Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 4 / c

A relativistic two-component spinor is an element of a two-dimensional complex representation of the special linear group $SL(2,\mathbb{C})$, the double cover of the proper orthochronous Lorentz group. A Weyl spinor transforms in either $(1/2,0)$ or $(0,1/2)$; parity interchanges the two chiralities. It describes a chiral spin-$1/2$ degree of freedom. A free massless Weyl field has one particle helicity and the opposite antiparticle helicity.

The number of components alone does not prove zero mass. A massive Dirac field has two coupled two-component chiral fields, and a nonrelativistic Pauli spinor describes the two spin states of a massive spin-$1/2$ particle. A Majorana mass term can also be written using a two-component field and its complex conjugate. Thus a single independent free massless Weyl equation is a dynamical assumption, not part of the bare algebraic definition of a two-component spinor.