The Weyl Algebra and the Canonical Commutation Relation

Abstract

This chapter gives an extensive treatment of Hilbert space representations of the canonical commutation relation and the Weyl algebra. After collecting algebraic properties of this algebra we treat the Bargmann–Fock representation and the corresponding uniqueness theorem. Then the Schrodinger representation is studied and the Stone–von Neumann uniqueness theorem is proved. The Segal–Bargmann transform establishes the unitary equivalence of both representations. Kato’s theorem on the characterization of Schrodinger pairs in terms of resolvents is derived. Further, the Heisenberg uncertainty principle and the Groenewold–van Hove “no-go theorem for quantization are developed.