Heisenberg group

In the three-dimensional case, the product of two Heisenberg matrices is given by:

${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}1&a'&c'\\0&1&b'\\0&0&1\\\end{pmatrix}}={\begin{pmatrix}1&a+a'&c+ab'+c'\\0&1&b+b'\\0&0&1\\\end{pmatrix}}\,.}$

As one can see from the term ab', the group is non-abelian.

The neutral element of the Heisenberg group is the identity matrix, and inverses are given by

${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}^{-1}={\begin{pmatrix}1&-a&ab-c\\0&1&-b\\0&0&1\\\end{pmatrix}}\,.}$

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

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

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,

${\displaystyle x={\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\\\end{pmatrix}},\ \ y={\begin{pmatrix}1&0&0\\0&1&1\\0&0&1\\\end{pmatrix}}}$

and relations

${\displaystyle z_{}^{}=xyx^{-1}y^{-1},\ xz=zx,\ yz=zy}$,

where

${\displaystyle z={\begin{pmatrix}1&0&1\\0&1&0\\0&0&1\\\end{pmatrix}}}$

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

${\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}=y^{b}z^{c}x^{a}\,.}$

The Lie algebra ${\displaystyle {\mathfrak {h}}}$ of the Heisenberg group ${\displaystyle H}$ (over the real numbers) is known as the Heisenberg algebra.[1] It may be represented using the space of 3×3 matrices of the form[2]

${\displaystyle {\begin{pmatrix}0&a&c\\0&0&b\\0&0&0\\\end{pmatrix}},}$

with ${\displaystyle a,b,c\in \mathbb {R} }$.

The following three elements form a basis for ${\displaystyle {\mathfrak {h}}}$,

${\displaystyle X={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\\\end{pmatrix}};\quad Y={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\\\end{pmatrix}};\quad Z={\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\\\end{pmatrix}}.}$

These basis elements satisfy the commutation relations,

${\displaystyle [X,Y]=Z;\quad [X,Z]=0;\quad [Y,Z]=0}$.

The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics,

${\displaystyle \left[{\hat {x}},{\hat {p}}\right]=i\hbar I;\quad \left[{\hat {x}},i\hbar I\right]=0;\quad \left[{\hat {p}},i\hbar I\right]=0,}$

where ${\displaystyle {\hat {x}}}$ is the position operator, ${\displaystyle {\hat {p}}}$ is the momentum operator, and ${\displaystyle \hbar }$ 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 ${\displaystyle {\mathfrak {h}}}$ to the group H,[3]

${\displaystyle \exp {\begin{pmatrix}0&a&c\\0&0&b\\0&0&0\\\end{pmatrix}}={\begin{pmatrix}1&a&c+{\frac {ab}{2}}\\0&1&b\\0&0&1\\\end{pmatrix}}.}$

More general Heisenberg groups ${\displaystyle H_{2n+1}}$ 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 ${\displaystyle 2n+1}$, for any integer ${\displaystyle n\geq 1}$. As a group of matrices, ${\displaystyle H_{2n+1}}$ (or ${\displaystyle H_{2n+1}(\mathbb {R} )}$ to indicate this is the Heisenberg group over the field ${\displaystyle \mathbb {R} }$ of real numbers) is defined as the group ${\displaystyle (n+2)\times (n+2)}$ matrices with entries in ${\displaystyle \mathbb {R} }$ and having the form:

${\displaystyle {\begin{bmatrix}1&\mathbf {a} &c\\\mathbf {0} &I_{n}&\mathbf {b} \\0&\mathbf {0} &1\end{bmatrix}}}$

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.

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 ${\displaystyle \hbar }$, we can define an irreducible unitary representation ${\displaystyle \Pi _{\hbar }}$ of ${\displaystyle H_{2n+1}}$ acting on the Hilbert space ${\displaystyle L^{2}(\mathbb {R} ^{n})}$ by the formula:[4]

${\displaystyle \left[\Pi _{\hbar }{\begin{pmatrix}1&\mathbf {a} &c\\0&I_{n}&\mathbf {b} \\0&0&1\end{pmatrix}}\psi \right](x)=e^{i\hbar c}e^{ib\cdot x}\psi (x+\hbar a)}$

This representation is known as the Schrödinger representation. The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics. The parameter ${\displaystyle a}$ describes translations in position space, the parameter ${\displaystyle b}$ describes translations in momentum space, and the parameter ${\displaystyle c}$ 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 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 ${\displaystyle \Pi _{\hbar }}$ for some ${\displaystyle \hbar }$.[5] 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 ${\displaystyle \mathbb {R} ^{2n}}$, its irreducible unitary representations can be viewed as irreducible unitary projective representations of ${\displaystyle \mathbb {R} ^{2n}}$. Conceptually, the representation given above constitutes the quantum mechanical counterpart to the group of translational symmetries on the classical phase space, ${\displaystyle \mathbb {R} ^{2n}}$. The fact that the quantum version is only a projective representation of ${\displaystyle \mathbb {R} ^{2n}}$ 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 Poisson bracket however, because ${\displaystyle \{x_{i},p_{j}\}=\delta _{i,j}.}$ 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 ${\displaystyle \mathbb {R} ^{2n}}$, isomorphic to the Lie algebra of the Heisenberg group.

