Helicity basis

Spinors

The two-component helicity eigenstates ξ λ satisfy

where σ are the Pauli matrices, p ^ is the direction of the fermion momentum, λ = ± 1 depending on whether spin is pointing in the same direction as p ^ or opposite.

To say more about the state, ξ λ 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 p ^ = + z ^

for then for when momentum is p ^ = − z ^

Fermion (spin 1/2) wavefunction

A fermion 4-component wave function, ψ may be decomposed into states with definite four-momentum:

where

Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is

and for an anti-fermion,

Dirac matrices

To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.

Spin-1 wavefunctions

The plane wave expansion is

For a Vector boson with mass 'm' and a four-momentum q μ = ( E , q x , q y , q z ) the polarization vectors quantized with respect to its momentum direction can be defined as

where