and, therefore, define projection operators into the space of positive and negative helicity two component spinors. They can be easily generalized to a reducible representation of operators acting on the four component spinors and have the form (see (2.71) or (3.119))
and it is straightforward to check from (3.158) and (3.165) that
which is the reason the spinors can be simultaneous eigenstates of chirality and helicity (when mass vanishes). In fact, from (3.161) as well as (3.165) we see that the right-handed spinors with chirality +1 are characterized by helicity +1 while the left-handed spinors with chirality −1 have helicity −1.
For completeness as well as for later use, let us derive some properties of these spinors. We note from (3.159) that we can write the positive and the negative energy solutions as (we will do this in detail for the right-handed spinors and only quote the results for the left-handed spinors)
Each of these spinors,
For example, we can choose
such that when p1 = p2 = 0, the helicity spinors simply reduce to eigenstates of σ3. Furthermore, we can also define normalized spinors u(+) and v(+). For example, with the choice of the basis in (3.169), the normalized spinors take the forms (here we are using the three dimensional notation so that pi = (p)i)
105
However, we do not need to use any particular representation for our discussions. In general, the positive helicity spinors satisfy
which can be checked from the explicit forms of the spinors in (3.170). Here we note that the second relation follows from the fact that a positive helicity spinor changes into an orthogonal negative helicity spinor when the direction of the momentum is reversed (which is also manifest in the projection operators in (3.162)).
Given the form of the right-handed spinors in (3.161), together with (3.171), it now follows in a straightforward manner that
The completeness relation in (3.172) can be simplified by noting the following identity. We note that with p0 = |p|, we can write
so that we can write the completeness relation in (3.172) as
which can also be derived using the methods in section 3.4. We conclude this section by noting (without going into details) that similar relations can be derived for the left-handed spinors and take the forms
3.8Non-relativistic limit of the Dirac equation
Let us recall that the positive energy solutions of the Dirac equation have the form (see (2.49))
while the negative energy solutions have the form
In (3.176) we have defined
and we emphasize that the subscript “L” here does not stand for the left-handed particles introduced in the last section. Similarly, in (3.177) we have denoted
It is clear that in the non-relativistic limit, when |p|
