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Heisenberg group
In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
\( \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix} \)
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. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.
The three-dimensional case
In the three-dimensional case, the product of two Heisenberg matrices is given by:
\( \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix} \begin{pmatrix} 1 & a' & c'\\ 0 & 1 & b'\\ 0 & 0 & 1\\ \end{pmatrix}= \begin{pmatrix} 1 & a+a' & c+c'+ab'\\ 0 & 1 & b+b'\\ 0 & 0 & 1\\ \end{pmatrix}\, .\)
The neutral element of the Heisenberg group is the identity matrix, and inverses are given by
\( \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}^{-1}= \begin{pmatrix} 1 & -a & ab-c\\ 0 & 1 & -b\\ 0 & 0 & 1\\ \end{pmatrix}\, .\)
There are several prominent examples of the three-dimensional case.
Continuous Heisenberg group
If a, b, c, 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 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 half-plane; it is so named for its connection with the theta functions.
Discrete Heisenberg group
A portion of the Cayley graph of the discrete Heisenberg group, with generators x,y,z as in the text. (The coloring is only for visual aid.)
If a, b, c, 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,
\( x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{pmatrix}\)
and relations
\( z^{}_{}=xyx^{-1}y^{-1},\ xz=zx,\ yz=zy ,\)
where
\( z=\begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}\)
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
\( \begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}=y^bz^cx^a\, .\)
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:
\( z^{}_{}=xyx^{-1}y^{-1},\ x^p=y^p=z^p=1,\ xz=zx,\ yz=zy. \)
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 skew-symmetric bilinear operator on abelian groups. However, requiring that G/Z 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
\( x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{pmatrix}.\)
Then
\( xy=\begin{pmatrix} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{pmatrix}, \)
and
\( yx=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{pmatrix}. \)
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).
Higher dimensions
More general Heisenberg groups Hn 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 2n+1, for any integer n ≥ 1. As a group of matrices, Hn (or Hn(R) to indicate this is the Heisenberg group over the ring R or real numbers) is defined as the group of square matrices of size n+2 with entries in R:
\( \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \)
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:
\( \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}1 & \mathbf a' & c' \\ 0 & I_n & \mathbf b' \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & \mathbf a+ \mathbf a' & c+c' +\mathbf a \cdot \mathbf b' \\ 0 & I_n & \mathbf b+\mathbf b' \\ 0 & 0 & 1 \end{bmatrix} \)
and
\( \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}1 & -\mathbf a & -c +\mathbf a \cdot \mathbf b\\ 0 & I_n & -\mathbf b \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & I_n & 0 \\ 0 & 0 & 1 \end{bmatrix}. \)
The Heisenberg group is a connected, simply-connected Lie group whose Lie algebra consists of matrices
\( \begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix}, \)
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.
Exponential Map
The exponential map is given by the following expression
\( \exp \begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix} = \sum_{k=0}^\infty \frac{1}{k!}\begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix}^k = \begin{bmatrix} 1 & \mathbf a & c + {1\over 2}\mathbf a \cdot \mathbf b\\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix}. \)
By letting e1, ..., en be the canonical basis of Rn, and setting
\( p_i = \begin{bmatrix} 0 & \operatorname{e}_i^{\mathrm{T}} & 0 \\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix},
q_j = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0_n & \operatorname{e}_j \\ 0 & 0 & 0 \end{bmatrix},
z = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix}, \)
the associated Lie algebra can be characterized by the canonical commutation relations,
\( \begin{bmatrix} p_i, q_j \end{bmatrix} = \delta_{ij}z, \qquad \begin{bmatrix} p_i, z \end{bmatrix} = 0, \qquad \begin{bmatrix} q_j, z \end{bmatrix} = 0~, \) (1)
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. The exponential map of a 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 (i.e. A could be a ring Z/p Z with an odd prime p or any field of characteristic 0).
On symplectic vector spaces
The general abstraction of a Heisenberg group is constructed from any symplectic vector space.[1] 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
\( (v,t)\cdot(v',t') =\left (v+v',t+t'+\tfrac{1}{2}\omega(v,v')\right).\)
The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence
\( 0\to\mathbf{R}\to H(V)\to V\to 0.\)
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
\( v=q^a\mathbf{e}_a+p_a\mathbf{f}^a.\)
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
\( v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+tE\)
and the group law becomes
\((p,q,t)\cdot(p',q',t') =\left (p+p',q+q',t+t'+\frac{1}{2}(p q'-p' q)\right).\)
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
\( \begin{bmatrix} (v_1,t_1),(v_2,t_2) \end{bmatrix} =\omega(v_1,v_2)\)
or written in terms of the Darboux basis
\( [\mathbf{e}_a,\mathbf{f}^b]=\delta_a^b\)
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
\( v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+uE\)
and the group law is
\( (p,q,u)\cdot(p',q',u')=(p+p',q+q',u+u'+p q').\)
An element of the group
\( v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+uE\)
can then be expressed as a matrix
\( \begin{bmatrix} 1 & p & u\\ 0 & I_n & q\\ 0 & 0 & 1 \end{bmatrix} ,\)
which gives a faithful matrix representation of H(V). The u in this formulation is related to t in our previous formulation by \( u=t+\tfrac{1}{2}pq, \)so that the t value for the product comes to
\( u+u'+p q'-\tfrac{1}{2}(p+p')(q+q')
=t+\tfrac{1}{2}p q+t'+\tfrac{1}{2}p' q'+p q'-\tfrac{1}{2}(p+p')(q+q')
=t+t'+\tfrac{1}{2}(p q'-p' q) ,\)
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 finite-dimensional representations; it is not isomorphic to any matrix group. It does however have a well-known family of infinite-dimensional unitary representations.
