Title: The Stone-von Neumann Theorem and Generalizations

Abstract: One of the most famous mathematical results in quantum mechanics is the Stone-von Neumann Theorem. Informally, the theorem establishes the physical equivalence of Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics, which was seen by Heisenberg to be an outstanding problem in the early days of quantum mechanics. The Theorem was first given a rigorous formulation by Marshall Stone in 1930, and it was this formulation that John von Neumann proved in 1931. The exponentiated form of the Heisenberg Commutation Relation, called the Weyl Commutation Relation, is investigated in these papers, because it involves only one-parameter unitary groups, rather than the self-adjoint unbounded operators these unitary groups generate.

Several generalizations of the Stone-von Neumann Theorem are non-trivial and interesting; however, their use of only Hilbert-space representations is a common limiting feature. In this talk, we provide not another incremental generalization of the Stone-von Neumann Theorem, but a complete paradigm shift that significantly augments the theorem’s range of applicability. By considering the Stone-von Neumann theorem on Hilbert C*-modules, we show that the Stone-von Neumann Theorem is not just about representations of locally compact Hausdorff abelian groups on Hilbert spaces, but is really about representations of C*-dynamical systems on Hilbert C*-modules.