Inmathematics, theHeisenberg group${\displaystyle H}$, named afterWerner Heisenberg, is thegroup of 3×3upper triangular matrices of the form

under the operation ofmatrix multiplication. Elementsa, b andc can be taken from anycommutative ring with identity, often taken to be the ring ofreal numbers (resulting in the "continuous Heisenberg group") or the ring ofintegers (resulting in the "discrete Heisenberg group").

The continuous Heisenberg group arises in the description of one-dimensionalquantum mechanical systems, especially in the context of theStone–von Neumann theorem. More generally, one can consider Heisenberg groups associated ton-dimensional systems, and most generally, to anysymplectic vector space.

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

As one can see from the termab′, the group isnon-abelian.

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

The group is a subgroup of the 2-dimensional affine group Aff(2): acting on${\displaystyle (\overrightarrow{x},1)}$ corresponds to the affine transform

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

Ifa,b,c, arereal numbers (in the ringR), then one has thecontinuous Heisenberg group H₃(R).

It is anilpotent realLie group of dimension 3.

In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several differentrepresentations in terms offunction spaces. ByStone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which itscentre acts by a given nontrivialcharacter. This representation has several important realizations, or models. In theSchrödinger model, the Heisenberg group acts on the space ofsquare integrable functions. In thetheta representation, it acts on the space ofholomorphic functions on theupper half-plane; it is so named for its connection with thetheta functions.

Ifa,b,c are integers (in the ringZ), then one has thediscrete Heisenberg group H₃(Z). It is anon-abeliannilpotent group. It has two generators:

and relations

${\displaystyle z=xy{x}^{-1}{y}^{-1},xz=zx,yz=zy,}$

where

is the generator of thecenter of H₃. (Note that the inverses ofx,y, andz replace the 1 above the diagonal with −1.)

ByBass's theorem, it has a polynomialgrowth rate of order 4.

One can generate any element through

If one takesa,b,c inZ/pZ for an oddprimep, then one has theHeisenberg group modulop. It is a group oforderp³ with generatorsx,y and relations

${\displaystyle z=xy{x}^{-1}{y}^{-1},x^p=y^p=z^p=1,xz=zx,yz=zy.}$

Analogues of Heisenberg groups overfinite fields of odd prime orderp are calledextra special groups, or more properly, extra special groups ofexponentp. More generally, if thederived subgroup of a groupG is contained in the centerZ ofG, then the mapG/Z ×G/Z →Z is a skew-symmetric bilinear operator on abelian groups.

However, requiring thatG/Z to be a finitevector space requires theFrattini subgroup ofG to be contained in the center, and requiring thatZ be a one-dimensional vector space overZ/pZ requires thatZ have orderp, so ifG is not abelian, thenG is extra special. IfG is extra special but does not have exponentp, then the general construction below applied to the symplectic vector spaceG/Z does not yield a group isomorphic toG.

The Heisenberg group modulo 2 is of order 8 and is isomorphic to thedihedral group D₄ (the symmetries of a square). Observe that if

then

and

The elementsx andy correspond to reflections (with 45° between them), whereasxy andyx correspond to rotations by 90°. The other reflections arexyx andyxy, and rotation by 180° isxyxy (= yxyx).

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]

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

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

These basis elements satisfy the commutation relations

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

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

${\displaystyle [\hat{x},\hat{p}]=i\hslash I,[\hat{x},i\hslash I]=0,[\hat{p},i\hslash I]=0,}$

where${\displaystyle \hat{x}}$ is the position operator,${\displaystyle \hat{p}}$ is the momentum operator, and${\displaystyle \hslash }$ is the Planck constant.

The Heisenberg groupH 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 groupH:^[3]

Inconformal field theory, the term Heisenberg algebra is used to refer to an infinite-dimensional generalization of the above algebra. It is spanned by elements${\displaystyle a_n,n\in \mathbb{Z}}$ with commutation relations

${\displaystyle [a_n,a_m]={\delta }_{n+m,0}.}$

Under a rescaling, this is simply a countably-infinite number of copies of the above algebra.

More general Heisenberg groups${\displaystyle H_{2n+1}}$ may be defined for higher dimensions in Euclidean space, and more generally onsymplectic vector spaces. The simplest general case is the real Heisenberg group of dimension${\displaystyle 2n+1}$, for any integer${\displaystyle n\ge 1}$. As a group of matrices,${\displaystyle H_{2n+1}}$ (or${\displaystyle H_{2n+1}(\mathbb{R})}$ to indicate that 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

where

a is arow vector of lengthn,

b is acolumn vector of lengthn,

I_n is theidentity matrix of sizen.

