Lectures on Quantum Field Theory. Ashok Das

to as chirality and these (4 component Dirac) spinors can be shown to be eigenstates of the γ₅ matrix which can also be understood more easily from the chiral symmetry associated with massless Dirac systems.)

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 ν_e is left-handed, namely, its helicity is In the hole theoretic language, then, the absence of a negative energy neutrino would appear as a “hole” with the momentum reversed. Therefore, the anti-neutrino, in this description, will have opposite helicity or will be right-handed. Alternatively, the neutrino is left-handed and hence satisfies the equation

      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

      A very heuristic way to conclude that parity is violated in processes involving neutrinos is as follows. The neutrino is described by the equation

      Under parity or space reflection,

      Since σ represents an angular momentum, we conclude that it must transform under parity like L, so that under a space reflection

      Consequently, the neutrino equation is not invariant under parity, and processes involving neutrinos, therefore, would violate parity. This has been experimentally verified in a number of processes.

3.7Chirality

      With the normalization for massless spinors discussed in (2.53) and (2.54), the solutions of the massless Dirac equation (m = 0)

      can be written as (see (2.53) and compare with the massive case (3.94))

      From the structure of the massless Dirac equation (3.149), we note that if u(p) (or v(p)) is a solution, then γ₅u(p) (or γ₅v(p)) is also a solution. Therefore, the solutions of the massless Dirac equation can be classified according to the eigenvalues of γ₅ also known as the chirality or the handedness.

      This can also be seen from the fact that the Hamiltonian for a massless Dirac fermion (see (1.100))

      commutes with γ₅ (in fact, in the Pauli-Dirac representation γ₅ = ρ defined in (2.60) and ρ commutes with α, see, for example, (2.61)). Since

      it follows that the eigenvalues of γ₅ are ±1 and spinors with the eigenvalue +1, namely,

      are known as right-handed (positive chirality) spinors while those with the eigenvalue −1, namely,

      are called left-handed (negative chirality) spinors. We note that if the fermion is massive (m ≠ 0), then the Dirac Hamiltonian (1.100) would no longer commute with γ₅ and in this case chirality would not be a good quantum number to label the states with.

      Given a general spinor, the right-handed and the left-handed components can be obtained through the projection operators ( denotes the identity matrix in the appropriate space)

      where we have defined

      We note that by definition these projection operators satisfy

      which implies that any four component spinor can be uniquely decomposed into a right-handed and a left-handed component. (In the Pauli-Dirac representation, these projection operators have the explicit forms (see (2.92))

      We note from (3.155) that in the massless Dirac theory, the four component spinors can be effectively described by two component spinors. This is connected with our earlier observation (see section 3.6) that in the massless limit, the Dirac equation reduces to two decoupled two component Weyl equations (recall that it is the mass term which generally couples these two spinors). The reducibility of the spinors is best seen in the Weyl representation for the Dirac matrices discussed in (2.120). However, we will continue our discussion in the Pauli-Dirac representation of the Dirac matrices which we have used throughout. From the definition of the helicity operator in (3.123) (for the two component spinors S = σ), we note that spinors of the form

      correspond