Dimensional reduction is the statement that a quantum field theory at high temperature behaves, for its long-distance (static) physics, like a lower-dimensional theory: a 4D theory at temperature \(T\) reduces to an effective 3D theory. It is the organizing principle behind EQCD, MQCD and the prediction of the spatial string tension.
At finite \(T\) the euclidean time is a circle of circumference \(\beta=1/T\). Fields are expanded in Matsubara modes with frequencies \(\omega_n=2\pi nT\) (bosons) or \((2n+1)\pi T\) (fermions). Each non-static mode behaves in the remaining three dimensions as a field of mass \(|\omega_n|\ge 2\pi T\). As \(T\to\infty\) these become infinitely heavy and decouple; only the static (\(n=0\)) bosonic modes survive. The dynamics along \(\tau\) is frozen — one spatial dimension has effectively "disappeared."
The reduction is a controlled effective field theory expansion because the relevant scales separate at weak coupling (asymptotic freedom):
- Hard modes \(\sim 2\pi T\): all non-static modes and fermions \(\to\) integrated out to give EQCD.
- Soft electric scale \(\sim gT\): the Debye mass of \(A_0\) \(\to\) integrated out to give MQCD.
- Ultrasoft magnetic scale \(\sim g^2 T\): the confining 3D gauge dynamics that sets \(\sigma_s\).
- It reorganizes hard thermal problems into a simpler 3D theory with a few matched couplings (\(g_E^2, m_E^2, \lambda_E, g_M^2\)).
- It isolates the non-perturbative magnetic sector (Linde problem) into a well-defined 3D lattice computation.
- It gives quantitative, testable predictions — most cleanly for the spatial string tension, as demonstrated by Cheng et al. See Why spatial string tension and dimensional reduction.