In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form

Elements a,b,c can be taken from some (arbitrary) commutative ring.

Examples

(i) If a,b,c are real numbers (in the ring R) then we get the continuous Heisenberg group. It is a nilpotent Lie group.

(ii) If a,b,c are integers (in the ring Z) then we get the discrete Heisenberg group H₃. It is a non-abelian nilpotent group. It has two generators

and relations

,

where z is the generator of the center of H₃. By Bass' theorem, it has a polynomial growth rate of order 4.

(iii) If one takes a,b,c in Z/p Z, then we get the Heisenberg group modulo p. It is a group of order p³ with two generators, x, y and relations

.

General Heisenberg group

There are more general Heisenberg groups H_n. We begin by discussing the Real Heisenberg group of dimension 2n+1, for any integer n ≥ 1. As a group of matrices, H_n (or H_n(R) to indicate this is the Heisenberg group over the ring R) is defined as the group of square matrices of size n+2 with entries in R:

where a is a row vector of length n, b is a column vector of length n and 1_n is the identity matrix of size n. This is indeed a group, as is shown by the multiplication:

and

The Heisenberg group is a connected, simply connected Lie group whose Lie algebra consists of matrices

where a is a row vector of length n, b is a column vector of length n and 0_n is the zero matrix of size n. The exponential map is given by the following expression

By choosing a basis e₁, ... , e_n of Rⁿ , and letting

the Lie algebra can also be characterized by the canonical commutation relations

where p₁, .., p_n, q₁, .., q_n, z are generators. In particular, z is a central element of the Heisenberg Lie algebra.

This group occurs not only in quantum mechanics but in the theory of theta functions; it is also used in Fourier analysis. This group is also used in some formulations of the Stone-von Neumann theorem.

The above discussion (aside from statements referring to dimension and Lie group) applies if we replace R by any commutative ring A. The corresponding group is denoted H_n(A). Moreover the exponential map is also defined, since it reduces to a finite sum and has the form above.

The connection with the Weyl algebra

The Lie algebra of the Heisenberg group was described above as a Lie algebra of matrices. We now apply the Poincaré-Birkhoff-Witt theorem, to determine the universal enveloping algebra . Among other properties, the universal enveloping algebra is an associative algebra into which injectively imbeds. By Poincaré-Birkhoff-Witt, it is the free vector space generated by the monomials

where the exponents are all non-negative. Thus consists of real polynomials

with the commutation relations

is closely related to the algebra of differential operators on Rⁿ 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_n is a quotient of . However, this also easy to see directly from the above representations; viz, by the mapping

Weyl's view of quantum mechanics

The application that led Hermann Weyl to an explicit introduction of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly there is a good explanation: the group H_n is a central extension of Rⁿ by a copy of R, and as such is a semidirect product. Its representation theory is relatively simple (a special case of the later Mackey theory), and in particular there is a uniqueness result for unitary representations with given action of the central element z (in the Lie algebra) or the one-parameter subgroup it creates under the exponential map, which is the central extension. This abstract uniqueness accounts for the equivalence of the two physical pictures.

The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of 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.