This is indeed a group, as is shown by the multiplication:

and

The Heisenberg group is asimply-connected Lie group whoseLie algebra consists of matrices

where

a is a row vector of lengthn,

b is a column vector of lengthn,

0_n is thezero matrix of sizen.

By letting e₁, ..., e_n be the canonical basis ofRⁿ and setting

the associatedLie algebra can be characterized by thecanonical commutation relations

${\displaystyle \left[\begin{array}{c}{p}_i,q_j\end{array}\right]={\delta }_{ij}z,\left[\begin{array}{c}{p}_i,z\end{array}\right]=0,\left[\begin{array}{c}{q}_j,z\end{array}\right]=0,}$

1

wherep₁, ..., p_n,q₁, ..., q_n,z are the algebra generators.

In particular,z is acentral element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent.

Let

which fulfills${\displaystyle u^3=0_{n+2}}$. Theexponential map evaluates to

The exponential map of any nilpotent Lie algebra is adiffeomorphism between the Lie algebra and the unique associatedconnected,simply-connected Lie group.

This discussion (aside from statements referring to dimension and Lie group) further applies if we replaceR by any commutative ringA. The corresponding group is denotedH_n(A).

Under the additional assumption that the prime 2 is invertible in the ringA, the exponential map is also defined, since it reduces to a finite sum and has the form above (e.g.A could be a ringZ/p Z with an odd primep or anyfield ofcharacteristic 0).

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

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

This representation is known as theSchrödinger representation. The motivation for this representation is the action of the exponentiatedposition andmomentum 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 theStone–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${\displaystyle {\mathrm{\Pi }}_{\hslash }}$ for some${\displaystyle \hslash }$.^[5] Alternatively, that they are all equivalent to theWeyl algebra (orCCR 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 unitaryprojective 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 aprojective 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 does not form a Lie algebra under thePoisson bracket, however, because${\displaystyle \{x_i,p_j\}={\delta }_{i,j}.}$ Rather, the span of the position and momentum functionsand 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 anysymplectic vector space.^[6] For example, let (V, ω) be a finite-dimensional real symplectic vector space (so ω is anondegenerateskew symmetricbilinear form onV). The Heisenberg group H(V) on (V,ω) (or simplyV for brevity) is the setV×R endowed with the group law

${\displaystyle (v,t)\cdot \left(v^{\prime },t^{\prime }\right)=\left(v+v^{\prime },t+t^{\prime }+\frac{1}{2}\omega \left(v,v^{\prime }\right)\right).}$

The Heisenberg group is acentral extension of the additive groupV. Thus there is anexact sequence

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

Any symplectic vector space admits aDarboux basis {e_j,f^k}_{1 ≤j,k ≤n} satisfyingω(e_j,f^k) =δ_j^k and where 2n is the dimension ofV (the dimension ofV is necessarily even). In terms of this basis, every vector decomposes as

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a.}$

Theq^a andp_a arecanonically conjugate coordinates.

If {e_j,f^k}_{1 ≤j,k ≤n} is a Darboux basis forV, then let {E} be a basis forR, and {e_j,f^k,E}_{1 ≤j,k ≤n} is the corresponding basis forV×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

${\displaystyle (p,q,t)\cdot \left(p^{\prime },q^{\prime },t^{\prime }\right)=\left(p+p^{\prime },q+q^{\prime },t+t^{\prime }+\frac{1}{2}(p{q}^{\prime }-p^{\prime }q)\right).}$

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

${\displaystyle \left[\begin{array}{c}(v_1,t_1),(v_2,t_2)\end{array}\right]=\omega (v_1,v_2)}$

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 useu instead oft, so a vector is given by

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a+uE}$

and the group law is

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

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

,

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

,

as before.

The isomorphism to the group using upper triangular matrices relies on the decomposition ofV into a Darboux basis, which amounts to a choice of isomorphismV ≅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 thepolarized Heisenberg group as a reminder that this group law relies on a choice of basis (a choice of a Lagrangian subspace ofV is apolarization).

To any Lie algebra, there is a uniqueconnected,simply connected Lie groupG. All other connected Lie groups with the same Lie algebra asG are of the formG/N whereN is a central discrete group inG. In this case, the center of H(V) isR and the only discrete subgroups are isomorphic toZ. 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. ThePoincaré–Birkhoff–Witt theorem applies to determine theuniversal enveloping algebra${\displaystyle U({\mathfrak{h}}_n)}$. Among other properties, the universal enveloping algebra is anassociative algebra into which${\displaystyle {\mathfrak{h}}_n}$ injectively imbeds.

