Manuscript V-B@1 · V-B-v1.tex · section · 3fadc1ad

Explicit signed kernels

A separate Newton–wave cubic form › Explicit signed kernels · lines 1399–1484

V-B@1#eq:numerators, V-B@1#eq:Z, V-B@1#eq:dcancel, V-B@1#eq:U

For $σ = \pm 1$ , put

u_{σ} (x; p) = N_{p} e^{iσ ω_{p} t - iσ p \cdot x}, a_{+} (p) = a^{†} (p), a_{-} (p) = a (p) .

For signed momenta define covectors

k_{a} = (σ_{a} ω_{a}, - σ_{a} p_{a}), e_{12} = σ_{1} ω_{1} + σ_{2} ω_{2}, q = σ_{1} p_{1} + σ_{2} p_{2}, r = ∣ q ∣,

and

n_{μν} = - k_{1 (μ} k_{2 ν)} + \frac{1}{2} η_{μν} (k_{1} \cdot k_{2} - m^{2}) .

Let $ρ_{12} = n_{00}$ , $J_{12, i} = n_{0 i}$ . Then

ρ_{12} J_{12, i} = \frac{1}{2} [m^{2} - σ_{1} σ_{2} (ω_{1} ω_{2} + p_{1} \cdot p_{2})], = \frac{σ _{1} σ _{2}}{2} (ω_{1} p_{2 i} + ω_{2} p_{1 i}),

and $e_{12} n_{0 ν} + q_{i} n_{i ν} = 0$ .

After removing $N_{p_{1}} N_{p_{2}} e^{- i q \cdot x}$ , the trace-reversed metric amplitudes are

Z_{00}^{in} (s) Z_{ij}^{in} (s) Z_{μν}^{src} (s) = \frac{2 ρ _{12}}{r ^{2}} cos (sr), = \frac{2 i}{r ^{2}} S_{w} (s, r) (q_{i} J_{12, j} + q_{j} J_{12, i} - δ_{ij} q \cdot J_{12}), = 2 n_{μν} \int_{0}^{s} S_{w} (s - u, r) e^{i e_{12} u} d u, Z_{0 i}^{in} (s) = 0,

where $S_{w} (s, r) = sin (sr) / r$ , continuously defined at zero.

Define

d_{ν} [Z] = - \partial_{s} Z_{0 ν} - i q_{i} Z_{i ν},

L_{3} [Z] = (r Z)^{μν} k_{3 μ} k_{3 ν} - i d [Z]^{ν} k_{3 ν} .

An explicit differentiation of (eq:Z) gives

d_{ν} [Z^{in}] = 2 S_{w} (s, r) n_{0 ν} = - d_{ν} [Z^{src}] .

The full $D (j)$ term is therefore retained before it cancels.

Write

F_{f} (t, w) = \int e^{- i w \cdot x} f (t, x) d^{3} x, w = q + σ_{3} p_{3},

and $S_{m} (τ, w) = sin (τ ω_{w}) / ω_{w}$ . The ordered kernel is

U_{f}^{σ} = N_{p_{1}} N_{p_{2}} N_{p_{3}} \int_{0}^{b} d t \int_{0}^{t} d s F_{f} (t, w) S_{m} (t - s, w) e^{i σ_{3} ω_{3} s} L_{3} [Z^{in} + Z^{src}] (s) .

Let $K_{f}^{σ}$ be the cyclic average over the three choices of stress pair. The first two arguments of $U$ are symmetric, so this average is fully symmetric in signed arguments.

Displayed formulas in this passage

V-B@1#eq:numerators · line 1420
$ρ_{12} J_{12, i} = \frac{1}{2} [m^{2} - σ_{1} σ_{2} (ω_{1} ω_{2} + p_{1} \cdot p_{2})], = \frac{σ _{1} σ _{2}}{2} (ω_{1} p_{2 i} + ω_{2} p_{1 i}),$
V-B@1#eq:Z · line 1434
$Z_{00}^{in} (s) Z_{ij}^{in} (s) Z_{μν}^{src} (s) = \frac{2 ρ _{12}}{r ^{2}} cos (sr), = \frac{2 i}{r ^{2}} S_{w} (s, r) (q_{i} J_{12, j} + q_{j} J_{12, i} - δ_{ij} q \cdot J_{12}), = 2 n_{μν} \int_{0}^{s} S_{w} (s - u, r) e^{i e_{12} u} d u, Z_{0 i}^{in} (s) = 0,$
V-B@1#eq:dcancel · line 1458
$d_{ν} [Z^{in}] = 2 S_{w} (s, r) n_{0 ν} = - d_{ν} [Z^{src}] .$
V-B@1#eq:U · line 1473
$U_{f}^{σ} = N_{p_{1}} N_{p_{2}} N_{p_{3}} \int_{0}^{b} d t \int_{0}^{t} d s F_{f} (t, w) S_{m} (t - s, w) e^{i σ_{3} ω_{3} s} L_{3} [Z^{in} + Z^{src}] (s) .$
unlabeled · line 1402
$u_{σ} (x; p) = N_{p} e^{iσ ω_{p} t - iσ p \cdot x}, a_{+} (p) = a^{†} (p), a_{-} (p) = a (p) .$
unlabeled · line 1407
$k_{a} = (σ_{a} ω_{a}, - σ_{a} p_{a}), e_{12} = σ_{1} ω_{1} + σ_{2} ω_{2}, q = σ_{1} p_{1} + σ_{2} p_{2}, r = ∣ q ∣,$
unlabeled · line 1413
$n_{μν} = - k_{1 (μ} k_{2 ν)} + \frac{1}{2} η_{μν} (k_{1} \cdot k_{2} - m^{2}) .$
unlabeled · line 1449
$d_{ν} [Z] = - \partial_{s} Z_{0 ν} - i q_{i} Z_{i ν},$
unlabeled · line 1452
$L_{3} [Z] = (r Z)^{μν} k_{3 μ} k_{3 ν} - i d [Z]^{ν} k_{3 ν} .$
unlabeled · line 1467
$F_{f} (t, w) = \int e^{- i w \cdot x} f (t, x) d^{3} x, w = q + σ_{3} p_{3},$