Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 35

In the extreme relativistic limit we expect the mass of the fermion to be negligible. For m = 0 the Dirac Hamiltonian operator commutes with γ₅ . Hence we may classify the eigenfunctions of the Hamiltonian by the eigenvalues of γ₅ . It is thus desirable to work in the Weyl representation, where γ₅ is diagonal. In this representation the Dirac operator becomes off-diagonal in the large momentum limit, and the Weyl equations (4.27) and (4.28) reduce to the form

or

The 2 × 2 matrix represents the projection of the angular momentum operator on the direction of motion of the particle and is called the helicity operator. Correspondingly one refers to its eigenvalues ±1/2 as helicity.

Equations (4.52) and (4.53) are just the Weyl equations for a massless particle. If parity is not conserved, we may confine ourselves to either one of the two equations, that is to either particles polarized in the direction of motion (positive helicity) or opposite to the direction of motion (negative helicity). This is the case for neutrinos (antineutrinos) participating in the parity-violating weak interactions, which carry helicity −1/2 (+1/2). If parity is conserved, both helicity states must exist.

The fact that the massive Dirac equation turns into Weyl equations in the “infinite momentum frame” shows that at high energies massive particles are polarized “parallel” or “anti-parallel” to the direction of motion. However, whereas the helicity of a massless particle is a Lorentz invariant, this is not the case for a massive particle: If a massive particle is polarized in the direction of motion in one inertial frame, its polarization will be a superposition of all possible spin projections in a different inertial system. Phrased in a different way: If the particle is massive one can always catch up with it and ultrapass it, so that the particle appears moving “backwards”, while continuing to be polarized in the original direction. With a zero mass particle you can never catch up since it is moving at the speed of light.