The general abstraction of a Heisenberg group is constructed from any symplectic vector space.[6] 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

${\displaystyle (v,t)\cdot \left(v',t'\right)=\left(v+v',t+t'+{\frac {1}{2}}\omega \left(v,v'\right)\right).}$

The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence

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

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

${\displaystyle v=q^{a}\mathbf {e} _{a}+p_{a}\mathbf {f} ^{a}.}$

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

${\displaystyle v=q^{a}\mathbf {e} _{a}+p_{a}\mathbf {f} ^{a}+tE}$

and the group law becomes

${\displaystyle (p,q,t)\cdot \left(p',q',t'\right)=\left(p+p',q+q',t+t'+{\frac {1}{2}}(pq'-p'q)\right).}$

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

${\displaystyle {\begin{bmatrix}(v_{1},t_{1}),(v_{2},t_{2})\end{bmatrix}}=\omega (v_{1},v_{2})}$

or written in terms of the Darboux basis

${\displaystyle \left[\mathbf {e} _{a},\mathbf {f} ^{b}\right]=\delta _{a}^{b}}$

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

${\displaystyle v=q^{a}\mathbf {e} _{a}+p_{a}\mathbf {f} ^{a}+uE}$

and the group law is

${\displaystyle (p,q,u)\cdot \left(p',q',u'\right)=\left(p+p',q+q',u+u'+pq'\right).}$

An element of the group

${\displaystyle v=q^{a}\mathbf {e} _{a}+p_{a}\mathbf {f} ^{a}+uE}$

can then be expressed as a matrix

${\displaystyle {\begin{bmatrix}1&p&u\\0&I_{n}&q\\0&0&1\end{bmatrix}}}$ ,

which gives a faithful matrix representation of H(V). The u in this formulation is related to t in our previous formulation by ${\displaystyle u=t+{\tfrac {1}{2}}pq}$, so that the t value for the product comes to

${\displaystyle {\begin{aligned}&u+u'+pq'-{\frac {1}{2}}\left(p+p'\right)\left(q+q'\right)\\={}&t+{\frac {1}{2}}pq+t'+{\frac {1}{2}}p'q'+pq'-{\frac {1}{2}}\left(p+p'\right)\left(q+q'\right)\\={}&t+t'+{\frac {1}{2}}\left(pq'-p'q\right)\end{aligned}}}$ ,

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 Lie algebra ${\displaystyle {\mathfrak {h}}_{n}}$ 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 ${\displaystyle U({\mathfrak {h}}_{n})}$. Among other properties, the universal enveloping algebra is an associative algebra into which ${\displaystyle {\mathfrak {h}}_{n}}$ injectively imbeds.

By the Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials

${\displaystyle z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell _{1}}q_{2}^{\ell _{2}}\cdots q_{n}^{\ell _{n}}~,}$

where the exponents are all non-negative.

Consequently, ${\displaystyle U({\mathfrak {h}}_{n})}$ consists of real polynomials

${\displaystyle \sum _{j,{\vec {k}},{\vec {\ell }}}c_{j{\vec {k}}{\vec {\ell }}}\,\,z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell _{1}}q_{2}^{\ell _{2}}\cdots q_{n}^{\ell _{n}}~,}$

with the commutation relations

${\displaystyle p_{k}p_{\ell }=p_{\ell }p_{k},\quad q_{k}q_{\ell }=q_{\ell }q_{k},\quad p_{k}q_{\ell }-q_{\ell }p_{k}=\delta _{k\ell }z,\quad zp_{k}-p_{k}z=0,\quad zq_{k}-q_{k}z=0~.}$

The algebra ${\displaystyle U({\mathfrak {h}}_{n})}$ 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

${\displaystyle P=\sum _{{\vec {k}},{\vec {\ell }}}c_{{\vec {k}}{\vec {\ell }}}\,\,\partial _{x_{1}}^{k_{1}}\partial _{x_{2}}^{k_{2}}\cdots \partial _{x_{n}}^{k_{n}}x_{1}^{\ell _{1}}x_{2}^{\ell _{2}}\cdots x_{n}^{\ell _{n}}~.}$

This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra W_n is a quotient of ${\displaystyle U({\mathfrak {h}}_{n})}$. However, this is also easy to see directly from the above representations; viz. by the mapping

${\displaystyle z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell _{1}}q_{2}^{\ell _{2}}\cdots q_{n}^{\ell _{n}}\,\mapsto \,\partial _{x_{1}}^{k_{1}}\partial _{x_{2}}^{k_{2}}\cdots \partial _{x_{n}}^{k_{n}}x_{1}^{\ell _{1}}x_{2}^{\ell _{2}}\cdots x_{n}^{\ell _{n}}~.}$

An animation of a geodesic in the Heisenberg group.

