One of the most important and celebrated mathematical results in quantum mechanics is the Stone-von Neumann Theorem. This theorem, in its modern formulation that is due to George Mackey and Marc Rieffel, says: If is a locally compact Hausdorff abelian group, and and are strongly-continuous unitary representations of and respectively on a Hilbert space that satisfy the Weyl Commutation Relation, i.e., for all and , then the triple must be unitarily equivalent to a direct sum of copies of the triple , where is the unitary representation of on by left translations, and is the unitary representation of on by phase modulations.
In this talk, we will present a new generalization of the Stone-von Neumann Theorem that, for any Hilbert space and any -dynamical system of the form , classifies up to unitary equivalence all quadruples where is a Hilbert -module, is a non-degenerate -representation of on by adjointable operators, and and are strongly-continuous unitary representations of and respectively on that satisfy the Weyl Commutation Relation as well as the following two commutation relations: (i) for all and , and (ii) for all and .
We will also present an infinitesimal (or unbounded) version of our result, which involves the Heisenberg Commutation Relation. We will conclude by elucidating a deep connection between our result and Takai-Takesaki Duality.
This is joint work with Lara Ismert of the University of Nebraska-Lincoln.
The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on C*-Algebras of Compact Operators Sponsored by the Meyer Fund