Consider the Dirac equation without potentials (free equation)
Its stationary states are found with the ansatz
where , and
We split up the 4-component spinor into two 2-component spinors and to give
Substituting our definitions into the Dirac equation we have
States with definite momentum can be written as
We recognize that these states are eigenfunctions of the momentum operator:
Substitution into equations 5.65 and 5.66 gives
Rearranging we have
For a solution we require
Using identity 2.24 we have
where we have been careful with the sign of when solving for both cases at ones.
We also have
Let us denote the two-component spinor by
where and are complex and is normalized according to .
The complete set of positive- and negative-energy free solutions is
The normalization is determined from
We now show that another quantum number, helicity, can be used to classify free one-particle states. Define
This is the four dimensional generalization of the spin vector operator.
and show that it commutes with the free Dirac Hamiltonian operator.
Since , we need only consider
Also , therefore can all be diagonalized together.
can be rewritten as the helicity operator
Helicity is the projection of the spin onto the direction of the momentum.
For an electron propagating in the -direction ,
with eigenvalues . The eigenvectors of are
We can write , where
We thus see that we have two independent states for a positive-energy electron:
We expect the absence of ``spin-up'' to be equivalent to the presence of ``spin-down''. This implies that a negative-energy solution with spin down is equivalent to a positive-energy solution with spin up. We choose
Why dose the Dirac spinor need four components to describe a spin-1/2 particle? The answer has to do with parity. in a relativistic theory, we would not be able to represent the effect of parity satisfactorily with only two components.