Stone–von Neumann theorem
Encyclopedia
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. The name is for Marshall Stone and .
, physical observable
s are represented mathematically by linear operators on Hilbert space
s.
For a single particle moving on the real line R, there are two important observables: position and momentum
. In the quantum-mechanical description of such a particle, the position operator
x and momentum operator
p are respectively given by
on the domain V of infinitely differentiable functions of compact support on R. Assume ħ to be a fixed non-zero real number — in quantum theory ħ is (up to a factor of 2π) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of units of the macroscopic world.
The operators x, p satisfy the canonical commutation relation
Lie algebra,
Already in his classic book, Hermann Weyl
observed that this commutation law was impossible to satisfy for linear operators P, Q acting on finite-dimensional spaces (as is clear by taking the trace of a matrix), unless ħ vanishes. Some analysis shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded. For notational convenience, the nonvanishing square root of ħ may be absorbed into the normalization of Q and P, so that, effectively, it amounts to 1 below. By exponentiating these operators, however, he showed that he could obtain braiding relations for the exponential operators U and V, (which, incidentally, may also be effectively realized on finite-dimensional spaces, through Sylvester
's celebrated clock and shift matrices).
Let Q and P be two self-adjoint operators satisfying the canonical commutation relation, and eit Q and eis P be the corresponding unitary groups given by functional calculus
. A formal computation with power series (degenerate Baker–Campbell–Hausdorff formula) shows that
Conversely, given two one parameter unitary groups U(t) and V(s) satisfying the relation
formally differentiating at 0 shows that the two infinitesmal generators satisfy the canonical commutation relation. These formal calculations can be made rigorous.
Therefore, there is a one-to-one correspondence between representations of the canonical commutation relation and two one parameter unitary groups U(t) and V(s) satisfying (*). This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the Weyl form of the CCR.
The problem thus becomes classifying two jointly irreducible one-parameter unitary groups U(t) and V(s) which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the Stone–von Neumann theorem: all such pairs of one-parameter unitary groups are unitarily equivalent. In other words, for any two such U(t) and V(s) acting jointly irreducibly on a Hilbert space H, there is a unitary operator
so that
where P and Q are the position and momentum operators from above.
Historically, this result was significant, because it was a key step in proving that Heisenberg
's matrix mechanics
, which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrödinger
's wave mechanical formulation (see Schrödinger picture
).
Informally stated, 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.
This was later generalized by Mackey theory – and was the motivation for the introduction of the Heisenberg group in quantum physics.
In detail:
These are thus all semidirect product
, and hence relatively easily understood.
In all cases, if one has a representation
where the center maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is Fourier theory.
If the center does not map to zero, one has a more interesting theory,
particularly if one restricts oneself to central representations.
Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, Planck's constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).
More formally, the group algebra
of the Heisenberg group
has center 
so rather than simply thinking of the group algebra as an algebra over the field of scalars K, one may think of it as an algebra over the commutative algebra 
As the center of a matrix algebra or operator algebra is the scalar matrices, a 
-structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of 
-algebras 
which is the formal way of saying that it sends the center to a chosen scale.
Then the Stone–von Neumann theorem is that, given a quantization value, every strongly continuous unitary representation is unitarily equivalent to the standard representation as position and momentum.
defined by
extends to a C*-isomorphism from the group C*-algebra
C*(G) of G and C0(G^), i.e. the spectrum of C*(G) is precisely G^. When G is the real line R, this is Stone's theorem characterizing one parameter unitary groups. The theorem of Stone-von Neumann can also be restated using similar language.
The group G acts on the C*-algebra C0(G) by right translation ρ: for s in G and f in C0(G),
Under the isomorphism given above, this action becomes the natural action of G on C*(G^):
So a covariant representation corresponding to the C*-crossed product
is a unitary representation U(s) of G and V(γ) of G^ such that
It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all irreducible representations of
are unitarily equivalent to the K(L(2G)), the compact operators
on L(2G)). Therefore all pairs {U(s), V(γ)} are unitarily equivalent. Specializing to the case where G = R yields the Stone–von Neumann theorem.
of general Heisenberg group Hn for n a positive integer. This is the Lie group
of (n+2) × (n+2) square matrices of the form
In fact, using the Heisenberg group, we can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the center of Hn consists of matrices M(0, 0, c).
Theorem. For each non-zero real number h there is an irreducible representation Uh acting on the Hilbert space L2(Rn)
by
All these representations are and any irreducible representation which is not trivial on the center of Hn is unitarily equivalent to exactly one of these.
Note that Uh is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by h a and multiplication by a function of absolute value
1. To show Uh is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness which is beyond the scope of the article. However, below we sketch a proof of the corresponding Stone–von Neumann theorem for certain finite Heisenberg groups.
In particular, irreducible representations π, π' of the Heisenberg group Hn which are non-trivial on the center of Hn are unitarily equivalent if and only if π(z) = π'(z) for any z in the center of Hn.
