Quantum Field Theory

4.4 Chiral SpinorsWhen we’ve needed an explicit form of the γ µ matrices, we’ve used the chiral representation( ) ( )0 10 σiγ 0 = , γ i =(4.57)1 0−σ i 0In this representation, the spinor rotation transformation S[Λ rot ] and boost transformationS[Λ boost ] were computed in (4.26) and (4.31). Both are block diagonal,S[Λ rot ] =( )e+i ⃗ϕ·⃗σ/200 e +i⃗ϕ·⃗σ/2and S[Λ boost ] =( )e+⃗χ·⃗σ/200 e −⃗χ·⃗σ/2(4.58)This means that the Dirac spinor representation of the Lorentz group is reducible. Itdecomposes into two irreducible representations, acting only on two-component spinorsu ± which, in the chiral representation, are defined byψ =(u+u −)(4.59)The two-component objects u ± are called Weyl spinors or chiral spinors. They transformin the same way under rotations,but oppositely under boosts,u ± → e i⃗ϕ·⃗σ/2 u ± (4.60)u ± → e ±⃗χ·⃗σ/2 u ± (4.61)In group theory language, u + is in the ( 1 , 0) representation of the Lorentz group,2while u − is in the (0, 1) representation. The Dirac spinor ψ lies in the 2 (1, 0) ⊕ (0, 1)2 2representation. (Strictly speaking, the spinor is a representation of the double cover ofthe Lorentz group SL(2,C)).4.4.1 The Weyl EquationLet’s see what becomes of the Dirac Lagrangian under the decomposition (4.59) intoWeyl spinors. We haveL = ¯ψ(i /∂ − m)ψ = iu † −σ µ ∂ µ u − + iu † +¯σ µ ∂ µ u + − m(u † +u − + u † −u + ) = 0 (4.62)– 91 –