7 - Two-Component Spinor Fields

Summary

The two two-dimensional representations of the Lorentz group give rise to left and right two-component spinor fields; on the basis of a thorough understanding of these representations, the procedure of perturbative canonical quantization developed for scalar fields can be applied to two-component spinor fields.

Introduction

Having classified the representations of the Lorentz group in the last chapter, this chapter returns to the theme of Chapter 4, namely perturbative canonical quantization. Here and in the two subsequent chapters, we extend the principles of Chapter 4 to two-component Fermi fields, four-component Fermi fields, and vector fields.

This chapter has a purpose to present the quantum theory of Weyl fields and make the connection between Weyl fields and Dirac fields. By the end of this chapter, we shall be able to compute scattering amplitudes for Feynman diagrams with external and internal Weyl lines.

We begin in Sections 7.1 and 7.2 by finding a convenient description of (0, 1/2) and (1/2, 0), the smallest non-trivial representations of the Lorentz group. These spinor representations respectively determine the right and left two-component spinor or Weyl fields. Section 7.3 then tabulates all the Lorentz-invariant Lagrangian terms which can be formed from these Weyl fields. The anti-symmetric character of these terms implies that the Weyl fields must be fermionic.