# In particular we can now identify ξ and ? as fields

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which provides the motivation for this basis choice. In particular, we can now identify ξ and η as fields of definite and opposite charge Q . Together, ξ and η constitute a single Dirac spin-1/2 fermion. Finally, we introduce the concept of off-shell degrees of freedom and contrast this with on-shell or physical degrees of freedom. Here, off-shell [on-shell] is shorthand for off- [on-] mass shell and distinguishes between virtual and real particle propagation, respectively. Given a particle with four-momentum p , real particle propagation must satisfy p 2 = m 2 (the mass-shell condition), whereas for virtual particles ( e.g. , particle exchange inside Feynman diagrams), the four-momentum p and the mass m are independent. The field operator corresponding to real particle propagation satisfies the field equations, whereas the field operator of a virtual particle is not constrained by the field equations. For scalar (spin-0) fields, there is no distinction between on-shell and off-shell degrees of freedom. The on-shell scalar field satisfies the Klein- Gordon equation, but this does not alter the fact that a real (complex) scalar field consists of one (two) degree(s) of freedom. For a spin- s field ( s > 0), the number of on-shell degrees of freedom is less than the number of off-shell degrees of freedom. Although all on-shell spin- s fields satisfy the Klein-Gordon equation, the spin- s field equations provide additional constraints that reduce the number of degrees of freedom originally present. We illustrate this in the case of free fermion field theory. A theory of a neutral self-conjugate fermion is described in terms of a single complex two-component fermion field χ α ( x ). This corresponds initially to four off-shell degrees of freedom, since χ α and χ ˙ α ( χ ) α are independent degrees of freedom. But, when the field equations are imposed [see eq. (1.100)], χ ( x ) is determined in terms of χ ( x ). Thus, the number of physical (“on-shell”) degrees of freedom for a neutral self- conjugate fermion is equal to two. A theory of a charged fermion is described in terms of a pair of mass- degenerate two-component fermions. We shall work in a basis, where the charged fermion is represented by a pair of two-component fermion fields ξ and η . The number of off-shell degrees of freedom for a charged fermion is eight. Applying the field equations determines ξ in terms of η and ¯ η in terms of ξ [see eqs. (1.106) and (1.107)]. Thus, the number of physical on-shell degrees of freedom for a charged fermion is equal to four.
26 1 Two-component formalism for Spin-1/2 Fermions Although we have illustrated the counting of degrees of freedom for free fermion field theory, the same counting applies in an interacting theory. 1.6 The fermion mass-matrix and its diagonalization We now generalize the discussion of Section 1.5 and consider a collection of n free anti-commuting two-component spin-1/2 fields, ζ αi ( x ), which transform as ( 1 2 , 0) fields under the Lorentz group. Here, α is the spinor index, and i is a ﬂavor index ( i = 1 , 2 , . . . , n ) that labels the distinct fields of the collection. The free-field Lagrangian is given by
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