Range of forces and broken symmetries in many-body systems

It is argued that for a many-body system with short range forces the commutators between local operators at different times will be fast decreasing for large spatial separations.This allows the adaptation of many discussions in relativistic field theories to the case of a many-body system with short range forces. In particular one has the result that a spontaneous breakdown in symmetry implies the existence of excitations of arbitrarily small energy. However this result has essentially only one application: We know that the Galilei invariance is always broken (in a medium of finite density). Therefore one concludes that in a many-body system with short range forces there can never be an energy gap.Supported by the National Science Foundation.