One representation of the Heisenberg group that is important in number theory
and the theory of modular form
s is the theta representation
, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group.
is an automorphism
of Hn which is the identity on the center of Hn. In particular, the representations Uh and Uh α are unitarily equivalent. This means that there is a unitary operator
W on L2(Rn) such that for any g in Hn,
Moreover, by irreducibility of the representations Uh, it follows that up to
a scalar, such an operator W is unique (cf. Schur's lemma
).
Theorem. The operator W is, up to a scalar multiple, the Fourier transform
on L2(Rn).
This means that (ignoring the factor of (2 π)n/2 in the definition of the Fourier transform)
The previous theorem can actually be used to prove the unitary
nature of the Fourier transform, also known as the Plancherel theorem
. Moreover, note that
Theorem. The operator W1 such that
is the reflection operator
From this fact the Fourier inversion formula easily follows.
into the circle group T. Note that Hn(K) is finite with cardinality |K|2 n+1. For finite Heisenberg group Hn(K) one can give a simple proof of the
Stone–von Neumann theorem using simple properties of character function
s of representations. These properties follow from the orthogonality relations for characters of representations of finite groups.
For any non-zero h in K define the representation Uh on the finite-dimensional inner product space l2(Kn) by
Theorem. For a fixed non-zero h, the character function χ of Uh is given by:
It follows that
By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups Hn(Z/p Z), particularly:
was directed at obtaining a formulation of the theory of induced representation
s developed originally by Frobenius
for finite groups to the context of unitary representations of locally compact topological groups.
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...
and in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, 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
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
operators. The name is for Marshall Stone and .
Trying to represent the commutation relations
In quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, physical observable
Observable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s are represented mathematically by linear operators on Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s.
For a single particle moving on the real line R, there are two important observables: position and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
. In the quantum-mechanical description of such a particle, the position operator
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...
x and momentum operator
Momentum operator
In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...
p are respectively given by
on the domain V of infinitely differentiable functions of compact support on R. Assume ħ to be a fixed non-zero real number — in quantum theory ħ is (up to a factor of 2π) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of units of the macroscopic world.
The operators x, p satisfy the canonical commutation relation
Canonical commutation relation
In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...
Lie algebra,
Already in his classic book, Hermann Weyl
Hermann 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...
observed that this commutation law was impossible to satisfy for linear operators P, Q acting on finite-dimensional spaces (as is clear by taking the trace of a matrix), unless ħ vanishes. Some analysis shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded. For notational convenience, the nonvanishing square root of ħ may be absorbed into the normalization of Q and P, so that, effectively, it amounts to 1 below. By exponentiating these operators, however, he showed that he could obtain braiding relations for the exponential operators U and V, (which, incidentally, may also be effectively realized on finite-dimensional spaces, through Sylvester
James Joseph Sylvester
James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
's celebrated clock and shift matrices).
Uniqueness of representation
One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, up to unitary equivalence. By Stone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one parameter unitary groups.Let Q and P be two self-adjoint operators satisfying the canonical commutation relation, and eit Q and eis P be the corresponding unitary groups given by functional calculus
Functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
. A formal computation with power series (degenerate Baker–Campbell–Hausdorff formula) shows that
Conversely, given two one parameter unitary groups U(t) and V(s) satisfying the relation
formally differentiating at 0 shows that the two infinitesmal generators satisfy the canonical commutation relation. These formal calculations can be made rigorous.
Therefore, there is a one-to-one correspondence between representations of the canonical commutation relation and two one parameter unitary groups U(t) and V(s) satisfying (*). This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the Weyl form of the CCR.
The problem thus becomes classifying two jointly irreducible one-parameter unitary groups U(t) and V(s) which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the Stone–von Neumann theorem: all such pairs of one-parameter unitary groups are unitarily equivalent. In other words, for any two such U(t) and V(s) acting jointly irreducibly on a Hilbert space H, there is a unitary operator
so that
where P and Q are the position and momentum operators from above.
Historically, this result was significant, because it was a key step in proving that 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...
's matrix mechanics
Matrix mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps...
, which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
's wave mechanical formulation (see 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...
).
Representation theory formulation
In terms of representation theory, the Stone–von Neumann theorem classifies certain unitary representations of the Heisenberg group. This is discussed in more detail in the Heisenberg group section, below.Informally stated, 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.
This was later generalized by Mackey theory – and was the motivation for the introduction of the Heisenberg group in quantum physics.
In detail:
- The continuous Heisenberg group is a central extension of the abelian Lie group R2n by a copy of R,
- the corresponding Heisenberg algebra is a central extension of the abelian Lie algebra R2n (with trivial bracket) by a copy of R,
- the discrete Heisenberg group is a central extension of the free abelian group Z2n by a copy of Z, and
- the discrete Heisenberg group module p is a central extension of the free abelian p-group (Z/pZ)2n by a copy of Z/pZ.
These are thus all 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...
, and hence relatively easily understood.
In all cases, if one has a representation
where the center maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is Fourier theory.If the center does not map to zero, one has a more interesting theory,
particularly if one restricts oneself to central representations.
Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, Planck's constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).
