In mathematics, the Heisenberg group , 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 n-dimensional systems, and most generally, to any symplectic vector space.
Three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by
As one can see from the term , the group is non-abelian.
The neutral element of the Heisenberg group is the identity matrix, and inverses are given by
The group is a subgroup of the 2-dimensional affine group : the action of the element on the vector corresponds to the affine transform
There are several prominent examples of the three-dimensional case.
Continuous Heisenberg group If , are real numbers (in the ring R), then one has the continuous Heisenberg group H3(R).
It is a nilpotent real Lie group of dimension 3.
In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. 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. This representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta, or holomorphic, model, the Heisenberg group acts on a Hilbert space of entire functions, with the model depending on a parameter in the upper half-plane.
Discrete Heisenberg group
If are integers (in the ring Z), then one has the discrete Heisenberg group H3(Z). It is a non-abelian nilpotent group. It has two generators:
and relations
where
is the generator of the center of H3. (Note that the inverses of x, y, and z replace the 1 above the diagonal with −1.)
By Bass's theorem, it has a polynomial growth rate of order 4.
One can generate any element through
Heisenberg group modulo an odd prime p If one takes a, b, c in Z/p Z for an odd prime p, then one has the Heisenberg group modulo p. It is a group of order p3 with generators x, y and relations
Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groups, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group G is contained in the center Z of G, then the map G/Z × G/Z → Z is a skew-symmetric bilinear operator on abelian groups.
However, requiring that G/Z to be a finite vector space requires the Frattini subgroup of G to be contained in the center, and requiring that Z be a one-dimensional vector space over Z/p Z requires that Z have order p, so if G is not abelian, then G is extra special. If G is extra special but does not have exponent p, then the general construction below applied to the symplectic vector space G/Z does not yield a group isomorphic to G.
Heisenberg group modulo 2 The Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D4 (the symmetries of a square). Observe that if
then
and
The elements x and y correspond to reflections (with 45° between them), whereas xy and yx correspond to rotations by 90°. The other reflections are xyx and yxy, and rotation by 180° is xyxy (= yxyx).
Heisenberg algebra The Lie algebra of the Heisenberg group (over the real numbers) is known as the Heisenberg algebra. It may be represented using the space of 3×3 matrices of the form. Proposition 3.26.
with .
The following three elements form a basis for :
These basis elements satisfy the commutation relations
The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics:
where is the position operator, is the momentum operator, and is the Planck constant.
The Heisenberg group has the special property that the exponential map is a one-to-one and onto map from the Lie algebra to the group :. Chapter 2, Exercise 9.
In conformal field theory In conformal field theory, the term Heisenberg algebra is used to refer to an infinite-dimensional generalization of the above algebra. It is spanned by elements with commutation relations
Under a rescaling, this is simply a countably-infinite number of copies of the above algebra.
Higher dimensions More general Heisenberg groups may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension , for any integer . As a group of matrices, (or to indicate that this is the Heisenberg group over the field of real numbers) is defined as the group matrices with entries in and having the form
where a is a row vector of length n, b is a column vector of length n, In is the identity matrix of size n.
Group structure This is indeed a group, as is shown by the multiplication:
and
Lie algebra The Heisenberg group is a 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, 0n is the zero matrix of size n.
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.
In particular, z is a central element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent.
Exponential map Let
which fulfills . The exponential map evaluates to
The exponential map of any nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connected, simply-connected Lie group.
This discussion (aside from statements referring to dimension and Lie group) further applies if we replace R by any commutative ring A. The corresponding group is denoted Hn(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).
Representation theory
The unitary representation theory of the Heisenberg group is fairly simple later generalized by Mackey theory and was the motivation for its introduction in quantum physics, as discussed below.
For each nonzero real number , we can define an irreducible unitary representation of acting on the Hilbert space by the formula. Proposition 14.7.
This representation is known as the Schrödinger representation. The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics. The parameter describes translations in position space, the parameter describes translations in momentum space, and the parameter gives an overall phase factor. The phase factor is needed to obtain a group of operators, since translations in position space and translations in momentum space do not commute.
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 to for some .. Theorem 14.8. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2n.
Since the Heisenberg group is a one-dimensional central extension of , its irreducible unitary representations can be viewed as irreducible unitary projective representations of . Conceptually, the representation given above constitutes the quantum-mechanical counterpart to the group of translational symmetries on the classical phase space, . The fact that the quantum version is only a projective representation of is suggested already at the classical level. 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, because Rather, the span of the position and momentum functions and the constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra , isomorphic to the Lie algebra of the Heisenberg group.
Plancherel theorem
The Plancherel theorem for the Heisenberg group describes the decomposition of under the noncommutative Fourier transform. Unlike the Fourier transform on an abelian group, the group Fourier transform of a function on is generally operator-valued.
Let denote the Schrödinger representation of with nontrivial central character . For a sufficiently nice function , for example , its group Fourier transform is the operator
Here denotes Haar measure on , which in exponential coordinates is ordinary Lebesgue measure.
With a common normalization of Haar measure and of the Schrödinger representations, the Plancherel formula is
where is the Hilbert–Schmidt norm. Equivalently, the Fourier transform extends from to a unitary map from onto the direct integral of Hilbert–Schmidt operator spaces
up to the same normalizing constant.
For suitable functions, the corresponding Fourier inversion formula is
The one-dimensional unitary representations of , which factor through the abelian quotient , do not appear in the Plancherel formula. Thus the Plancherel measure of the Heisenberg group is concentrated on the infinite-dimensional Schrödinger representations with nonzero central character.
The Plancherel theorem describes how the representations classified by the Stone–von Neumann theorem occur in the regular representation of on .
On symplectic vector spaces The general abstraction of a Heisenberg group is constructed from any symplectic vector space.Hans Tilgner, "A class of solvable Lie groups and their relation to the canonical formalism ", Annales de l'institut Henri Poincaré (A) Physique théorique, 13 no. 2 (1970), pp. 103-127. For example, let (V, ω) be a finite-dimensional real symplectic vector space (so ω is a nondegenerate skew symmetric bilinear form on V). The Heisenberg group H(V) on (V, ω) (or simply V for brevity) is the set V×R endowed with the group law
The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence
Any symplectic vector space admits a Darboux basis {ej, fk}1 ≤ j,k ≤ n satisfying ω(ej, fk) = δjk and where 2n is the dimension of V (the dimension of V is necessarily even). In terms of this basis, every vector decomposes as
The qa and pa are canonically conjugate coordinates.
If {ej, fk}1 ≤ j,k ≤ n is a Darboux basis for V, then let {E} be a basis for R, and {ej, fk, E}1 ≤ j,k ≤ n is the corresponding basis for V×R. A vector in H(V) is then given by
and the group law becomes
Because the underlying manifold of the Heisenberg group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg group is given by the commutation relation
or written in terms of the Darboux basis
and all other commutators vanish.
It is also possible to define the group law in a different way but which yields a group isomorphic to the group we have just defined. To avoid confusion, we will use u instead of t, so a vector is given by
and the group law is
An element of the group
can then be expressed as a matrix ,