Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

Foundations of Quantum Field Theory - Klaus D Rothe

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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.

       Chirality

      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:

figure

      Let us define the “spin” operator

figure

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

figure

      The Dirac equation may then be written in the form

figure

      where figure. Thus figure 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

figure

      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

figure

      We have

figure

      where

figure

      Recalling that in the Weyl representation

figure

      we have

figure

      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).

       Solution of Weyl equations

      Experiment shows that neutrinos (antineutrinos) only occur with negative (positive) helicity. One thus refers to figure 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

figure

      with

figure

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

figure

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

figure

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

figure

      Hence

figure

      This determines θ as a function of |

|:

figure

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

figure

      The result is

figure

      where figure is the matrix (2.50):

figure

      Here

) in the massive case. Correspondingly we have from (4.60) and (2.34) for the 2-component spinors

figure

      Similarly we have

figure

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

figure

      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),

figure

      where we have chosen κ = m.

      So far we have considered the Dirac representation, particularly suited for discussing the non-relativistic limit as we shall see, and the Weyl

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