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

Ambient scalar edge

Setting, explicit interfaces, and normalization › Ambient scalar edge · lines 386–429

XVI-K@1#eq:W0, XVI-K@1#eq:S

Define

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

The integral is distributional. The ambient advanced inverse of $□$ is specified by the spatial Fourier multiplier

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

with the continuous value of the quotient at zero momentum. Equivalently, for compact sources,

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

For sources compactly supported in the slab interior, this restricts to the supplied upper limit $T$ . The graph edges use the ambient kernel, not an arbitrary slab operator.

Set

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

The scalar $d$ is the massless positive-frequency time-ordered kernel in this normalization. It is the scalar kernel denoted by $Δ_{F}^{FR}$ in XVI-J.

Displayed formulas in this passage

XVI-K@1#eq:W0 · line 389
$W_{0} (z) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ z^{0} + i p \cdot z}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p, h_{0} = - W_{0} .$
XVI-K@1#eq:S · line 418
$S = - W_{0} + i E_{0}^{A}, d = W_{0} - i E_{0}^{A} = - S, D_{F} = - P S = P d .$
unlabeled · line 401
$E_{0}^{A} (t, p) = - θ (- t) \frac{sin ( t ∣ p ∣ )}{∣ p ∣},$
unlabeled · line 407
$(E_{0}^{A} F) (t) = \int_{t}^{\infty} \frac{sin (( s - t ) - Δ )}{- Δ} F (s) d s .$

Ambient scalar edge

Displayed formulas in this passage

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