A fermion is a particle (or field; basically particles are excitations of fields) with half-integer spin \(s=\tfrac{1}{2},\tfrac{3}{2},\dots\) that obeys Fermi–Dirac statistics. All matter fields of the Standard Model — the quarks and the leptons — are spin-\(\tfrac{1}{2}\) fermions, described by 4-component Dirac spinors.

The defining property of fermions is that their multiparticle state is totally antisymmetric under exchange of any two identical particles:

\[ \psi(x_1,x_2) = -\,\psi(x_2,x_1). \]

An immediate consequence is the Pauli exclusion principle: two identical fermions cannot occupy the same single-particle quantum state, since setting \(x_1=x_2\) forces \(\psi=0\). This is precisely the constraint that made the color quantum number necessary in QCD.

The connection "half-integer spin \(\Leftrightarrow\) antisymmetric statistics" is not an assumption but a theorem in relativistic QFT: causality (microcausality of local observables) together with a positive-definite energy spectrum force spin-\(\tfrac{1}{2}\) fields to be quantized with anticommutators,

\[ \{\psi_\alpha(\vec x,t),\psi_\beta^{\dagger}(\vec y,t)\} = \delta_{\alpha\beta}\,\delta^{3}(\vec x-\vec y), \]

rather than the commutators used for a boson. Quantizing a spinor with commutators instead would violate causality and give a Hamiltonian unbounded below.

In the path integral formulation a fermionic field cannot be an ordinary c-number, precisely because of antisymmetry. It is represented by Grassmann numbers \(\theta,\eta\) obeying

\[ \theta\eta = -\eta\theta,\qquad \theta^{2}=0, \]

with Berezin integration rules \(\int d\theta\,1 = 0\), \(\int d\theta\,\theta = 1\). Gaussian integration over Grassmann fields yields a determinant rather than the inverse determinant of the bosonic case:

\[ \int [D\overline{\psi}][D\psi]\ e^{-\overline{\psi}M\psi} = \det M . \]

This fermion determinant is the object that must be evaluated (or stochastically estimated) when the quarks are integrated out in lattice QCD simulations.

At finite temperature the trace \(Z=\mathrm{Tr}\,e^{-\beta H}\) imposes anti-periodic boundary conditions in euclidean time for fermions,

\[ \psi(\vec x,\tau+\beta) = -\,\psi(\vec x,\tau), \]

in contrast to the periodic conditions for bosonic fields. This is the origin of the \(-\psi(\tau_i)\) boundary term appearing in the QCD partition function.