The partition function \(Z\) is the central object of Statistical Mechanics: it encodes the thermodynamics of a system in equilibrium with a heat bath at temperature \(T\). For a quantum system with Hamiltonian \(H\),

\[ Z = \mathrm{Tr}\,e^{-\beta H},\qquad \beta = \frac{1}{k_B T}. \]

All equilibrium quantities follow from \(Z\): the free energy \(F=-T\ln Z\), the energy \(\langle E\rangle=-\partial_\beta \ln Z\), the pressure, entropy, and susceptibilities as further derivatives.

The exponential \(e^{-\beta H}\) is formally the time-evolution operator evaluated at imaginary (Euclidean) time \(t=-i\tau\) with \(\tau\) running over an interval \(\beta\). Writing the trace as a sum over field eigenstates and inserting the path integral formulation gives, for a scalar field,

\[ Z = \int d\phi\,\langle\phi|e^{-\beta H}|\phi\rangle = \int d\phi \int_{\phi(\tau_i)}^{\phi(\tau_i+\beta)=\phi(\tau_i)} [D\phi]\ e^{-S_E[\phi]} . \]

The trace is what forces the field to return to itself after Euclidean time \(\beta\) — i.e. periodic boundary conditions for bosons.

For QCD one integrates over the gauge field \(A\) and the quarks \(\overline\psi,\psi\):

\[ Z = \int d\overline\psi\, d\psi\, dA \int_{A(\tau_i),\psi(\tau_i)}^{\substack{A(\tau_i+\beta)=A(\tau_i)\\ \psi(\tau_i+\beta)=-\psi(\tau_i)}} [D\overline\psi][D\psi][DA]\ e^{-S_E[\overline\psi,\psi,A]} , \]

with the Euclidean action

\[ S_E[\overline\psi,\psi,A] = \int_{\tau_i}^{\tau_i+\beta} d\tau \int d^3x\ \mathcal{L}^E_{QCD}. \]

The bosonic gauge field is periodic and the fermionic quark field anti-periodic in \(\tau\). This Euclidean \(Z\) is exactly what lattice QCD evaluates numerically: discretizing spacetime turns \(Z\) into a very high-dimensional integral sampled by importance-sampling Monte Carlo.