Heisenberg group
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 onedimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ndimensional systems, and most generally, to any symplectic vector space.
The threedimensional case
In the threedimensional case, the product of two Heisenberg matrices is given by:
As one can see, the group is nonabelian.
The neutral element of the Heisenberg group is the identity matrix, and inverses are given by
The group is a subgroup of the 2dimensional affine group Aff(2): acting on corresponds to the affine transform .
There are several prominent examples of the threedimensional case.
Continuous Heisenberg group
If a, b, c, are real numbers (in the ring R) then one has the continuous Heisenberg group H_{3}(R).
It is a nilpotent real Lie group of dimension 3.
In addition to the representation as real 3x3 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 representation, it acts on the space of holomorphic functions on the upper halfplane; it is so named for its connection with the theta functions.
Discrete Heisenberg group
If a, b, c, are integers (in the ring Z) then one has the discrete Heisenberg group H_{3}(Z). It is a nonabelian nilpotent group. It has two generators,
and relations
 ,
where
is the generator of the center of H_{3}. (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 p^{3} 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 from G/Z × G/Z → Z is a skewsymmetric 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 onedimensional 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 D_{4} (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).
Lie algebra
The Lie algebra of the Heisenberg group (over the real numbers) is the space of matrices of the form[1]
 ,
with . The following three elements form a basis for :
 .
The 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 Planck's constant.
The Heisenberg group has the special property that the exponential map is a onetoone and onto map from the Lie algebra to the group .[2]
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 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,
 I_{n} 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 simplyconnected 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,
 0_{n} is the zero matrix of size n.
By letting e_{1}, ..., e_{n} be the canonical basis of R^{n}, and setting
the associated Lie algebra can be characterized by the canonical commutation relations,

(1)
where p_{1}, ..., p_{n}, q_{1}, ..., q_{n}, 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, simplyconnected 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 H_{n}(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 (i.e. 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:[3]
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 .[4] 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 onedimensional 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 do 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 onedimensional central extension of the commutative Lie algebra , isomorphic to the Lie algebra of the Heisenberg group.
On symplectic vector spaces
The general abstraction of a Heisenberg group is constructed from any symplectic vector space.[5] For example, let (V,ω) be a finitedimensional 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 {e_{j},f^{k}}_{1 ≤ j,k ≤ n} satisfying ω(e_{j},f^{k}) = δ_{j}^{k} 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 q^{a} and p_{a} are canonically conjugate coordinates.
If {e_{j}, f^{k}}_{1 ≤ j,k ≤ n} is a Darboux basis for V, then let {E} be a basis for R, and {e_{j}, f^{k}, 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
 ,
which gives a faithful matrix representation of H(V). The u in this formulation is related to t in our previous formulation by , so that the t value for the product comes to
 ,
as before.
The isomorphism to the group using upper triangular matrices relies on the decomposition of V into a Darboux basis, which amounts to a choice of isomorphism V ≅ U ⊕ U*. Although the new group law yields a group isomorphic to the one given higher up, the group with this law is sometimes referred to as the polarized Heisenberg group as a reminder that this group law relies on a choice of basis (a choice of a Lagrangian subspace of V is a polarization).
To any Lie algebra, there is a unique connected, simply connected Lie group G. All other connected Lie groups with the same Lie algebra as G are of the form G/N where N is a central discrete group in G. In this case, the center of H(V) is R and the only discrete subgroups are isomorphic to Z. Thus H(V)/Z is another Lie group which shares this Lie algebra. Of note about this Lie group is that it admits no faithful finitedimensional representations; it is not isomorphic to any matrix group. It does however have a wellknown family of infinitedimensional unitary representations.
The connection with the Weyl algebra
The Lie algebra of the Heisenberg group was described above, (1), as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theorem applies to determine the universal enveloping algebra . Among other properties, the universal enveloping algebra is an associative algebra into which injectively imbeds.
By the Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials
where the exponents are all nonnegative.
Consequently, consists of real polynomials
with the commutation relations
The algebra is closely related to the algebra of differential operators on ℝ^{n} 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 is also easy to see directly from the above representations; viz. by the mapping
Applications
Weyl's parameterization of quantum mechanics
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 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.
Thus, the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this essentially unique representation.
Theta representation
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 simplest case is the theta representation of the Heisenberg group, of which the discrete case gives the theta function.
Fourier analysis
The Heisenberg group also occurs in Fourier analysis, where it is used in some formulations of the Stone–von Neumann theorem. In this case, the Heisenberg group can be understood to act on the space of square integrable functions; the result is a representation of the Heisenberg groups sometimes called the Weyl representation.
