The path integral (Feynman) formulation expresses quantum amplitudes as a sum over all field configurations, each weighted by a phase \(e^{iS}\) (Minkowski) or a Boltzmann-like factor \(e^{-S_E}\) (Euclidean). It is the formulation of choice for gauge theories and the direct basis of lattice QCD.

The transition amplitude between position eigenstates is

\[ \langle q_f,t_f | q_i,t_i\rangle = \int_{q(t_i)=q_i}^{q(t_f)=q_f} [Dq]\ e^{\,iS[q]/\hbar},\qquad S=\int dt\,L, \]

a weighted sum over every path connecting the endpoints. The classical trajectory is recovered as the stationary-phase configuration \(\delta S=0\).

For fields the integration is over field configurations,

\[ Z = \int [D\phi]\ e^{\,iS[\phi]}. \]

A Wick rotation to imaginary time \(t=-i\tau\) turns the oscillatory weight into a real, positive, convergent one:

\[ e^{\,iS}\ \longrightarrow\ e^{-S_E},\qquad S_E = \int d\tau\, d^3x\ \mathcal{L}_E . \]

This euclidean form makes the integral resemble a statistical-mechanical Boltzmann sum, enabling Monte Carlo evaluation.

Comparing \(Z=\mathrm{Tr}\,e^{-\beta H}\) with the Euclidean path integral shows that thermal field theory is a path integral on a compact time circle: the euclidean time is periodic with period \(\beta=1/T\), giving

\[ Z = \int_{\text{(anti)periodic}} [D\phi]\ e^{-S_E[\phi]} . \]

Bosonic fields are periodic, fermionic fields anti-periodic. This is exactly the representation of the QCD partition function used in Continuum QCD. For fermions the fields are Grassmann-valued, and integrating them out produces the fermion determinant that dominates the cost of lattice simulations.