By Samiou E.
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The corresponding Lagrangian is given by eq. 105). Together, ξ and η constitute a single Dirac spin-1/2 fermion. 51) and all other anticommutators vanish. 52) p, λ; F ≡ (2π)3/2 (2Ep)1/2 a† (p, λ) |0 . 53) 56 2 Feynman Rules for Fermions Note that both ξ(x) and η¯(x) can create |p, λ; F from the vacuum, while ¯ ξ(x) and η(x) can create p, λ; F . 57) and the eight other single-particle matrix elements vanish. Note that the two-component wave functions, xα (p) and yα (p) that appear in the Fourier expansions of the two-component fermion ﬁelds [eqs.
In deriving eq. 30), we made use of eq. 29) and the completeness of the ζλ . The product of three dot-products was simpliﬁed by noting that p·s = 0 implies s·σ p·σ = −p·σ s·σ. The other spin projection formulas can be similarly derived. 37) yβ˙ (p, λ) = − 12 mδα˙ β˙ − [s·σ p·σ]α˙ β˙ x ¯α˙ (p, λ)¯ . 38) 54 2 Feynman Rules for Fermions For the case of massless spin-1/2 fermions, we must use helicity spinor wave functions. 41) yα (p, λ)¯ yβ˙ (p, λ) = ( 12 + λ)p·σαβ˙ , x ¯α˙ (p, λ)¯ yβ˙ (p, λ) = 0 .
Charge conjugation is less trivial in a theory of charged fermions. Consider a theory of a pair of twocomponent fermion ﬁelds of equal (non-zero) mass. The corresponding free-fermion Lagrangian is given by eq. 102) with m1 = m2 . As before, we assume that the phases of the ﬁelds have been chosen such that m1 and m2 are real and positive. In this case, the Lagrangian exhibits a global O(2) symmetry, χi → Cij χj , where C ∈O(2). Corresponding to this symmetry is an hermitian conserved Noether current [eq.
2-step nilpotent Lie groups of higher rank by Samiou E.