Foundations of Quantum Field Theory. Klaus D Rothe
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Название: Foundations of Quantum Field Theory

Автор: Klaus D Rothe

Издательство: Ingram

Жанр: Физика

Серия: World Scientific Lecture Notes In Physics

isbn: 9789811221941

isbn:

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       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 (or chiral) representation particularly suited for discussing the relativistic limit, or the case of zero mass particles. There exists another choice of basis for the Gamma matrices called the Majorana representation which is particularly suited for the case of charge self-conjugate fermions, referred to as Majorana fermions.

      Spin zero, charge neutral particles are called “self-conjugate”, and are described by real fields satisfying the Klein–Gordon equation, which itself is real. In the case of “self-conjugate” spin one-half particles, the analogon is provided by the “Majorana” representation.

      The Majorana representation is, however, also useful in the case where fermions and anti-fermions are distinct particles if the symmetry group in question is for instance the orthogonal group O(N) rather than the unitary group U(N). The reason is that in the Majorana representation the Dirac equation is real. We therefore discuss separately the notion of Majorana representation and Majorana fermions.

       Majorana representation

      There exists a choice of basis in which all Dirac matrices are purely imaginary. This is called the Majorana representation of the Gamma-matrices. They are obtained from the corresponding ones in the Dirac representation via the unitary transformation

figure

      with

figure

      and have the property

figure

      One explicitly computes

figure

      Now, the Dirac wave function and its charge conjugate (in the Dirac and Weyl representations) are related by (see Chapter 7, Eq. (7.48)),

figure

      Applying the unitary operator (4.65) to both sides of the equation, we obtain for the Dirac wave function in the Majorana representation

figure

      where

figure

      We thus conclude that

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