rel="nofollow" href="#ulink_ab5ee148-79ad-5740-9fde-74f2d2c416df">3.112) or (3.113) can now be written as
3.6Massless Dirac particle
Let us consider the free Dirac equation for a massive spin
where we are not assuming any relation between p0 and p as yet. Let us represent the four component spinor (as before) as
where u1(p) and u2(p) are two component spinors. In terms of u1(p) and u2(p), the Dirac equation takes the form
Explicitly, this leads to the two (2-component) coupled equations
which can also be written as
Taking the sum and the difference of the two equations in (3.132), we obtain
We note that if we define two new (2-component) spinors as
then, the equations in (3.133) can be rewritten as a set of two coupled (2-component) spinor equations of the form
This shows that it is the mass term which couples the two equations.
Let us note that in the limit m → 0, the two equations in (3.135) reduce to two (2-component) spinor equations which are decoupled and have the simpler forms
These two equations, like the Dirac equation, can be shown to be covariant under proper Lorentz transformations (as they should be, since vanishing of the mass which is a Lorentz scalar should not change the behavior of the equation under proper Lorentz transformations). These equations, however, are not invariant under parity or space reflection and are known as the Weyl equations. The corresponding two component spinors uL and uR are also known as Weyl spinors.
Let us note that, in the massless limit,
Similarly, we can show that uL(p) also satisfies
Thus, for a nontrivial solution of these equations to exist, we must have
which is the Einstein relation for a massless particle. It is clear, therefore, that for such solutions, we must have
For p0 = |p|, namely, for the positive energy solutions, we note that
while
In other words, the two different Weyl equations really describe particles with opposite helicity. Recalling that
Figure 3.1: Right-handed and left-handed particles with spins parallel and anti-parallel to the direction of motion.
As we know, the electron neutrino emitted in a beta decay
is massless (present experiments suggest they are almost massless) and, therefore, can be described by a two component Weyl equation. We also know, experimentally, that νe is left-handed, namely, its helicity is
and has negative helicity. It is helicity which is the conserved quantum number and, hence, the absence of a negative energy neutrino would appear as a “hole” with opposite helicity. That the anti-neutrino is right-handed is, of course, observed in experiments such as
