In mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the form For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...

or its generalizations. Elements a,b,c can be taken from some (arbitrary) commutative ring, often taken to be the ring of real numbers or the ring of integers. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The integers are commonly denoted by the above symbol. ...

The real Heisenberg group arises in the description of one-dimensional quantum mechanical systems. More generally, one can consider groups associated to n-dimensional systems, and most generally, to any symplectic vector space. In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ...

The three-dimensional case

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 Lie group. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

In addition to the representation as real 3x3 matricies, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stone-von Neumann theorem, there is a unique irreducible unitary representation of H in which its center 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 to its connection with the theta functions. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... this article is useless. ... In mathematics, theta functions are special functions of several complex variables. ...

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 non-abelian nilpotent group. It has two generators Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...

and relations

,

where

is the generator of the center of H₃. By Bass' theorem, it has a polynomial growth rate of order 4. In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. ...

Heisenberg group modulo p

If one takes a,b,c in Z/p Z, then one has the Heisenberg group modulo p. It is a group of order p³ with two generators, x, y and relations In group theory, a branch of mathematics, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

.

Analogues of Heisenberg groups over finite fields of prime order are called extra special groups.

Higher dimensions

More general Heisenberg groups H_n 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, H_n (or H_n(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: In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ...

where a is a row vector of length n, b is a column vector of length n and I_n is the identity matrix of size n. This is indeed a group, as is shown by the multiplication: In linear algebra, a row vector is a 1 × n matrix, that is, a matrix consisting of a single row: The transpose of a row vector is a column vector. ... In linear algebra, a column vector is an m × 1 matrix, i. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

and

The Heisenberg group is a connected, simply-connected Lie group whose Lie algebra consists of matrices Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

where a is a row vector of length n, b is a column vector of length n and 0_n is the zero matrix of size n. The exponential map is given by the following expression In mathematics, a zero matrix is a matrix with all its entries being zero. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...

By let e₁, ..., e_n be the canonical basis of Rⁿ, and setting

the Lie algebra can also be characterized by the canonical commutation relations In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

where p₁, .., p_n, q₁, .., q_n, z are 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. There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

The above discussion (aside from statements referring to dimension and Lie group) 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/pZ with an odd prime p or any field of characteristic 0). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

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 In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

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. 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) may be identified with the matrix

which gives a faithful matrix representation of H(V). Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

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

[(v₁,t₁),(v₂,t₂)] = ω(v₁,v₂)

or written in terms of the Darboux basis

and all other commutators vanish.

The isomorphism to the group of upper triangular matrices relies on a decomposition of V into a Darboux basis, which amounts to a choice of isomorphism V ≅ U ⊕ U*. By means of this isomorphism, another group law may be introduced:

Although this group law yields an isomorphic group to the one given above, 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). In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be embedded into projective space. ...

To any Lie algebra, there is a unique connected, simply connected Lie group G. All other 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. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

The connection with the Weyl algebra

The Lie algebra of the Heisenberg group was described above as a Lie algebra of matrices. We now apply the Poincaré-Birkhoff-Witt theorem, to determine the universal enveloping algebra . Among other properties, the universal enveloping algebra is an associative algebra into which injectively imbeds. By Poincaré-Birkhoff-Witt, it is the free vector space generated by the monomials In the theory of Lie algebras, the Poincaré-Birkhoff-Witt theorem is a fundamental result characterizing the universal enveloping algebra of a Lie algebra. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a free module is a module having a free basis. ...

where the exponents are all non-negative. Thus consists of real polynomials

with the commutation relations

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:

This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra W_n is a quotient of . However, this also easy to see directly from the above representations; viz, by the mapping In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X... Abstract nonsense is a popular term used by mathematicians to describe certain kinds of arguments and concepts in category theory. ... In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X...

Weyl's view of quantum mechanics

See main article Weyl quantization.

The application that led Hermann Weyl to an explicit introduction of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly there is a good explanation: the group H_n is a central extension of R²ⁿ by a copy of R, and as such is a semidirect product. Its representation theory is relatively simple (a special case of the later Mackey theory), and in particular there is a uniqueness result for unitary representations with given action of the central element z (in the Lie algebra) or the one-parameter subgroup it creates under the exponential map, which is the central extension. This abstract uniqueness accounts for the equivalence of the two physical pictures. In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for associating a quantum mechanical Hermitian operator with a classical distribution in phase space. ... Hermann Klaus Hugo Weyl (November 9, 1885 – December 9, 1955) was a German mathematician. ... Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ... The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...

The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of 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. David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. ... In mathematics, theta functions are special functions of several complex variables. ...

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. Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...

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 On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. ... (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

.

This one-form belongs to the cotangent bundle of R³; that is, In linear algebra, a one-form on a vector space is the same as a linear functional on the space. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

is a map on the tangent bundle. Let In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

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 = (v₁,v₂,v₃) and w = (w₁,w₂,w₃) in TR³, the inner product is given by In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces Ux of the fibers Vx of V at x in X, that make up a vector bundle in their own right. ...

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 Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

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 In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...

γ(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

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. In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

See also

• Wigner quasi-probability distribution

The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ...

References

1. ^ 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.
2. ^ 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.