At finite temperature the euclidean time is compact, \(\tau\in[0,\beta]\) with \(\beta=1/T\), so every field is periodic (bosons) or anti-periodic (fermions) in \(\tau\) and can be Fourier-expanded in discrete Matsubara frequencies. These discrete modes are the technical origin of dimensional reduction.

\[ \phi(\vec x,\tau)=T\sum_{n=-\infty}^{\infty} e^{i\omega_n\tau}\,\tilde\phi_n(\vec x), \]

with the frequencies fixed by the (anti)periodic boundary conditions:

\[ \omega_n = 2\pi n T\quad(\text{[[boson|bosons]]}),\qquad \omega_n = (2n+1)\pi T\quad(\text{[[fermion|fermions]]}). \]

Bosons have a static \(n=0\) mode; fermions do not (their lowest frequency is \(\pi T\)).

In the 3D theory of the static modes, a Matsubara mode of frequency \(\omega_n\) appears as a field of mass \(|\omega_n|\). Thus every non-static mode carries a "thermal mass" \(\ge 2\pi T\):

• as \(T\to\infty\) these modes become heavy and decouple;
• only the static (\(n=0\)) bosonic modes remain light and dynamical.

Integrating out the heavy (\(n\neq0\)) modes is exactly the first matching step that produces EQCD. The surviving static gauge field then splits into the magnetic part \(A_i\) and the electric part \(A_0\), whose subsequent removal gives MQCD.

Thermodynamic quantities in the partition function are computed as sums over Matsubara frequencies,

\[ T\sum_n \int\!\frac{d^3k}{(2\pi)^3}\ f(\omega_n,\vec k), \]

which reduce to the usual loop integrals at \(T=0\) and encode all thermal effects at \(T>0\).