Manuscript XV-A@1 · XV-A.tex · section · f53f3515

The fixed retarded pole

Internal kernels are not the joint quantum operation › The fixed retarded pole · lines 952–1019

XV-A@1#eq:pole

The copied XIV-I pole chart uses

ρ > 0, s \in {+ 1, - 1}, λ_{s} = ω - s ρ, ∣ λ_{s} ∣ < c ρ, 0 < c < 1,

and

B_{-} (λ) = (λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ) .

On this chart the source states

M_{R} = - \frac{s}{2 ρ} B_{-} (λ_{s}) + \frac{s}{2 ρ ( ω + s ρ )} .

For an explicit convention check, take

F_{t} u (ω) = \int e^{- iω t} u (t) d t, F_{t}^{- 1} v (t) = \frac{1}{2 π} \int e^{iω t} v (ω) d ω .

Then the partial fraction expression is

M_{R} = - \frac{1}{2 ρ} [(ω - ρ - i 0)^{- 1} - (ω + ρ - i 0)^{- 1}] .

For $s = + 1$ , the second denominator is smooth on the selected chart; for $s = - 1$ , the first is smooth there. This proves (eq:pole) algebraically. The first coefficient has homogeneous degree $- 1$ , and the regular remainder has degree $- 2$ .

For $η > 0$ ,

F_{t} (i Θ (t) e^{ia t - η t}) = (ω - a - i η)^{- 1} .

Passing to the distributional boundary value yields

F_{t}^{- 1} (ω - a - i 0)^{- 1} = i Θ (t) e^{ia t} .

Consequently

F_{t}^{- 1} M_{R} = Θ (t) \frac{sin ( ρt )}{ρ} .

This checks the temporal sign convention in the supplied pole formula. It does not independently prove the source's complete amplitude estimates or spatial wavefront theorem Ipole.

The delta contribution is fixed by this retarded boundary value. Discarding or adjusting it changes the represented internal line. The damping above computes the Fourier transform of an existing kernel; it is not a regulator for the unresolved graph product.

Displayed formulas in this passage

XV-A@1#eq:pole · line 967
$M_{R} = - \frac{s}{2 ρ} B_{-} (λ_{s}) + \frac{s}{2 ρ ( ω + s ρ )} .$
unlabeled · line 955
$ρ > 0, s \in {+ 1, - 1}, λ_{s} = ω - s ρ, ∣ λ_{s} ∣ < c ρ, 0 < c < 1,$
unlabeled · line 961
$B_{-} (λ) = (λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ) .$
unlabeled · line 975
$F_{t} u (ω) = \int e^{- iω t} u (t) d t, F_{t}^{- 1} v (t) = \frac{1}{2 π} \int e^{iω t} v (ω) d ω .$
unlabeled · line 982
$M_{R} = - \frac{1}{2 ρ} [(ω - ρ - i 0)^{- 1} - (ω + ρ - i 0)^{- 1}] .$
unlabeled · line 997
$F_{t} (i Θ (t) e^{ia t - η t}) = (ω - a - i η)^{- 1} .$
unlabeled · line 1002
$F_{t}^{- 1} (ω - a - i 0)^{- 1} = i Θ (t) e^{ia t} .$
unlabeled · line 1007
$F_{t}^{- 1} M_{R} = Θ (t) \frac{sin ( ρt )}{ρ} .$