Heisenberg group

In, the Heisenberg group $$H$$, named after , is the of 3&times;3  of the form


 * $$\begin{pmatrix}

1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}$$

under the operation of. Elements a, b and c can be taken from any with identity, often taken to be the ring of s (resulting in the "continuous Heisenberg group") or the ring of s (resulting in the "discrete Heisenberg group").

The continuous Heisenberg group arises in the description of one-dimensional systems, especially in the context of the. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any.

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}\, .$$ As one can see, the group is.

The neutral element of the Heisenberg group is the, 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}\, .$$

The group is a subgroup of the 2-dimensional affine group Aff(2): $$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{pmatrix}$$ acting on $$(\vec{x},1)$$ corresponds to the affine transform $$\begin{pmatrix} 1 & a\\ 0 & 1 \end{pmatrix}{\vec x}+\begin{pmatrix} c\\ b \end{pmatrix}$$.

There are several prominent examples of the three-dimensional case.

Continuous Heisenberg group
If $a, b, c$, are s (in the ring R) then one has the continuous Heisenberg group H3(R).

It is a real  of dimension 3.

In addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different in terms of s. By, there is, up to isomorphism, a unique irreducible unitary representation of H in which its  acts by a given nontrivial. This representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of functions. In the, it acts on the space of s on the ; it is so named for its connection with the s.

Discrete Heisenberg group
If $a, b, c$, are integers (in the ring Z) then one has the discrete Heisenberg group H3(Z). It is a. 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 of H3. (Note that the inverses of x, y, and z replace the 1 above the diagonal with −1.)

By, it has a polynomial 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 for an odd  p, then one has the Heisenberg group modulo p. It is a group of 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 s, or more properly, extra special groups of p. More generally, if the 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 to be a finite requires the  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 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).

Lie algebra
The Lie algebra $$\mathfrak h$$ of the Heisenberg group $$H$$ (over the real numbers) is the space of $$3\times 3$$ matrices of the form
 * $$\begin{pmatrix}

0 & a & b\\ 0 & 0 & c\\ 0 & 0 & 0\\ \end{pmatrix}$$, with $$a,b,c\in\mathbb R$$. The following three elements form a basis for $$\mathfrak h$$:
 * $$X=\begin{pmatrix}

0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{pmatrix};\quad Y=\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\\ \end{pmatrix};\quad Z=\begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{pmatrix} $$. The basis elements satisfy the commutation relations:
 * $$[X,Y]=Z;\quad [X,Z]=0;\quad [Y,Z]=0$$.

The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the s in quantum mechanics:
 * $$[\hat x,\hat p]=i\hbar I;\quad [\hat x,i\hbar I]=0;\quad [\hat p,i\hbar I]=0$$,

where $$\hat x$$ is the position operator, $$\hat p$$ is the momentum operator, and $$\hbar$$ is Planck's constant.

The Heisenberg group $$H$$ has the special property that the exponential map is a one-to-one and onto map from the Lie algebra $$\mathfrak h$$ to the group $$H$$.

Higher dimensions
More general Heisenberg groups $$H_{2n+1}$$ may be defined for higher dimensions in Euclidean space, and more generally on s. The simplest general case is the real Heisenberg group of dimension $$2n+1$$, for any integer $$n\geq 1$$. As a group of matrices, $$H_{2n+1}$$ (or $$H_{2n+1}(\mathbb R)$$ to indicate this is the Heisenberg group over the field $$\mathbb R$$ of real numbers) is defined as the group $$(n+2)\times (n+2)$$ matrices with entries in $$\mathbb R$$ and having the form:


 * $$ \begin{bmatrix} 1 & \mathbf a & c \\ \mathbf 0 & I_n & \mathbf b \\ 0 & \mathbf 0 & 1 \end{bmatrix} $$

where
 * a is a of length n,
 * b is a of length n,
 * In is the 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


 * $$ \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}1 & -\mathbf a & -c +\mathbf a \cdot \mathbf b\\ 0 & I_n & -\mathbf b \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & I_n & 0 \\ 0 & 0 & 1 \end{bmatrix}.  $$

Lie algebra
The Heisenberg group is a Lie group whose  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 of size n.

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 can be characterized by the ,

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
 * $$u = \begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix},$$

which fulfills $$u^3 = 0_{n+2}$$. The evaluates to


 * $$ \exp (u) = \sum_{k=0}^\infty \frac{1}{k!}u^k = I_{n+2} + u + \tfrac{1}{2}u^2 = \begin{bmatrix} 1 & \mathbf a & c + {1\over 2}\mathbf a \cdot \mathbf b\\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix}. $$

The exponential map of any nilpotent Lie algebra is a between the Lie algebra and the unique associated,  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 of  0).

Representation theory
The unitary of the Heisenberg group is fairly simple – later generalized by  – and was the motivation for its introduction in quantum physics, as discussed below.

For each nonzero real number $$\hbar$$, we can define an irreducible unitary representation $$\Pi_\hbar$$ of $$H_{2n+1}$$ acting on the Hilbert space $$L^2(\mathbb R^n)$$ by the formula:


 * $$ \left[\Pi_\hbar\begin{pmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{pmatrix}\psi\right](x)=e^{i\hbar c}e^{ib\cdot x}\psi(x+\hbar a) $$

This representation is known as the. The motivation for this representation is the action of the exponentiated and s in quantum mechanics. The parameter $$a$$ describes translations in position space, the parameter $$b$$ describes translations in momentum space, and the parameter $$c$$ 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, which states that every (strongly continuous) irreducible unitary representation of the Heisenberg group in which the center acts nontrivially is equivalent to $$\Pi_\hbar$$ for some $$\hbar$$. Alternatively, that they are all equivalent to the (or ) on a symplectic space of dimension 2n.