By the Poincaré–Birkhoff–Witt theorem, it is thus thefree 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

${\displaystyle \sum _{j,\overrightarrow{k},\overrightarrow{\ell }}{c}_{j\overrightarrow{k}\overrightarrow{\ell }}{z}^j{p}_1^{k_1}{p}_2^{k_2}\cdots p_n^{k_n}{q}_1^{{\ell }_1}{q}_2^{{\ell }_2}\cdots q_n^{{\ell }_n},}$

with the commutation relations

${\displaystyle p_k{p}_{\ell }=p_{\ell }{p}_k,q_k{q}_{\ell }=q_{\ell }{q}_k,p_k{q}_{\ell }-q_{\ell }{p}_k={\delta }_{k\ell }z,z{p}_k-p_{k}z=0,z{q}_k-q_{k}z=0.}$

The algebra${\displaystyle U({\mathfrak{h}}_n)}$ is closely related to the algebra of differential operators on${\displaystyle {\mathbb{R}}^n}$ with polynomial coefficients, since any such operator has a unique representation in the form

${\displaystyle P=\sum _{\overrightarrow{k},\overrightarrow{\ell }}{c}_{\overrightarrow{k}\overrightarrow{\ell }}{\mathrm{\partial }}_{x_1}^{k_1}{\mathrm{\partial }}_{x_2}^{k_2}\cdots {\mathrm{\partial }}_{x_n}^{k_n}{x}_1^{{\ell }_1}{x}_2^{{\ell }_2}\cdots x_n^{{\ell }_n}.}$

This algebra is called theWeyl algebra. It follows fromabstract nonsense that theWeyl algebraW_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

${\displaystyle z^j{p}_1^{k_1}{p}_2^{k_2}\cdots p_n^{k_n}{q}_1^{{\ell }_1}{q}_2^{{\ell }_2}\cdots q_n^{{\ell }_n}\mapsto {\mathrm{\partial }}_{x_1}^{k_1}{\mathrm{\partial }}_{x_2}^{k_2}\cdots {\mathrm{\partial }}_{x_n}^{k_n}{x}_1^{{\ell }_1}{x}_2^{{\ell }_2}\cdots x_n^{{\ell }_n}.}$

The application that ledHermann Weyl to an explicit realization of the Heisenberg group was the question of why theSchrödinger picture andHeisenberg picture are physically equivalent. Abstractly, the reason is theStone–von Neumann theorem: there is a uniqueunitary representation with given action of the central Lie algebra elementz, 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.

The same uniqueness result was used byDavid Mumford for discrete Heisenberg groups, in his theory ofequations defining abelian varieties. This is a large generalization of the approach used inJacobi's elliptic functions, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is thetheta representation of the Heisenberg group, of which the discrete case gives thetheta function.

The Heisenberg group also occurs inFourier analysis, where it is used in some formulations of theStone–von Neumann theorem. In this case, the Heisenberg group can be understood to act on the space ofsquare integrable functions; the result is a representation of the Heisenberg groups sometimes called the Weyl representation.

The three-dimensional Heisenberg groupH₃(R) on the reals can also be understood to be a smoothmanifold, and specifically, a simple example of asub-Riemannian manifold.^[7] Given a pointp = (x,y,z) inR³, define a differential1-form Θ at this point as

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

Thisone-form belongs to thecotangent bundle ofR³; that is,

${\displaystyle {\mathrm{\Theta }}_p:T_p{\mathbf{R}}^3\to \mathbf{R}}$

is a map on thetangent bundle. Let

${\displaystyle H_p=\left\{v\in T_p{\mathbf{R}}^3\mid {\mathrm{\Theta }}_p(v)=0\right\}.}$

It can be seen thatH is asubbundle of the tangent bundle TR³. Acometric onH is given by projecting vectors to the two-dimensional space spanned by vectors in thex andy 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

${\displaystyle ⟨v,w⟩=v_1{w}_1+v_2{w}_2.}$

The resulting structure turnsH into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lievector fields

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. Thegeodesics 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 thecircular arc, which follows byGreen's theorem.

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

In more detail, theHilbert space${\displaystyle L^2(K)}$ consists of square-integrable complex-valued functions${\displaystyle f}$ onK. The translations inK form aunitary representation ofK 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 withK is a type ofcentral 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 thecohomology 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${\displaystyle L^{\mathrm{\infty }}}$multiplier. But commuting with translations implies that the multiplier is constant.^[10]

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

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

• 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 WikiUnitriangular matrix group UT(3,p)

• Group theory
• Lie groups
• Mathematical quantization
• Mathematical physics
• Werner Heisenberg
• 3-manifolds

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