A (Dirac) spinor is the 4-component field \(\psi_\alpha(x)\), \(\alpha=1,2,3,4\), that describes a spin-\(\tfrac12\) fermion. It transforms in the \((\tfrac12,0)\oplus(0,\tfrac12)\) representation of the Lorentz group and is the object on which the \(\gamma\)-matrices act. In QCD each quark is such a spinor for every color \(a\) and flavour \(f\), so \(\psi_{\substack{\alpha \\ a}}^{f}\) has \(4\times 3\) components per flavour.

The \(\gamma^\mu\) are \(4\times4\) matrices satisfying the Clifford algebra

\[ \{\gamma^\mu,\gamma^\nu\} = 2\,\eta^{\mu\nu}\,\mathbb{1}, \]

with \(\eta^{\mu\nu}=\mathrm{diag}(+,-,-,-)\). The Dirac conjugate is \(\overline{\psi}=\psi^\dagger\gamma^0\), chosen so that \(\overline\psi\psi\) is a Lorentz scalar and \(\overline\psi\gamma^\mu\psi\) a Lorentz vector.

Under a Lorentz transformation \(\Lambda\) the spinor rotates as

\[ \psi(x)\ \to\ S(\Lambda)\,\psi(\Lambda^{-1}x),\qquad S(\Lambda)=\exp\!\Big(-\tfrac{i}{4}\,\omega_{\mu\nu}\,\sigma^{\mu\nu}\Big),\quad \sigma^{\mu\nu}=\tfrac{i}{2}[\gamma^\mu,\gamma^\nu]. \]

The bilinear \(\overline\psi\,(i\gamma^\mu\partial_\mu - m)\,\psi\) is therefore Lorentz invariant, which is why the free quark Lagrangian takes the Dirac form.

Extremizing the free Lagrangian gives the Dirac equation

\[ (i\gamma^\mu\partial_\mu - m)\,\psi(x) = 0 . \]

Its plane-wave solutions come in two positive-energy (\(u\)) and two negative-energy (\(v\)) spinors, describing particle and antiparticle with the two spin projections.

Using \(\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3\) one splits a Dirac spinor into left- and right-handed Weyl components,

\[ \psi_{L,R} = \tfrac12(1\mp\gamma_5)\,\psi , \]

a decomposition central to the chiral structure of the weak interaction and to the discussion of chiral symmetry breaking in QCD. On the lattice, faithfully representing chirality is subtle (the Nielsen–Ninomiya no-go theorem), which motivates improved fermion actions such as HISQ.