Single-Particle Electrodynamics - Assassination Science

= U γ Σ β{ }
∂ γ F α β + ∂ α F βγ
= U γ Σ β{ −∂ β F } α
γ
= −U γ (Σ · ∂)F γα
= +U γ (Σ · ∂)F αγ
= (Σ · ∂)F · U;
hence,
(U · ∂)F ·Σ + ∂(Σ ·F · U) = (Σ · ∂)F · U;
this is equation (B.28).
(U · ∂) ˜F ·Σ + ∂(Σ · ˜F · U) = U γ ∂ γ ˜F α βΣ β + ∂ α Σ β ˜F βγ U γ
= U γ Σ β{ ∂ γ ˜F α β + ∂ α ˜F βγ
}
= U γ Σ β{ −∂ β ˜F γ α − ε α βγµJ µ}
= −U γ (Σ · ∂)F γα − ε α βγµΣ β U γ J µ
= +U γ (Σ · ∂)F αγ + ε α βγµΣ β J γ U µ
= (Σ · ∂)F · U + Σ ×J ×U;
hence,
(U · ∂) ˜F ·Σ + ∂(Σ · ˜F · U) = (Σ · ∂) ˜F · U + Σ ×J ×U;
this is equation (B.29).
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