Spinors of Spin-one-half Fields

TL;DR

This paper reviews the transformation properties of spin-one-half fields and spinors under rotations and discrete symmetries, deriving explicit forms of spinors from the Dirac equation and discussing their decomposition and symmetry behaviors.

Contribution

It provides a detailed derivation of spinors for spin-one-half fields from fundamental principles and explores their transformation properties under various symmetries, including charge conjugation, parity, and time reversal.

Findings

Explicit forms of momentum-zero and finite-momentum spinors derived from the Dirac equation.

Demonstration of proper transformation of Dirac fields under discrete symmetries.

Description of field decompositions into Majorana and chiral components.

Abstract

This paper reviews how a two-state, spin-one-half system transforms under rotations. It then uses that knowledge to explain how momentum-zero, spin-one-half annihilation and creation operators transform under rotations. The paper then explains how a spin-one-half field transforms under rotations. The momentum-zero spinors are found from the way spin-one-half systems transform under rotations and from the Dirac equation. Once the momentum-zero spinors are known, the Dirac equation immediately yields the spinors at finite momentum. The paper then shows that with these spinors, a Dirac field transforms appropriately under charge conjugation, parity, and time reversal. The paper also describes how a Dirac field may be decomposed either into two 4-component Majorana fields or into a 2-component left-handed field and a 2-component right-handed field. Wigner rotations and Weinberg's derivation of the properties of spinors are also discussed.