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Lectures on Quantum Field Theory. Ashok Das
Читать онлайн.Название Lectures on Quantum Field Theory
Год выпуска 0
isbn 9789811220883
Автор произведения Ashok Das
Жанр Биология
Издательство Ingram
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
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 (
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 =
correspond