Heisenberg group Heisenberg group In mathematics, the Heisenberg group formula_1, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements "a, b" and "c" can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to
Heisenberg group of differential operators on ℝ with polynomial coefficients, since any such operator has a unique representation in the form This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra "W" is a quotient of formula_79. However, this is also easy to see directly from the above representations; viz. by the mapping The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with