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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.

A two-component complex wave function describing the two spin states of a nonrelativistic spin-$1/2$ particle. Under a spatial rotation by angle $\theta$ about $\hat{\mathbf{n}}$ it transforms as $\psi \mapsto e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{\sigma }/2}\psi$, with Pauli matrices $\mathbf{\sigma }$. A Pauli spinor may describe a massive particle; two components alone do not imply the massless Weyl equation.