"Euclidean" refers to the formulation of field theory obtained by rotating from real (Minkowski) time to imaginary time, \(t=-i\tau\). This Wick rotation changes the spacetime metric from Lorentzian to Euclidean and is the technical device that makes both thermal field theory and lattice QCD well defined.

Substituting \(t\to -i\tau\) turns the Minkowski line element into a Euclidean one:

\[ ds^2 = dt^2 - d\vec x^{\,2}\ \longrightarrow\ -\big(d\tau^2 + d\vec x^{\,2}\big), \]

so the metric becomes \(\delta_{\mu\nu}=\mathrm{diag}(+,+,+,+)\) and all four directions are on equal footing. Correspondingly the weight in the path integral formulation rotates,

\[ e^{\,iS_M}\ \longrightarrow\ e^{-S_E}, \]

with \(S_E\) real and bounded below. The oscillatory Minkowski integrand becomes a convergent, positive Boltzmann-like weight — essential for numerical (Monte Carlo) evaluation.

• Convergence: \(e^{-S_E}\) is a genuine probability weight, so importance sampling is possible; \(e^{iS}\) is not.
• Thermal physics: the identity \(e^{-\beta H}\leftrightarrow\) evolution through Euclidean time \(\beta\) makes temperature equal to the inverse length of the compact \(\tau\) direction. The partition function is a Euclidean path integral with \(\tau\in[0,\beta]\).
• Lattice regularization: a Euclidean lattice is a finite, real, positive-weight statistical system that can be simulated directly.

On the thermal Euclidean circle of circumference \(\beta\), bosonic fields are periodic and fermionic fields anti-periodic:

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

This is the origin of the boundary conditions in the QCD partition function of Continuum QCD. Physical (Minkowski) results — spectral functions, transport — must ultimately be recovered by analytic continuation back from Euclidean data.