As a subRiemannian manifold
The threedimensional Heisenberg group H_{3}(R) on the reals can also be understood to be a smooth manifold, and specifically, a simple example of a subRiemannian manifold.[6] Given a point p=(x,y,z) in R^{3}, define a differential 1form Θ at this point as
This oneform belongs to the cotangent bundle of R^{3}; that is,
is a map on the tangent bundle. Let
It can be seen that H is a subbundle of the tangent bundle TR^{3}. A cometric on H is given by projecting vectors to the twodimensional space spanned by vectors in the x and y direction. That is, given vectors and in TR^{3}, the inner product is given by
The resulting structure turns H into the manifold of the Heisenberg group. 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 leftinvariant basis for the group action. The geodesics on the manifold are spirals, projecting down to circles in two dimensions. That is, if
is a geodesic curve, then the curve is an arc of a circle, and
with the integral limited to the twodimensional plane. That is, the height of the curve is proportional to the area of the circle subtended by the circular arc, which follows by Stokes' theorem.
Heisenberg group of a locally compact abelian group
It is more generally possible to define the Heisenberg group of a locally compact abelian group K, equipped with a Haar measure.[7] Such a group has a Pontrjagin dual , consisting of all continuous valued characters on K, which is also a locally compact abelian group if endowed with the compactopen topology. The Heisenberg group associated with the locally compact abelian group K is the subgroup of the unitary group of generated by translations from K and multiplications by elements of .
In more detail, the Hilbert space consists of squareintegrable complexvalued functions on K. The translations in K form a unitary representation of K as operators on :
for . So too do the multiplications by characters:
for . These operators do not commute, and instead satisfy
multiplication by a fixed unit modulus complex number.
So the Heisenberg group associated with K is a type of central extension of , via an exact sequence of groups:
More general Heisenberg groups are described by 2cocyles in the cohomology group . The existence of a duality between and gives rise to a canonical cocycle, but there are generally others.
The Heisenberg group acts irreducibly on . Indeed, the continuous characters separate points[8] so any unitary operator of that commutes with them is an multiplier. But commuting with translations implies that the multiplier is constant.[9]
A version of the Stone–von Neumann theorem, proved by George Mackey, holds for the Heisenberg group .[10][11] The Fourier transform is the unique intertwiner between the representations of and . See the discussion at Stone–von Neumann theorem#Relation to the Fourier transform for details.
See also
Notes
 Hall 2015 Proposition 3.26
 Hall 2015 Chapter 2, Exercise 9
 Hall 2013 Proposition 14.7
 Hall 2013 Theorem 14.8
 Hans Tilgner, "A class of solvable Lie groups and their relation to the canonical formalism Archived 20110605 at the Wayback Machine", Annales de l'institut Henri Poincaré (A) Physique théorique, 13 no. 2 (1970), pp. 103127.
 Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0821813919.
 David Mumford (1991), "Tata lectures on theta III", Progress in Mathematics, Birkhauser, 97
 Karl Heinrich Hofmann, Sidney A. Morris (2006), The structure of compact groups: a primer for students, a handbook for the expert, De Gruyter studies in mathematics 25, (2nd rev. ed, ed.), Walter de Gruyter, ISBN 9783110190069CS1 maint: extra punctuation (link)
 This argument appears in a slightly different setting in Roger Howe (1980), "On the role of the Heisenberg group in harmonic analysis", Bulletin of the American Mathematical Society, 3 (2): 821–844, doi:10.1090/S027309791980148259, MR 0578375
 George Mackey (1949), "On a theorem of Stone and von Neumann", Duke Mathematical Journal, 16 (2): 313–326, doi:10.1215/s0012709449016312
 A Prasad (2009), An easy proof of the Stone–von Neumann–Mackey theorem, arXiv:0912.0574, doi:10.1016/j.exmath.2010.06.001
References
 Binz, Ernst; Pods, Sonja (2008). Geometry of Heisenberg Groups. American Mathematical Society. ISBN 9780821844953.
 Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 9781461471158
 Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. 222 (second ed.). Springer. ISBN 9783319134666.
 Howe, Roger (1980). "On the role of the Heisenberg group in harmonic analysis". Bulletin of the American Mathematical Society. 3 (2): 821–843. doi:10.1090/s027309791980148259. MR 0578375.
 Kirillov, Alexandre A. (2004). "Ch. 2: "Representations and Orbits of the Heisenberg Group". Lectures on the Orbit Method. American Mathematical Society. ISBN 0821835300.
 Mackey, George (1976). The theory of Unitary Group Representations. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 9780226500522.
External links
 Groupprops, The Group Properties Wiki Unitriangular matrix group UT(3,p)