Appendix

Notation for Fermion Fields

We describe spin- fermions by two-component fields of definite chirality: lefthanded fields are denoted and right-handed fields For massless fermions, chirality and helicity are equivalent and the two chirality states are independent. Only one of the states need therefore be present (for massless neutrinos is absent).

For the two-component fields, denotes the left-handed antiparticle of , while denotes the right-handed antiparticle of . For fields in which both helicity states are present, parity () serves to interchange L and R components, while charge conjugation () interchanges particles with antiparticles, according to

where is the Pauli matrix. These transformations are summarized in fig. 27. Note the important feature that while the separate operations of and interchange L and R components, the combined transformation does not modify the helicity state. Hence while the definition of individual and transformation properties require the presence of both L and R states, transformation properties may be defined for massless particles with only a single helicity state.

[ Figure 27 ] Action of and transformations on two-component fermion fields.

The two-component fermion fields may be collected into a four-component Dirac spinor describing a fermion of arbitrary helicity: It is convenient to take the Dirac gamma matrices which act on this spinor in the Weyl representation:

with the usual Pauli matrices. (This representation differs from the more usual Dirac representation simply by the interchange .)

The kinetic energy term in the fermion lagrangian is given by

with , .

Fermion fields for which both helicity states are present may give a Dirac mass term

If only one helicity is present, say , no Dirac mass term may be constructed, but a Majorana mass term is still possible:

Here the charge-conjugate four-component spinor is given by

For a fermion field with only a single helicity state, it is sometimes convenient to define a four-component Majorana spinor

in terms of which the Majorana mass term becomes .

Note that fields with Majorana mass terms may not carry any charges since the mass term is not invariant under gauge transformations .

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