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

Frozen component arrays, source, and crossed kernels

Frozen component arrays, source, and crossed kernels · lines 2174–2248

The advanced root is

F (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot x} \int_{t}^{\infty} \frac{sin ( ω _{p} ( s - t ))}{ω _{p}} f (s, p) d s, ω_{p} = ∣ p ∣^{2} + m^{2} .

The zero-frequency quotient is $s - t$ . Differentiation gives $P_{m} F = f$ and zero future data.

The retarded massless measure and its convolution square are

R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x, B (x) = \frac{1}{8 π} 1_{{x^{0} \geq ∣ x ∣}} .

They obey

□_{η} R = δ_{D}^{(4)}, R * R = B, \partial_{x^{μ}} B (x) = \frac{x _{μ}}{2} R (x) (x^{0} > 0, x \neq = 0) .

Their spatial transforms are

R (t, p) = Θ (t) \frac{sin ( ∣ p ∣ t )}{∣ p ∣}, B (t, p) = Θ (t) \frac{sin ( ∣ p ∣ t ) - ∣ p ∣ t cos ( ∣ p ∣ t )}{2∣ p ∣ ^{3}} .

The scalar lines are

W_{m} (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} t + i p \cdot x},

D_{T, m} (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} ∣ t ∣ + i p \cdot x} .

The conventions are

P_{m} D_{T, m} = - i δ_{D}^{(4)}, W_{m} (x) - W_{m} (- x) = - i Δ_{m} (x), P_{m} Δ_{m} = 0.

Set

Π_{α β ρ σ} = \frac{1}{2} (η_{α ρ} η_{β σ} + η_{α σ} η_{β ρ} - η_{α β} η_{ρ σ}),

N_{I}^{μν} = η^{μα} η^{ν β} Π_{α β ρ σ} (e^{I})^{ρ σ}, ϑ_{I} = η_{ρ σ} (e^{I})^{ρ σ} .

The grouped crossed array is

C_{I}^{μν} = ℏ^{2} N_{I}^{μν} (D_{T, 0} \otimes D_{T, m} - W_{0} \otimes W_{m}) .

Displayed formulas in this passage

unlabeled · line 2177
$F (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3}} e^{i p \cdot x} \int_{t}^{\infty} \frac{sin ( ω _{p} ( s - t ))}{ω _{p}} f (s, p) d s, ω_{p} = ∣ p ∣^{2} + m^{2} .$
unlabeled · line 2189
$R (ψ) = \frac{1}{4 π} \int_{R^{3}} \frac{ψ ( ∣ x ∣ , x )}{∣ x ∣} d^{3} x, B (x) = \frac{1}{8 π} 1_{{x^{0} \geq ∣ x ∣}} .$
unlabeled · line 2197
$□_{η} R = δ_{D}^{(4)}, R * R = B, \partial_{x^{μ}} B (x) = \frac{x _{μ}}{2} R (x) (x^{0} > 0, x \neq = 0) .$
unlabeled · line 2203
$R (t, p) = Θ (t) \frac{sin ( ∣ p ∣ t )}{∣ p ∣}, B (t, p) = Θ (t) \frac{sin ( ∣ p ∣ t ) - ∣ p ∣ t cos ( ∣ p ∣ t )}{2∣ p ∣ ^{3}} .$
unlabeled · line 2213
$W_{m} (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} t + i p \cdot x},$
unlabeled · line 2218
$D_{T, m} (t, x) = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 ω _{p}} e^{- i ω_{p} ∣ t ∣ + i p \cdot x} .$
unlabeled · line 2224
$P_{m} D_{T, m} = - i δ_{D}^{(4)}, W_{m} (x) - W_{m} (- x) = - i Δ_{m} (x), P_{m} Δ_{m} = 0.$
unlabeled · line 2229
$Π_{α β ρ σ} = \frac{1}{2} (η_{α ρ} η_{β σ} + η_{α σ} η_{β ρ} - η_{α β} η_{ρ σ}),$
unlabeled · line 2235
$N_{I}^{μν} = η^{μα} η^{ν β} Π_{α β ρ σ} (e^{I})^{ρ σ}, ϑ_{I} = η_{ρ σ} (e^{I})^{ρ σ} .$
unlabeled · line 2242
$C_{I}^{μν} = ℏ^{2} N_{I}^{μν} (D_{T, 0} \otimes D_{T, m} - W_{0} \otimes W_{m}) .$