Since the Heisenberg group is a one-dimensional central extension of $$\mathbb R^{2n}$$, its irreducible unitary representations can be viewed as irreducible unitary s of $$\mathbb R^{2n}$$. Conceptually, the representation given above constitutes the quantum mechanical counterpart to the group of translational symmetries on the classical phase space, $$\mathbb R^{2n}$$. The fact that the quantum version is only a projective representation of $$R^{2n}$$ 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 however, because $$\{x_i,p_j\}=\delta_{i,j}.$$ 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 $$\mathbb R^{2n}$$, isomorphic to the Lie algebra of the Heisenberg group.

On symplectic vector spaces
The general abstraction of a Heisenberg group is constructed from any. For example, let (V,ω) be a finite-dimensional real symplectic vector space (so ω is a   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 of the additive group V. Thus there is an


 * $$0\to\mathbf{R}\to H(V)\to V\to 0.$$

Any symplectic vector space admits a {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.

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}$$ , which gives a faithful of H(V). The u in this formulation is related to t in our previous formulation by $$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 ).

To any Lie algebra, there is a unique, 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
The Lie algebra $$\mathfrak{h}_n$$ of the Heisenberg group was described above, (1), as a Lie algebra of matrices. The applies to determine the  $$U(\mathfrak{h}_n)$$. Among other properties, the universal enveloping algebra is an into which $$\mathfrak{h}_n$$ injectively imbeds.

By the Poincaré–Birkhoff–Witt theorem, it is thus the 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. It follows from that the  Wn is a quotient of $$U(\mathfrak{h}_n)$$. However, this is also easy to see directly from the above representations; by the mapping
 * $$ z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n} \, \mapsto \, \partial_{x_1}^{k_1} \partial_{x_2}^{k_2} \cdots \partial_{x_n}^{k_n} x_1^{\ell_1} x_2^{\ell_2} \cdots x_n^{\ell_n}~.$$

Weyl's parameterization of quantum mechanics
The application that led to an explicit realization of the Heisenberg group was the question of why the  and  are physically equivalent. Abstractly, the reason is the : there is a unique 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 for discrete Heisenberg groups, in his theory of. This is a large generalization of the approach used in, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is the of the Heisenberg group, of which the discrete case gives the.

Fourier analysis
The Heisenberg group also occurs in, where it is used in some formulations of the. In this case, the Heisenberg group can be understood to act on the space of 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, and specifically, a simple example of a. Given a point p=(x,y,z) in R3, define a differential Θ at this point as


 * $$\Theta_p=dz -\frac{1}{2}\left(xdy - ydx\right).$$

This belongs to the  of R3; that is,


 * $$\Theta_p:T_p\mathbf{R}^3\to\mathbf{R}$$

is a map on the. Let


 * $$H_p = \{ v\in T_p\mathbf{R}^3 \mid \Theta_p(v) = 0 \}.$$

It can be seen that H is a of the tangent bundle TR3. A 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 s


 * $$X=\frac{\partial}{\partial x} - \frac{1}{2} y\frac{\partial}{\partial z},$$
 * $$Y=\frac{\partial}{\partial y} + \frac{1}{2} x\frac{\partial}{\partial z},$$
 * $$Z=\frac{\partial}{\partial z},$$

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 s 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, which follows by.

Heisenberg group of a locally compact abelian group
It is more generally possible to define the Heisenberg group of a K, equipped with a. Such a group has a $$\hat{K}$$, consisting of all continuous $$U(1)$$-valued characters on K, which is also a locally compact abelian group if endowed with the. The Heisenberg group associated with the locally compact abelian group K is the subgroup of the unitary group of $$L^2(K)$$ generated by translations from K and multiplications by elements of $$\hat{K}$$.

In more detail, the $$L^2(K)$$ consists of square-integrable complex-valued functions $$f$$ on K. The translations in K form a of K as operators on $$L^2(K)$$:
 * $$(T_xf)(y) = f(x+y)$$

for $$x,y\in K$$. So too do the multiplications by characters:
 * $$(M_\chi f)(y) = \chi(y)f(y)$$

for $$\chi\in\hat{K}$$. These operators do not commute, and instead satisfy
 * $$(T_xM_\chi T^{-1}_x M_\chi^{-1}f)(y) = \overline{\chi(x)}f(y)$$

multiplication by a fixed unit modulus complex number.

So the Heisenberg group $$H(K)$$ associated with K is a type of of $$K\times\hat{K}$$, via an exact sequence of groups:
 * $$1\to U(1) \to H(K) \to K\times\hat{K}\to 0.$$

More general Heisenberg groups are described by 2-cocyles in the $$H^2(K,U(1))$$. The existence of a duality between $$K$$ and $$\hat{K}$$ gives rise to a canonical cocycle, but there are generally others.

The Heisenberg group acts irreducibly on $$L^2(K)$$. Indeed, the continuous characters separate points so any unitary operator of $$L^2(K)$$ that commutes with them is an $$L^\infty$$. But commuting with translations implies that the multiplier is constant.

A version of the, proved by , holds for the Heisenberg group $$H(K)$$. The is the unique intertwiner between the representations of $$L^2(K)$$ and $$L^2(\hat{K})$$. See the discussion at for details.