Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero.
Dimensional reduction also refers to a specific cancellation of divergences in Feynman diagrams.
It was put forward by Amnon Aharony, Yoseph Imry, and Shang-keng Ma who proved in 1976 that "to all orders in perturbation expansion, the critical exponents in a d-dimensional (4 < d < 6) system with short-range exchange and a random quenched field are the same as those of a (d − 2)-dimensional pure system".
[2] Their arguments indicated that the "Feynman diagrams which give the leading singular behavior for the random case are identically equal, apart from combinatorial factors, to the corresponding Feynman diagrams for the pure case in two fewer dimensions.
"[3] This dimensional reduction was investigated further in the context of supersymmetric theory of Langevin stochastic differential equations by Giorgio Parisi and Nicolas Sourlas [4] who "observed that the most infrared divergent diagrams are those with the maximum number of random source insertions, and, if the other diagrams are neglected, one is left with a diagrammatic expansion for a classical field theory in the presence of random sources ... Parisi and Sourlas explained this dimensional reduction by a hidden supersymmetry.