Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Exact radial reduction of the entire crossed trace

Exact radial reduction of the entire crossed trace · lines 765–843

Em01-G@5#eq:scalar-trace, Em01-G@5#eq:radial-parameters, Em01-G@5#eq:angular-amplitude, Em01-G@5#eq:radial-amplitude

On the actual relative time region,

D = W_{0} \otimes Δ_{m}, C_{I}^{μν} = i ℏ^{2} N_{I}^{μν} D .

For a compact smooth profile $ν$ , let $d_{δ}^{ν} = ν_{δ} * D$ locally. Admissible relative cutoffs are eventually irrelevant.

Fix $α > 0$ , $d = β - α > 0$ , and the frame. For a compact amplitude $A (v, ξ)$ , supported in the separated time region, set

J_{δ}^{ν} [A] = \int A (v, ξ) d_{δ}^{ν} (α ℓ + v k + ξ E, - d ℓ + v k + ξ E) d v d^{2} ξ .

Compact smooth spectator dependence is allowed.

Write the smoothing displacement as

H_{i} = A_{i} ℓ + B_{i} k + Z_{i} E, ν_{n} (H) = 4 ν (H_{1}, H_{2}) .

The factor four is the joint frame Jacobian. Below $d H$ is frame-coordinate measure.

Define

a_{δ} Z_{δ} κ_{δ} L_{δ} = α - δ A_{1}, = (b_{δ} Z_{1} + a_{δ} Z_{2}) / e_{δ}, = 4 a_{δ} b_{δ} / e_{δ}, = κ_{δ} (B_{2} - B_{1}) - δ D_{δ} . b_{δ} D_{δ} = d + δ A_{2}, = a_{δ} b_{δ} ∣ Z_{1} - Z_{2} ∣^{2} / e_{δ}^{2}, e_{δ} = a_{δ} + b_{δ}, c_{δ} = a_{δ} / e_{δ},

All denominators have fixed positive lower bounds on the compact parameter ranges under consideration.

Set

A_{δ, H} (t, z) = \frac{1}{2 π} \int_{0}^{2 π} A (δ B_{2} - \frac{∣ t ω _{φ} + δ ( Z _{δ} - Z _{2} ) ∣ ^{2} + z}{4 b _{δ}}, t ω_{φ} + δ Z_{δ}) d φ,

where $ω_{φ} = (cos φ, sin φ)$ . Let

h_{m} (z) = - \frac{m ^{2}}{8 π} j = 0 \sum^{\infty} \frac{( - m ^{2} z /4 ) ^{j}}{j ! ( j + 1 )!},

and

F_{δ, H} (t) = A_{δ, H} (t, 0) + \frac{2 π}{c _{δ}} \int_{0}^{t} h_{m} (\frac{t - r}{c _{δ}}) A_{δ, H} (r, \frac{t - r}{c _{δ}}) d r .

Displayed formulas in this passage

Em01-G@5#eq:scalar-trace · line 780
$J_{δ}^{ν} [A] = \int A (v, ξ) d_{δ}^{ν} (α ℓ + v k + ξ E, - d ℓ + v k + ξ E) d v d^{2} ξ .$
Em01-G@5#eq:radial-parameters · line 798
$a_{δ} Z_{δ} κ_{δ} L_{δ} = α - δ A_{1}, = (b_{δ} Z_{1} + a_{δ} Z_{2}) / e_{δ}, = 4 a_{δ} b_{δ} / e_{δ}, = κ_{δ} (B_{2} - B_{1}) - δ D_{δ} . b_{δ} D_{δ} = d + δ A_{2}, = a_{δ} b_{δ} ∣ Z_{1} - Z_{2} ∣^{2} / e_{δ}^{2}, e_{δ} = a_{δ} + b_{δ}, c_{δ} = a_{δ} / e_{δ},$
Em01-G@5#eq:angular-amplitude · line 813
$A_{δ, H} (t, z) = \frac{1}{2 π} \int_{0}^{2 π} A (δ B_{2} - \frac{∣ t ω _{φ} + δ ( Z _{δ} - Z _{2} ) ∣ ^{2} + z}{4 b _{δ}}, t ω_{φ} + δ Z_{δ}) d φ,$
Em01-G@5#eq:radial-amplitude · line 833
$F_{δ, H} (t) = A_{δ, H} (t, 0) + \frac{2 π}{c _{δ}} \int_{0}^{t} h_{m} (\frac{t - r}{c _{δ}}) A_{δ, H} (r, \frac{t - r}{c _{δ}}) d r .$
unlabeled · line 769
$D = W_{0} \otimes Δ_{m}, C_{I}^{μν} = i ℏ^{2} N_{I}^{μν} D .$
unlabeled · line 790
$H_{i} = A_{i} ℓ + B_{i} k + Z_{i} E, ν_{n} (H) = 4 ν (H_{1}, H_{2}) .$
unlabeled · line 828
$h_{m} (z) = - \frac{m ^{2}}{8 π} j = 0 \sum^{\infty} \frac{( - m ^{2} z /4 ) ^{j}}{j ! ( j + 1 )!},$