Heisenberg group

, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication.The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem.In the three-dimensional case, the product of two Heisenberg matrices is given by As one can see from the term ab′, the group is non-abelian.By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character.In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions.with commutation relations Under a rescaling, this is simply a countably-infinite number of copies of the above algebra.The simplest general case is the real Heisenberg group of dimensionand having the form where This is indeed a group, as is shown by the multiplication: and The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices where By letting e1, ..., en be the canonical basis of Rn and setting the associated Lie algebra can be characterized by the canonical commutation relations where p1, ..., pn, q1, ..., qn, z are the algebra generators.This discussion (aside from statements referring to dimension and Lie group) further applies if we replace R by any commutative ring A.Under the additional assumption that the prime 2 is invertible in the ring A, the exponential map is also defined, since it reduces to a finite sum and has the form above (e.g. A could be a ring Z/p Z with an odd prime p or any field of characteristic 0).The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics.The key result is the Stone–von Neumann theorem, which states that every (strongly continuous) irreducible unitary representation of the Heisenberg group in which the center acts nontrivially is equivalent toConceptually, the representation given above constitutes the quantum-mechanical counterpart to the group of translational symmetries on the classical phase space,The Hamiltonian generators of translations in phase space are the position and momentum functions.The span of these functions does not form a Lie algebra under the Poisson bracket, however, becauseRather, the span of the position and momentum functions and the constants forms a Lie algebra under the Poisson bracket.The general abstraction of a Heisenberg group is constructed from any symplectic vector space.In terms of this basis, every vector decomposes as The qa and pa are canonically conjugate coordinates.By the Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials where the exponents are all non-negative.Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with given action of the central Lie algebra element z, up to a unitary equivalence: the nontrivial elements of the algebra are all equivalent to the usual position and momentum operators.The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of equations defining abelian varieties.This is a large generalization of the approach used in Jacobi's elliptic functions, which is the case of the modulo 2 Heisenberg group, of order 8.The Heisenberg group also occurs in Fourier analysis, where it is used in some formulations of the Stone–von Neumann theorem.An orthonormal frame on the manifold is given by the Lie vector fields which obey the relations [X, Y] = Z and [X, Z] = [Y, Z] = 0.Being Lie vector fields, these form a left-invariant basis for the group action.That is, the height of the curve is proportional to the area of the circle subtended by the circular arc, which follows by Green's theorem.-valued characters on K, which is also a locally compact abelian group if endowed with the compact-open topology.These operators do not commute, and instead satisfy multiplication by a fixed unit modulus complex number.[10] A version of the Stone–von Neumann theorem, proved by George Mackey, holds for the Heisenberg groupSee the discussion at Stone–von Neumann theorem#Relation to the Fourier transform for details.