The connection with the Weyl algebra
Main article: Weyl algebra
The Lie algebra \mathfrak{h}_n 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 \(U(\mathfrak{h}_n). Among other properties, the universal enveloping algebra is an associative algebra into which \( \mathfrak{h}_n injectively imbeds.
By the Poincaré–Birkhoff–Witt thorem, it is thus the free vector space generated by the monomials
\( z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n} ~,\)
where the exponents are all non-negative.
Consequently, U(\mathfrak{h}_n) consists of real polynomials
\( \sum_{j, \vec{k}, \vec{\ell}} c_{j \vec{k} \vec{\ell}} \,\, z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n} ~,\)
with the commutation relations
\( p_k p_\ell = p_\ell p_k, \quad q_k q_\ell = q_\ell q_k, \quad p_k q_\ell - q_\ell p_k = \delta_{k \ell} z, \quad z p_k - p_k z =0, \quad z q_k - q_k z =0~.\)
The algebra \( U(\mathfrak{h}_n) \) 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
\( P=\sum_{\vec{k}, \vec{\ell}} c_{\vec{k} \vec{\ell}} \,\, \partial_{x_1}^{k_1} \partial_{x_2}^{k_2} \cdots \partial_{x_n}^{k_n} x_1^{\ell_1} x_2^{\ell_2} \cdots x_n^{\ell_n} ~.\)
This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra Wn is a quotient of \( U(\mathfrak{h}_n). However, this is also easy to see directly from the above representations; viz. by the mapping
\( z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n} \, \mapsto \, \partial_{x_1}^{k_1} \partial_{x_2}^{k_2} \cdots \partial_{x_n}^{k_n} x_1^{\ell_1} x_2^{\ell_2} \cdots x_n^{\ell_n}~.\)
Representation theory
Main article: Stone–von Neumann theorem
The 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.
The key result is the Stone–von Neumann theorem, which, informally stated, says that (with certain technical assumptions) every representation of the Heisenberg group H2n+1 is equivalent to the position operators and momentum operators on Rn. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2n.
More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation.
Further, as the Heisenberg group is a semidirect product, its representation theory can be studied in terms of ergodic theory, via ergodic actions of the group, as in the work of George Mackey.
Applications
Weyl's parameterization of quantum mechanics
Main article: Wigner–Weyl transform
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
Main article: 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 sub-Riemannian manifold
The three-dimensional Heisenberg group H3(R) on the reals can also be understood to be a smooth manifold, and specifically, a simple example of a sub-Riemannian manifold.[2] Given a point p=(x,y,z) in R3, define a differential 1-form Θ at this point as
\( \Theta_p=dz -\frac{1}{2}\left(xdy - ydx\right).\)
This one-form belongs to the cotangent bundle of R3; that is,
\( \Theta_p:T_p\mathbf{R}^3\to\mathbf{R}\)
is a map on the tangent bundle. Let
H_p = \{ v\in T_p\mathbf{R}^3 \mid \Theta_p(v) = 0 \}.\)
It can be seen that H is a subbundle of the tangent bundle TR3. A cometric on H is given by projecting vectors to the two-dimensional space spanned by vectors in the x and y direction. That is, given vectors \(v=(v_1,v_2,v_3) and w=(w_1,w_2,w_3) in TR3, the inner product is given by
\( \langle v,w\rangle = v_1w_1+v_2w_2.\)
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
\( X=\frac{\partial}{\partial x} - \frac{1}{2} y\frac{\partial}{\partial z},\)
\( Y=\frac{\partial}{\partial y} + \frac{1}{2} x\frac{\partial}{\partial z},\)
\( Z=\frac{\partial}{\partial z},\)
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. The geodesics on the manifold are spirals, projecting down to circles in two dimensions. That is, if
\( \gamma(t)=(x(t),y(t),z(t))\)
is a geodesic curve, then the curve c(t)=(x(t),y(t)) is an arc of a circle, and
\( z(t)=\frac{1}{2}\int_c xdy-ydx\)
with the integral limited to the two-dimensional 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.
See also
Wigner–Weyl transform
Stone–von Neumann theorem
Notes
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.
Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9.
References
Binz, Ernst; Pods, Sonja (2008). Geometry of Heisenberg Groups. American Mathematical Society. ISBN 978-0-8218-4495-3.
Hall, Brian C. (2004). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Berlin: Springer. ISBN 0-387-40122-9.
Howe, Roger (1980). "On the role of the Heisenberg group in harmonic analysis". Bull. Amer. Math. Soc. 3 (2): 821–843. doi:10.1090/s0273-0979-1980-14825-9. MR 578375.
Kirilov, A.A. (2004). "Ch. 2: "Representations and Orbits of the Heisenberg Group". Lectures on the Orbit Method. American Mathematical Society. ISBN 0-8218-3530-0.
Mackey, G. (1976). The theory of Unitary Group Representations. Chicago Lectures in Mathematics. University Of Chicago Press. ISBN 978-0226500522.
External links
Groupprops, The Group Properties Wiki Unitriangular matrix group UT(3,p)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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