Heisenberg group
Encyclopedia
In mathematics
, the Heisenberg group, named after Werner Heisenberg
, is the group
of 3×3 upper triangular matrices
of the form
or its generalizations under the operation of matrix multiplication. Elements a, b, c can be taken from some (arbitrary) commutative ring
with identity, often taken to be the ring of real number
s or the ring of integer
s.
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
.
s (in the ring R) then one has the continuous Heisenberg group H3(R). It is a nilpotent
Lie group
.
In addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different representations
in terms of function space
s. 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 function
s on the upper half-plane; it is so named for its connection with the theta functions.
nilpotent group
. It has two generators
and relations
,
where
is the generator of the center of H3. By Bass' theorem, it has a polynomial growth rate
of order 4.
p3 with two generators, x, y and relations
Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special group
s, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group is contained in the center Z, of the group 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/pZ 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.
Observe that if
.
Then
and
may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector space
s. 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:
where a is a row vector of length n, b is a column vector of length n and
is
the identity matrix
of size n. This is indeed a group, as is shown by the multiplication:
and
The Heisenberg group is a connected
, simply-connected Lie group whose Lie algebra
consists of matrices
where a is a row vector of length n, b is a column vector of length n and 0n is
the zero matrix of size n.
The exponential map
is given by the following expression
By letting e1, ..., en be the canonical basis of Rn, and setting
the Lie algebra can also be characterized by the canonical commutation relations
where p1, ..., pn, q1, ..., qn, 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.
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 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/pZ with an odd prime p or any field
of characteristic
0).
. 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
The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence
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
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
is then given by
and the group law becomes
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
or written in terms of the Darboux basis
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
and the group law is
An element of the group :
can then be expressed as a matrix
which gives a faithful matrix representation
of H(V).
The u in this formulation is related to t in our previous formulation by
, so that the t value for the product comes to
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.
of the Heisenberg group was described above as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theorem
applies 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
where the exponents are all non-negative.
Thus
consists of real polynomials
with the commutation relations
The algebra
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 Wn is a quotient of
. However, this also easy to see directly from the above representations; viz. by the mapping
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
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
.
to an explicit introduction 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 element z, up to a unitary equivalence: they are all equivalent to the position and momentum operators. Thus the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this unique representation.
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.
. 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.
, and specifically, a simple example of a sub-Riemannian manifold
. Given a point p=(x,y,z) in R3, define a differential 1-form Θ at this point as
This one-form
belongs to the cotangent bundle
of R3; that is,
is a map on the tangent bundle
. Let
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
and 
in TR3, the inner product is given by
The resulting structure turns H into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie vector field
s
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 geodesic
s on the manifold are spirals, projecting down to circles in two dimensions. That is, if
is a geodesic curve, then the curve
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Heisenberg group, named after Werner Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
, is the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of 3×3 upper triangular matrices
Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...
of the form
or its generalizations under the operation of matrix multiplication. Elements a, b, c can be taken from some (arbitrary) commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with identity, often taken to be the ring of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s or the ring of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
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
Symplectic vector space
In mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...
.
Continuous Heisenberg group
If a, b, c are real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (in the ring R) then one has the continuous Heisenberg group H3(R). It is a nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
Lie group
Lie group
In mathematics, a Lie group 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 matrices, the continuous Heisenberg group also has several different representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
in terms of function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s. By Stone–von Neumann theorem
Stone–von Neumann theorem
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...
, 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
Theta representation
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group...
, it acts on the space of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on the upper half-plane; it is so named for its connection with the theta functions.
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-abelianNonabelian group
In mathematics, a non-abelian group, also sometimes called a non-commutative group, is a group in which there are at least two elements a and b of G such that a * b ≠ b * a...
nilpotent group
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
. It has two generators
and relations
,where
is the generator of the center of H3. By Bass' theorem, it has a polynomial growth rate
Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
of order 4.
