Symmetry breaking in non-commutative cut-off field theories

The relation between symmetry breaking in non-commutative cut-off field theories and transitions to inhomogeneous phases in condensed matter and in finite density QCD is discussed. The non-commutative dynamics, with its peculiar infrared-ultraviolet mixing, can be regarded as an effective description of the mechanisms which lead to inhomogeneous phase transitions and a roton-like excitation spectrum.