Manuscript V@1 · V.tex · section · e7ddf37d

Explicit signed kernels

Contrast: the Newton–wave cubic form › Explicit signed kernels · lines 3216–3322

V@1#eq:numerators, V@1#eq:Z, V@1#eq:dcancel, V@1#eq:cubickernel

Let

N_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2}, u_{σ} (x; p) = N_{p} e^{iσ ω_{p} t - iσ p \cdot x},

with $a_{+} = a^{†}, a_{-} = a$ . For a signed triple define

k_{a μ} = (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 ∣, w = q + σ_{3} p_{3} .

The stress numerator is

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

In particular,

ρ_{12} = n_{00} J_{12, i} = n_{0 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}), e_{12} n_{0 ν} + q_{i} n_{i ν} = 0.

Remove the common mode factors and the spatial wave $e^{- i q \cdot x}$ from the trace-reversed metric. Writing $E_{0} = e^{i e_{12} t_{0}}$ and $s_{0} (τ, r) = sin (τ r) / r$ , with value $τ$ at zero, the remaining amplitudes are

Z^{init}_{00} (s) Z^{init}_{ij} (s) Z^{src}_{μν} (s) = \frac{2 E _{0} ρ _{12}}{r ^{2}} cos ((s - t_{0}) r), Z^{init}_{0 i} = 0, = \frac{2 i E _{0}}{r ^{2}} s_{0} (s - t_{0}, r) (q_{i} J_{12, j} + q_{j} J_{12, i} - δ_{ij} q \cdot J_{12}), = 2 n_{μν} \int_{t_{0}}^{s} s_{0} (s - u, r) e^{i e_{12} u} d u .

For either part set

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} .

These retain the full $D (j)$ term before summation. Directly from (eq:numerators),

d_{ν} [Z^{init}] = 2 E_{0} s_{0} (s - t_{0}, r) n_{0 ν} = - d_{ν} [Z^{src}] .

For the spatial part use $q_{i} (q_{i} J_{j} + q_{j} J_{i} - δ_{ij} q \cdot J) = r^{2} J_{j}$ ; for the source part integrate once in $u$ . Thus the individual divergence terms are bounded and cancel in the total.

Let

F_{f} (t, \bk) = \int e^{- i \bk \cdot x} f (t, x) d^{3} x, s_{m} (τ, w) = \frac{sin ( τ ω _{w} )}{ω _{w}} .

For each of the eight signs define

U_{f}^{σ} = K_{f}^{σ} = N_{p_{1}} N_{p_{2}} N_{p_{3}} \int_{t_{0}}^{T} d t \int_{t_{0}}^{t} d s F_{f} (t, w) s_{m} (t - s, w) e^{i σ_{3} ω_{3} s} \times L_{3} [Z^{init} + Z^{src}] (s), \frac{1}{3} (U_{f}^{σ_{1} σ_{2} σ_{3}} (p_{1}, p_{2}, p_{3}) + U_{f}^{σ_{2} σ_{3} σ_{1}} (p_{2}, p_{3}, p_{1}) + U_{f}^{σ_{3} σ_{1} σ_{2}} (p_{3}, p_{1}, p_{2})) .

This is a finite-integral definition of the complete symmetric kernel, including all mode normalizations. The first two arguments of $U$ are symmetric; the cyclic average is therefore fully symmetric in signed arguments.

Displayed formulas in this passage

V@1#eq:numerators · line 3238
$ρ_{12} = n_{00} J_{12, i} = n_{0 i} = \frac{1}{2} [m^{2} - σ_{1} σ_{2} (ω_{1} ω_{2} + p_{1} \cdot p_{2})], = \frac{σ _{1} σ _{2}}{2} (ω_{1} p_{2 i} + ω_{2} p_{1 i}), e_{12} n_{0 ν} + q_{i} n_{i ν} = 0.$
V@1#eq:Z · line 3257
$Z^{init}_{00} (s) Z^{init}_{ij} (s) Z^{src}_{μν} (s) = \frac{2 E _{0} ρ _{12}}{r ^{2}} cos ((s - t_{0}) r), Z^{init}_{0 i} = 0, = \frac{2 i E _{0}}{r ^{2}} s_{0} (s - t_{0}, r) (q_{i} J_{12, j} + q_{j} J_{12, i} - δ_{ij} q \cdot J_{12}), = 2 n_{μν} \int_{t_{0}}^{s} s_{0} (s - u, r) e^{i e_{12} u} d u .$
V@1#eq:dcancel · line 3282
$d_{ν} [Z^{init}] = 2 E_{0} s_{0} (s - t_{0}, r) n_{0 ν} = - d_{ν} [Z^{src}] .$
V@1#eq:cubickernel · line 3301
$U_{f}^{σ} = K_{f}^{σ} = N_{p_{1}} N_{p_{2}} N_{p_{3}} \int_{t_{0}}^{T} d t \int_{t_{0}}^{t} d s F_{f} (t, w) s_{m} (t - s, w) e^{i σ_{3} ω_{3} s} \times L_{3} [Z^{init} + Z^{src}] (s), \frac{1}{3} (U_{f}^{σ_{1} σ_{2} σ_{3}} (p_{1}, p_{2}, p_{3}) + U_{f}^{σ_{2} σ_{3} σ_{1}} (p_{2}, p_{3}, p_{1}) + U_{f}^{σ_{3} σ_{1} σ_{2}} (p_{3}, p_{1}, p_{2})) .$
unlabeled · line 3219
$N_{p} = (2 π)^{- 3/2} (2 ω_{p})^{- 1/2}, u_{σ} (x; p) = N_{p} e^{iσ ω_{p} t - iσ p \cdot x},$
unlabeled · line 3225
$k_{a μ} = (σ_{a} ω_{a}, - σ_{a} p_{a}), e_{12} = σ_{1} ω_{1} + σ_{2} ω_{2}, q = σ_{1} p_{1} + σ_{2} p_{2}, r = ∣ q ∣, w = q + σ_{3} p_{3} .$
unlabeled · line 3232
$n_{μν} = - k_{1 (μ} k_{2 ν)} + \frac{1}{2} η_{μν} (k_{1} \cdot k_{2} - m^{2}) .$
unlabeled · line 3273
$d_{ν} [Z] = - \partial_{s} Z_{0 ν} - i q_{i} Z_{i ν}, L_{3} [Z] = (r Z)^{μν} k_{3 μ} k_{3 ν} - i d [Z]^{ν} k_{3 ν} .$
unlabeled · line 3295
$F_{f} (t, \bk) = \int e^{- i \bk \cdot x} f (t, x) d^{3} x, s_{m} (τ, w) = \frac{sin ( τ ω _{w} )}{ω _{w}} .$