This unitary map is the Segal–Bargmann transform. Representations of finite Heisenberg groups The Heisenberg group is defined for any commutative ring . In this section let us specialize to the field for a prime. This field has the property that there is an embedding of as an additive group into the circle group . Note that is finite with cardinality . For finite Heisenberg group one can give a simple proof of the Stone–von Neumann theorem using simple properties of character functions of representations. These properties follow from the orthogonality relations for characters of representations of finite groups. For any non-zero in define the representation on the finite-dimensional inner product space by It follows that By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups , particularly: Irreducibility of Pairwise inequivalence of all the representations . Actually, all irreducible representations of on which the center acts nontrivially arise in this way. Generalizations The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups. See also Oscillator representation Wigner–Weyl transform CCR and CAR algebras (for bosons and fermions respectively) Segal–Bargmann space Moyal product Weyl algebra Stone's theorem on one-parameter unitary groups Hille–Yosida theorem C0-semigroup Notes References Rosenberg, Jonathan (2004) "A Selective History of the Stone–von Neumann Theorem" Contemporary Mathematics 365. American Mathematical Society. Summers, Stephen J. (2001). "On the Stone–von Neumann Uniqueness Theorem and Its Ramifications." In John von Neumann and the foundations of quantum physics, pp. 135-152. Springer, Dordrecht, 2001, online. Functional analysis Mathematical quantization Theorems in functional analysis Theorems in mathematical physics John von Neumann