47 The Dirac Equation and Spin 12 Fields

The space of solutions to the Klein-Gordon equation $\mathrm{gi}$ ves an irreducible representation of the Poincar´e group corresponding to a relativistic particle of mass m and spin zero. Elementary matter particles (quarks and leptons) are spin $\begin{array}{l}\frac{1}{2}\end{array}$ particles, and we would like to have a relativistic wave equation that describes them, suitable for building a quantum field theory.

This is provided by a remarkable construction that uses the Cliford algebra and its action on spinors to find a square root of the Klein-Gordon equation, the Dirac equation. We will begin with the case of real-valued spinor fields, for which the quantum field theory describes spin $\begin{array}{l}\frac{1}{2}\end{array}$ neutral massive relativistic fermions, known as Majorana fermions. In the massless case it turns out that the Dirac equation decouples into two separate equations for two-component complex fields, the Weyl equations, and quantization leads to a relativistic theory of massless helicity $\pm \frac{1}{2}$ particles which can carry a charge, the Weyl fermions. Pairs of Weyl fermions of opposite helicity can be coupled together to form a theory of charged, massive, spin $\frac{1}{2}$ particles, the Dirac fermions. In the low energy, non-relativistic limit, the Dirac fermion theory becomes the Pauli-Schr¨odinger theory discussed in chapter 34.

中文目录

Chapter contents

• 47.1 The Dirac equation in Minkowski space
• 47.2 Majorana spinors and the Majorana field
  • 47.2.1 Majorana spinor fields in momentum space
  • 47.2.2 Quantization of the Majorana field
• 47.3 Weyl spinors
• 47.4 Dirac spinors
• 47.5 For further reading

Previous chapter: 46 Quantization of the Electromagnetic Field: the Photon

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