Manuscript XVI-J@1 · XVI-J.tex · section · 6c1f306d

Definition and normalization

Ambient propagator and the punctured numerical square › Definition and normalization · lines 402–444

XVI-J@1#eq:W, XVI-J@1#eq:EA, XVI-J@1#eq:DF

Define the massless positive-frequency distribution by

W_{0} (z) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ z^{0} + i p \cdot z}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p .

The integral is distributional. Set $h_{0} = - W_{0}$ .

The ambient advanced fundamental solution has spatial Fourier kernel

E_{0}^{A} (t, p) = - θ (- t) \frac{sin ( t ∣ p ∣ )}{∣ p ∣},

with the continuous value at zero momentum. It satisfies $□ E_{0}^{A} = δ$ . For a source compactly supported in the interior of the slab, its restriction is

(E_{0}^{A} F) (t) = \int_{t}^{T} \frac{sin (( s - t ) - Δ )}{- Δ} F (s) d s .

Thus the supplied slab sine formula is recovered on that source class. The graph itself uses the ambient, translation-invariant kernel.

Put

d = W_{0} - i E_{0}^{A}, S = h_{0} + i E_{0}^{A} = - d, D_{F} = - P S = P d .

Displayed formulas in this passage

XVI-J@1#eq:W · line 405
$W_{0} (z) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ z^{0} + i p \cdot z}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p .$
XVI-J@1#eq:EA · line 416
$E_{0}^{A} (t, p) = - θ (- t) \frac{sin ( t ∣ p ∣ )}{∣ p ∣},$
XVI-J@1#eq:DF · line 436
$d = W_{0} - i E_{0}^{A}, S = h_{0} + i E_{0}^{A} = - d, D_{F} = - P S = P d .$
unlabeled · line 425
$(E_{0}^{A} F) (t) = \int_{t}^{T} \frac{sin (( s - t ) - Δ )}{- Δ} F (s) d s .$

Definition and normalization

Displayed formulas in this passage

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