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In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann.

The above canonical commutation relations for P, Q are identical to the commutation relations that specify the Lie algebra of the general Heisenberg group H2n+1 for n a positive integer. This is the Lie group of (n + 2) × (n + 2) square matrices of the form

In fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory. Note that the center of H2n+1 consists of matrices M(0, 0, c). However, this center is not the identity operator in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e. g. for , are

and the central generator 1=z = log M(0, 0, 1) = exp(z) − 1 is not the identity. All these representations are unitarily inequivalent; and any irreducible representation which is not trivial on the center of Hn is unitarily equivalent to exactly one of these. Note that Uh is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by ha and multiplication by a function of absolute value 1. To show Uh is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness; this claim, nevertheless, follows easily from the Stone–von Neumann theorem as stated above. We will sketch below a proof of the corresponding Stone–von Neumann theorem for certain finite Heisenberg groups. In particular, irreducible representations π, π′ of the Heisenberg group Hn which are non trivial on the center of Hn are unitarily equivalent if and only if 1=π(z) = π′(z) for any z in the center of Hn. One representation of the Heisenberg group which is important in number theory and the theory of modular forms is the theta representation, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group.

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.