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 orderOrder (group theory)
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.e., the number of its elements....
p3 with two generators, x, y and relations
Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special group
Extra special group
In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two extra special groups of order p1+2n. Extra special groups often occur in centralizers of...
s, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group is contained in the center Z, of the group 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
Frattini subgroup
In mathematics, the Frattini subgroup Φ of a group G is the intersection of all maximal subgroups of G. For the case that G is the trivial group e, which has no maximal subgroups, it is defined by Φ = e...
of G to be contained in the center, and requiring that Z be a one dimensional vector space over Z/pZ 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
2k The Heisenberg group of order 8 is a dihedral group of order 8.Observe that if
.Then
and
Higher dimensions
More general Heisenberg groups may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaceSymplectic vector space
In mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...
s. 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:
where a is a row vector of length n, b is a column vector of length n and
isthe identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
of size n. This is indeed a group, as is shown by the multiplication:
and
The Heisenberg group is a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, simply-connected Lie group whose Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
consists of matrices
where a is a row vector of length n, b is a column vector of length n and 0n is
the zero matrix of size n.
The exponential map
Exponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
is given by the following expression
By letting e1, ..., en be the canonical basis of Rn, and setting
the Lie algebra can also be characterized by the canonical commutation relations
where p1, ..., pn, q1, ..., qn, 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
Exponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
of a nilpotent Lie algebra is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
between the Lie algebra and the unique associated connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, simply-connected Lie group.
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 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/pZ with an odd prime p or any field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0).
On symplectic vector spaces
The general abstraction of a Heisenberg group is constructed from any symplectic vector spaceSymplectic vector space
In mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...
. 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
The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
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
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
is then given byand the group law becomes
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
or written in terms of the Darboux basis
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
and the group law is
An element of the group :
can then be expressed as a matrixwhich gives a faithful matrix representation
Matrix representation
Matrix representation is a method used by a computer language to store matrices of more than one dimension in memory.Fortran and C use different schemes. Fortran uses "Column Major", in which all the elements for a given column are stored contiguously in memory...
of H(V).
The u in this formulation is related to t in our previous formulation by
, so that the t value for the product comes toas 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
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, 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
The Lie algebra of the Heisenberg group was described above as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theoremPoincaré–Birkhoff–Witt theorem
In the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem is a result giving an explicit description of the universal enveloping algebra of a Lie algebra...
applies to determine the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
. Among other properties, the universal enveloping algebra is an associative algebraAssociative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
into which
injectively imbeds. By Poincaré–Birkhoff–Witt, it is the free vector spaceFree module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
generated by the monomials
where the exponents are all non-negative.
Thus
consists of real polynomialswith the commutation relations
The algebra
is closely related to the algebra of differential operators on Rn with polynomial coefficients, since any such operator has a unique representation in the formThis algebra is called the Weyl algebra. It follows from abstract nonsense
Abstract nonsense
In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used facetiously by some mathematicians to describe certain kinds of arguments and methods related to category theory. roughly speaking, category theory is the study of the general form of mathematical...
that the Weyl algebra Wn is a quotient of
. However, this also easy to see directly from the above representations; viz. by the mappingRepresentation theory
The representation theoryRepresentation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
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
Stone–von Neumann theorem
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...
, which, informally stated, says that (with certain technical assumptions) every representation of the Heisenberg group
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
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
, its representation theory can be studied in terms of ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, via ergodic actions of the group, as in the work of George Mackey
George Mackey
George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...
.
Weyl's view of quantum mechanics
The application that led Hermann WeylHermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
to an explicit introduction of the Heisenberg group was the question of why the Schrödinger picture
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
and Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem
Stone–von Neumann theorem
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...
: there is a unique unitary representation
Unitary representation
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 is a unitary operator for every g ∈ G...
with given action of the central element z, up to a unitary equivalence: they are all equivalent to the position and momentum operators. Thus the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this unique representation.
Theta representation
The same uniqueness result was used by David MumfordDavid Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
for discrete Heisenberg groups, in his theory of equations defining abelian varieties
Equations defining abelian varieties
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety...
. This is a large generalization of the approach used in Jacobi's elliptic functions
Jacobi's elliptic functions
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...
, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is the theta representation
Theta representation
In mathematics, the theta representation is a particular representation of the Heisenberg group of quantum mechanics. It gains its name from the fact that the Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group...
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 theoremStone–von Neumann theorem
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 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 manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, and specifically, a simple example of a sub-Riemannian manifold
Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces....
. Given a point p=(x,y,z) in R3, define a differential 1-form Θ at this point as
This one-form
One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
belongs to the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
of R3; that is,
is a map on the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
. Let
It can be seen that H is a subbundle
Subbundle
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....
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
and in TR3, the inner product is given byThe resulting structure turns H into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s
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 geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s on the manifold are spirals, projecting down to circles in two dimensions. That is, if
is a geodesic curve, then the curve
is an arc of a circle, andwith 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
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
.