The effort to put physical theories on a mathematically rigorous footing not only developed physics but also has influenced developments of some mathematical areas. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum mechanics, quantum field theory, and quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous mathematical formulation of quantum field theory has also brought about some progress in fields such as representation theory.