{i} \partial_{j} ϕ_{0} d^{4} x .$
unlabeled · line 945
$α_{g} = \int_{t_{0}}^{T} cos ((t - t_{0}) Ω_{m}) g (t) d t, β_{g} = \int_{t_{0}}^{T} \frac{sin (( t - t _{0} ) Ω _{m} )}{Ω _{m}} g (t) d t .$
unlabeled · line 954
$ℓ_{g} = \int (α_{g} φ + β_{g} ϖ) d x .$

Three scalar one-particle vectors

Displayed formulas in this passage

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