232 Appendix F: Heavy Quark Production Radiation Derivation of the integration over all space. Therefore we have that Note that we have implicitly taken k 0 = | ⃗ k | and will continue to do so for the rest of the calculation. If we define A $^{µ}$( x ) = Re ∫ d $^{3}$k (2 π ) $_{3}$A $^{µ}$( k ) e − ik · $^{x}$, (F.39) A $^{µ}$( k ) = − e | ⃗ k | ( p ′ µ k · p ′ − p µ k · p$^{)}$ . (F.40) E $^{i}$( x ) = Re ∫ d $^{3}$k (2 π ) $_{3}$E $^{i}$( x ) e − ik · x (F.41) B $^{i}$( x ) = Re ∫ d $^{3}$k (2 π ) $_{3}$B $^{i}$( x ) e − ik · x (F.42) E i = − F 0 i = ∂ $^{i}$A 0 − ∂ $^{0}$A i (F.43) B = ∗ F ⇒ B i = 1 2 ϵ $^{i}$$_{jk}$F jk = 1 2 ϵ i jk $^{(}$∂ $^{j}$Ak − ∂ $^{k}$Aj ) = ϵ $^{i}$$_{jk}$∂$^{j}$Ak = ( ⃗ ∇ × ⃗ A ) $^{i}$. (F.44) and recall that