Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 37

Experiment shows that neutrinos (antineutrinos) only occur with negative (positive) helicity. One thus refers to as being left (right) handed. This is reflected by the so-called V − A (vector minus axial vector) coupling of the neutrino sector. Since parity is violated, the absence of right-handed neutrinos and left-handed antineutrinos is admissible. The 4-component Dirac field (4.9) of the massive case is thus replaced in this case by

with

where the spin projection now refers to helicity. The Weyl equations thus reduce to solving the eigenvalue problems

For the momentum pointing in the z-direction, the eigenvalue problems are solved by

with . Now perform the following transformation: First boost the momentum kμ to the momentum with the matrix (see (2.15))

Hence

This determines θ as a function of ||:

We now rotate the vector pμ thus obtained in the desired direction of the final vector pμ with the rotation matrix ,

The result is

where is the matrix (2.50):

Here plays the role of the boost L() in the massive case. Correspondingly we have from (4.60) and (2.34) for the 2-component spinors

Similarly we have

The fields (4.57) and (4.58) now take the form

In this form, the Fourier decomposition resembles closely that of a massive field except for the fact that in the massless case U = V. Alternatively we have using (4.61) and (4.62),

where we have chosen κ = m.