Stone–von Neumann theorem

In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. The name is for Marshall Stone and John von Neumann (1931).

In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces.

For a single particle moving on the real line R, there are two important observables: position and momentum. In the quantum-mechanical description of such a particle, the position operator x and momentum operator p are respectively given by

on the domain V of infinitely differentiable functions of compact support on R. Assume ℏ to be a fixed non-zero real number — in quantum theory ℏ is (up to a factor of 2π) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of (action) units of the macroscopic world.

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