Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 36

As the last argument above shows, the m = 0 case has to be treated separately, and cannot be obtained as the zero-mass limit of massive case discussed so far, which was based on the existence of a rest frame of the particle. According to our discussion in the chapter on Lorentz transformations, zero-mass 1-particle states indeed transform quite differently from the massive ones.

In the zero mass case, the Dirac equations for the U and V spinors reduce to one and the same equation:

Let us define the “spin” operator

In terms of the Gamma matrices (Dirac or Weyl basis) this operator reads

The Dirac equation may then be written in the form

where . Thus is just the helicity operator in the 4-component representation.

The helicity operator (4.56) commutes with the free Dirac Hamiltonian. The same applies to γ₅, if the mass of the particle is zero. Since furthermore

we may classify the eigenstates of the zero-mass Dirac Hamiltonian according to their helicity and chirality, the latter being defined as the corresponding eigenvalue ±1 of γ⁵. Such states are obtained from the solutions U to the Dirac equation with the aid of the projection operator

We have

where

Recalling that in the Weyl representation

we have

The eigenvalue of γ₅ thus coincides with twice the eigenvalue of the helicity operator: particles of positive (negative) chirality, carry helicity +1/2(−1/2).