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.[7] Given a point p=(x,y,z) in R³, define a differential 1-form Θ at this point as

${\displaystyle \Theta _{p}=dz-{\frac {1}{2}}\left(xdy-ydx\right).}$

This one-form belongs to the cotangent bundle of R³; that is,

${\displaystyle \Theta _{p}:T_{p}\mathbf {R} ^{3}\to \mathbf {R} }$

is a map on the tangent bundle. Let

${\displaystyle H_{p}=\left\{v\in T_{p}\mathbf {R} ^{3}\mid \Theta _{p}(v)=0\right\}.}$

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 ${\displaystyle v=(v_{1},v_{2},v_{3})}$ and ${\displaystyle w=(w_{1},w_{2},w_{3})}$ in TR³, the inner product is given by

${\displaystyle \langle v,w\rangle =v_{1}w_{1}+v_{2}w_{2}.}$

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

${\displaystyle {\begin{aligned}X&={\frac {\partial }{\partial x}}-{\frac {1}{2}}y{\frac {\partial }{\partial z}},\\Y&={\frac {\partial }{\partial y}}+{\frac {1}{2}}x{\frac {\partial }{\partial z}},\\Z&={\frac {\partial }{\partial z}},\end{aligned}}}$

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

${\displaystyle \gamma (t)=(x(t),y(t),z(t))}$

is a geodesic curve, then the curve ${\displaystyle c(t)=(x(t),y(t))}$ is an arc of a circle, and

${\displaystyle z(t)={\frac {1}{2}}\int _{c}xdy-ydx}$

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.

It is more generally possible to define the Heisenberg group of a locally compact abelian group K, equipped with a Haar measure.[8] Such a group has a Pontrjagin dual ${\displaystyle {\hat {K}}}$, consisting of all continuous ${\displaystyle U(1)}$-valued characters on K, which is also a locally compact abelian group if endowed with the compact-open topology. The Heisenberg group associated with the locally compact abelian group K is the subgroup of the unitary group of ${\displaystyle L^{2}(K)}$ generated by translations from K and multiplications by elements of ${\displaystyle {\hat {K}}}$.

In more detail, the Hilbert space ${\displaystyle L^{2}(K)}$ consists of square-integrable complex-valued functions ${\displaystyle f}$ on K. The translations in K form a unitary representation of K as operators on ${\displaystyle L^{2}(K)}$:

${\displaystyle (T_{x}f)(y)=f(x+y)}$

for ${\displaystyle x,y\in K}$. So too do the multiplications by characters:

${\displaystyle (M_{\chi }f)(y)=\chi (y)f(y)}$

for ${\displaystyle \chi \in {\hat {K}}}$. These operators do not commute, and instead satisfy

${\displaystyle \left(T_{x}M_{\chi }T_{x}^{-1}M_{\chi }^{-1}f\right)(y)={\overline {\chi (x)}}f(y)}$

multiplication by a fixed unit modulus complex number.

So the Heisenberg group ${\displaystyle H(K)}$ associated with K is a type of central extension of ${\displaystyle K\times {\hat {K}}}$, via an exact sequence of groups:

${\displaystyle 1\to U(1)\to H(K)\to K\times {\hat {K}}\to 0.}$

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

The Heisenberg group acts irreducibly on ${\displaystyle L^{2}(K)}$. Indeed, the continuous characters separate points[9] so any unitary operator of ${\displaystyle L^{2}(K)}$ that commutes with them is an ${\displaystyle L^{\infty }}$ multiplier. But commuting with translations implies that the multiplier is constant.[10]

A version of the Stone–von Neumann theorem, proved by George Mackey, holds for the Heisenberg group ${\displaystyle H(K)}$.[11][12] The Fourier transform is the unique intertwiner between the representations of ${\displaystyle L^{2}(K)}$ and ${\displaystyle L^{2}\left({\hat {K}}\right)}$. See the discussion at Stone–von Neumann theorem#Relation to the Fourier transform for details.

• Canonical commutation relations
• Wigner–Weyl transform
• Stone–von Neumann theorem
• Projective representation

• Binz, Ernst; Pods, Sonja (2008). Geometry of Heisenberg Groups. American Mathematical Society. ISBN 978-0-8218-4495-3.
• Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158
• Hall, Brian C. (2015). Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (second ed.). Springer. ISBN 978-3319134666.
• 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/s0273-0979-1980-14825-9. MR 0578375.
• Kirillov, Alexandre 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, George (1976). The theory of Unitary Group Representations. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0226500522.

• Groupprops, The Group Properties Wiki Unitriangular matrix group UT(3,p)