More formally, the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
of the Heisenberg group
has center so rather than simply thinking of the group algebra as an algebra over the field of scalars K, one may think of it as an algebra over the commutative algebra As the center of a matrix algebra or operator algebra is the scalar matrices, a -structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of -algebras which is the formal way of saying that it sends the center to a chosen scale.Then the Stone–von Neumann theorem is that, given a quantization value, every strongly continuous unitary representation is unitarily equivalent to the standard representation as position and momentum.
Reformulation via Fourier transform
Let G be a locally compact abelian group and G^ be the Pontryagin dual of G. The Fourier-Plancherel transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
defined by
extends to a C*-isomorphism from the group C*-algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
C*(G) of G and C0(G^), i.e. the spectrum of C*(G) is precisely G^. When G is the real line R, this is Stone's theorem characterizing one parameter unitary groups. The theorem of Stone-von Neumann can also be restated using similar language.
The group G acts on the C*-algebra C0(G) by right translation ρ: for s in G and f in C0(G),
Under the isomorphism given above, this action becomes the natural action of G on C*(G^):
So a covariant representation corresponding to the C*-crossed product
Crossed product
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed productis a basic method of constructing a new von Neumann algebra froma von Neumann algebra acted on by a group. It is related to...
is a unitary representation U(s) of G and V(γ) of G^ such that
It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all irreducible representations of
are unitarily equivalent to the K(L(2G)), the compact operators
Compact operator on Hilbert space
In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
on L(2G)). Therefore all pairs {U(s), V(γ)} are unitarily equivalent. Specializing to the case where G = R yields the Stone–von Neumann theorem.
The Heisenberg group
The commutation relations for P, Q look very similar to the commutation relations that define the Lie algebraLie 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...
of general Heisenberg group Hn for n a positive integer. This is the 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...
of (n+2) × (n+2) square matrices of the form
In fact, using the Heisenberg group, we can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the center of Hn consists of matrices M(0, 0, c).
Theorem. For each non-zero real number h there is an irreducible representation Uh acting on the Hilbert space L2(Rn)
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
by
All these representations are and any irreducible representation which is not trivial on the center of Hn is unitarily equivalent to exactly one of these.
Note that Uh is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by h a and multiplication by a function of absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1. To show Uh is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness which is beyond the scope of the article. However, below we sketch a proof of the corresponding Stone–von Neumann theorem for certain finite Heisenberg groups.
In particular, irreducible representations π, π' of the Heisenberg group Hn which are non-trivial on the center of Hn are unitarily equivalent if and only if π(z) = π'(z) for any z in the center of Hn.
One representation of the Heisenberg group that is important in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and the theory of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s 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...
, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group.
Relation to the Fourier transform
For any non-zero h, the mappingis an automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of Hn which is the identity on the center of Hn. In particular, the representations Uh and Uh α are unitarily equivalent. This means that there is a unitary operator
W on L2(Rn) such that for any g in Hn,
Moreover, by irreducibility of the representations Uh, it follows that up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
a scalar, such an operator W is unique (cf. Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
).
Theorem. The operator W is, up to a scalar multiple, the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
on L2(Rn).
This means that (ignoring the factor of (2 π)n/2 in the definition of the Fourier transform)
The previous theorem can actually be used to prove the unitary
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
nature of the Fourier transform, also known as the Plancherel theorem
Plancherel theorem
In mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
. Moreover, note that
Theorem. The operator W1 such that
is the reflection operator
From this fact the Fourier inversion formula easily follows.
Representations of finite Heisenberg groups
The Heisenberg group Hn(K) is defined for any commutative ring K. In this section let us specialize to the field K = Z/p Z for p a prime. This field has the property that there is an imbedding ω of K as an additive groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
into the circle group T. Note that Hn(K) is finite with cardinality |K|2 n+1. For finite Heisenberg group Hn(K) one can give a simple proof of the
Stone–von Neumann theorem using simple properties of character function
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
s of representations. These properties follow from the orthogonality relations for characters of representations of finite groups.
For any non-zero h in K define the representation Uh on the finite-dimensional inner product space l2(Kn) by
Theorem. For a fixed non-zero h, the character function χ of Uh is given by:
It follows that
By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups Hn(Z/p Z), particularly:
- Irreducibility of Uh
- Pairwise inequivalence of all the representations Uh.
Generalizations
The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George MackeyGeorge 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...
was directed at obtaining a formulation of the theory of induced representation
Induced representation
In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...
s developed originally by Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
for finite groups to the context of unitary representations of locally compact topological groups.
See also
- Weyl quantizationWeyl quantizationIn mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
- CCR algebra
- Moyal productMoyal productIn mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
- Weyl algebra
- Stone's theorem on one-parameter unitary groupsStone's theorem on one-parameter unitary groupsIn mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators...
- Hille–Yosida theoremHille–Yosida theoremIn functional analysis, the Hille–Yosida theorem characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. It is sometimes stated for the special case of contraction semigroups, with the general case being called the Feller–Miyadera–Phillips...