In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example).
In this basis, the spin is quantized along the axis in the direction of motion of the particle.
The two-component helicity eigenstates
we will use the generic form of fermion four-momentum: Then one can say the two helicity eigenstates are and These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather: In this situation the helicity eigenstates are for when the particle momentum is
A fermion 4-component wave function,
may be decomposed into states with definite four-momentum: Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is and for an anti-fermion, To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.
The plane wave expansion is For a vector boson with mass m and a four-momentum
, the polarization vectors quantized with respect to its momentum direction can be defined as