Manuscript XVI-I@1 · XVI-I.tex · section · a7046a8d

Ambient advanced kernel versus a slab inverse

An ambient propagator in the selected normalization › Ambient advanced kernel versus a slab inverse · lines 610–655

XVI-I@1#eq:W, XVI-I@1#eq:EAamb

This section makes an explicit propagator choice. It is not inferred merely from a free-shell equation.

Let $W_{0}$ be the massless Minkowski positive-frequency distribution

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

The integral is distributional. Set $h_{0} = - W_{0}$ , in the reported XI-E convention.

Let $E_{0, amb}^{A}$ be the ambient advanced fundamental solution of $□$ . Its spatial Fourier kernel is

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

with the continuous value at $p = 0$ . The retarded kernel is $θ (t) sin (t ∣ p ∣) /∣ p ∣$ . These distributions are translation invariant on ambient $M$ .

For a source compactly supported in $M_{T}^{\circ}$ , its ambient advanced solution restricted to the slab is

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

The upper limit is $T$ because the zero-extended source vanishes above $T$ . This identifies the restriction with the supplied slab sine integral on that source class. It does not declare an arbitrary slab operator translation invariant.

All graph edges below use the ambient kernel. Thus an interaction point outside the slab is not evaluated by an unjustified slab-only prescription.

Displayed formulas in this passage

XVI-I@1#eq:W · line 617
$W_{0} (r) = \int_{R^{3}} \frac{e ^{- i ∣ p ∣ r^{0} + i p \cdot r}}{( 2 π ) ^{3} 2∣ p ∣} d^{3} p .$
XVI-I@1#eq:EAamb · line 629
$E_{0, amb}^{A} (t, p) = - θ (- t) \frac{sin ( t ∣ p ∣ )}{∣ p ∣},$
unlabeled · line 641
$(E_{0, amb}^{A} F) (u) = \int_{u}^{T} \frac{sin (( s - u ) - Δ )}{- Δ} F (s) d s .$