Manuscript XVI-K@1 · XVI-K.tex · lemma · aa7553bb

Lemma: Global definition and inverse normalization

Setting, explicit interfaces, and normalization › Ambient scalar edge · lines 430–497 · proof included or linked

XVI-K@1#lem:families, XVI-K@1#eq:W-fam, XVI-K@1#eq:E-fam

For

ψ (t, p) = \int_{R^{3}} e^{i p \cdot z} ψ (t, z) d^{3} z,

the two summands are defined by

⟨ W_{0}, ψ ⟩ ⟨ E_{0}^{A}, ψ ⟩ = \int_{R} d t \int_{R^{3}} \frac{e ^{- i ∣ p ∣ t} ψ ( t , p )}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p, = \int_{- \infty}^{0} d t \int_{R^{3}} \frac{sin ( - t ∣ p ∣ )}{( 2 π ) ^{3} ∣ p ∣} ψ (t, p) d^{3} p .

Both pairings converge absolutely. Moreover,

□ S = i δ, D_{F}^{T} = D_{F}, K D_{F} = i id .

Proof. The partial Fourier transform of a compact smooth test is rapidly decreasing in $p$ , uniformly for $t$ in a compact interval. The weight $1/∣ p ∣$ is locally integrable in three dimensions, and

\frac{sin ( - t ∣ p ∣ )}{∣ p ∣} \leq ∣ t ∣.

This proves absolute convergence. The time restriction in (eq:E-fam) is a restriction of this integral; it does not require multiplying distributions at coincidence.

The temporal Fourier families of $d$ are

d (t, p) = {e^{- i ∣ p ∣ t} / (2∣ p ∣), e^{i ∣ p ∣ t} / (2∣ p ∣), t > 0, t < 0.

For $t < 0$ , this follows from $e^{- i ω t} + 2 i sin (ω t) = e^{i ω t}$ . The jump of its temporal derivative is $- i$ , so $□ d = - i δ$ and $□ S = i δ$ . The two branches, together with spatial parity, give symmetry under $z \mapsto - z$ . Finally,

(- □ P) (- P S) = □ S = i id .

Displayed formulas in this passage

XVI-K@1#eq:W-fam · line 439
$⟨ W_{0}, ψ ⟩ ⟨ E_{0}^{A}, ψ ⟩ = \int_{R} d t \int_{R^{3}} \frac{e ^{- i ∣ p ∣ t} ψ ( t , p )}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p, = \int_{- \infty}^{0} d t \int_{R^{3}} \frac{sin ( - t ∣ p ∣ )}{( 2 π ) ^{3} ∣ p ∣} ψ (t, p) d^{3} p .$
XVI-K@1#eq:E-fam · line 439
$⟨ W_{0}, ψ ⟩ ⟨ E_{0}^{A}, ψ ⟩ = \int_{R} d t \int_{R^{3}} \frac{e ^{- i ∣ p ∣ t} ψ ( t , p )}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p, = \int_{- \infty}^{0} d t \int_{R^{3}} \frac{sin ( - t ∣ p ∣ )}{( 2 π ) ^{3} ∣ p ∣} ψ (t, p) d^{3} p .$
unlabeled · line 433
$ψ (t, p) = \int_{R^{3}} e^{i p \cdot z} ψ (t, z) d^{3} z,$
unlabeled · line 457
$□ S = i δ, D_{F}^{T} = D_{F}, K D_{F} = i id .$
unlabeled · line 471
$\frac{sin ( - t ∣ p ∣ )}{∣ p ∣} \leq ∣ t ∣.$
unlabeled · line 480
$d (t, p) = {e^{- i ∣ p ∣ t} / (2∣ p ∣), e^{i ∣ p ∣ t} / (2∣ p ∣), t > 0, t < 0.$
unlabeled · line 494
$(- □ P) (- P S) = □ S = i id .$

Lemma: Global definition and inverse normalization

Displayed formulas in this passage

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