Oscillator representation

In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself.

The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.

Historical overviewEdit

The mathematical formulation of quantum mechanics by Werner Heisenberg and Erwin Schrödinger was originally in terms of unbounded self-adjoint operators on a Hilbert space. The fundamental operators corresponding to position and momentum satisfy the Heisenberg commutation relations. Quadratic polynomials in these operators, which include the harmonic oscillator, are also closed under taking commutators.

A large amount of operator theory was developed in the 1920s and 1930s to provide a rigorous foundation for quantum mechanics. Part of the theory was formulated in terms of unitary groups of operators, largely through the contributions of Hermann Weyl, Marshall Stone and John von Neumann. In turn these results in mathematical physics were subsumed within mathematical analysis, starting with the 1933 lecture notes of Norbert Wiener, who used the heat kernel for the harmonic oscillator to derive the properties of the Fourier transform.

The uniqueness of the Heisenberg commutation relations, as formulated in the Stone–von Neumann theorem, was later interpreted within group representation theory, in particular the theory of induced representations initiated by George Mackey. The quadratic operators were understood in terms of a projective unitary representation of the group SU(1,1) and its Lie algebra. Irving Segal and David Shale generalized this construction to the symplectic group in finite and infinite dimensions—in physics, this is often referred to as bosonic quantization: it is constructed as the symmetric algebra of an infinite-dimensional space. Segal and Shale have also treated the case of fermionic quantization, which is constructed as the exterior algebra of an infinite-dimensional Hilbert space. In the special case of conformal field theory in 1+1 dimensions, the two versions become equivalent via the so-called "boson-fermion correspondence." Not only does this apply in analysis where there are unitary operators between bosonic and fermionic Hilbert spaces, but also in the mathematical theory of vertex operator algebras. Vertex operators themselves originally arose in the late 1960s in theoretical physics, particularly in string theory.

André Weil later extended the construction to p-adic Lie groups, showing how the ideas could be applied in number theory, in particular to give a group theoretic explanation of theta functions and quadratic reciprocity. Several physicists and mathematicians observed the heat kernel operators corresponding to the harmonic oscillator were associated to a complexification of SU(1,1): this was not the whole of SL(2,C), but instead a complex semigroup defined by a natural geometric condition. The representation theory of this semigroup, and its generalizations in finite and infinite dimensions, has applications both in mathematics and theoretical physics.[1]

Semigroups in SL(2,C)Edit

The group:

 

is a subgroup of Gc = SL(2,C), the group of complex 2 × 2 matrices with determinant 1. If G1 = SL(2,R) then

 

This follows since the corresponding Möbius transformation is the Cayley transform which carries the upper half plane onto the unit disk and the real line onto the unit circle.

The group SL(2,R) is generated as an abstract group by

 

and the subgroup of lower triangular matrices

 

Indeed, the orbit of the vector

 

under the subgroup generated by these matrices is easily seen to be the whole of R2 and the stabilizer of v in G1 lies in inside this subgroup.

The Lie algebra   of SU(1,1) consists of matrices

 

The period 2 automorphism σ of Gc

 

with

 

has fixed point subgroup G since

 

Similarly the same formula defines a period two automorphism σ of the Lie algebra   of Gc, the complex matrices with trace zero. A standard basis of   over C is given by

 

Thus for −1 ≤ m, n ≤ 1

 

There is a direct sum decomposition

 

where   is the +1 eigenspace of σ and   the –1 eigenspace.

The matrices X in   have the form

 

Note that

 

The cone C in   is defined by two conditions. The first is   By definition this condition is preserved under conjugation by G. Since G is connected it leaves the two components with x > 0 and x < 0 invariant. The second condition is  

The group Gc acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D. A semigroup H of Gc, first considered by Olshanskii (1981), can be defined by the geometric condition:

 

The semigroup can be described explicitly in terms of the cone C:[2]

 

In fact the matrix X can be conjugated by an element of G to the matrix

 

with

 

Since the Möbius transformation corresponding to exp Y sends z to e−2yz, it follows that the right hand side lies in the semigroup. Conversely if g lies in H it carries the closed unit disk onto a smaller closed disk in its interior. Conjugating by an element of G, the smaller disk can be taken to have centre 0. But then for appropriate y, the element   carries D onto itself so lies in G.

A similar argument shows that the closure of H, also a semigroup, is given by

 

From the above statement on conjugacy, it follows that

 

where

 

If

 

then

 

since the latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1. Hence H also contains

 

which gives the inverse matrix if the original matrix lies in SU(1,1).

A further result on conjugacy follows by noting that every element of H must fix a point in D, which by conjugation with an element of G can be taken to be 0. Then the element of H has the form

 

The set of such lower triangular matrices forms a subsemigroup H0 of H.

Since

 

every matrix in H0 is conjugate to a diagonal matrix by a matrix M in H0.

Similarly every one-parameter semigroup S(t) in H fixes the same point in D so is conjugate by an element of G to a one-parameter semigroup in H0.

It follows that there is a matrix M in H0 such that

 

with S0(t) diagonal. Similarly there is a matrix N in H0 such that

 

The semigroup H0 generates the subgroup L of complex lower triangular matrices with determinant 1 (given by the above formula with a ≠ 0). Its Lie algebra consists of matrices of the form

 

In particular the one parameter semigroup exp tZ lies in H0 for all t > 0 if and only if   and  

This follows from the criterion for H or directly from the formula

 

The exponential map is known not to be surjective in this case, even though it is surjective on the whole group L. This follows because the squaring operation is not surjective in H. Indeed, since the square of an element fixes 0 only if the original element fixes 0, it suffices to prove this in H0. Take α with |α| < 1 and

 

If a = α2 and

 

with

 

then the matrix

 

has no square root in H0. For a square root would have the form

 

On the other hand,

 

The closed semigroup   is maximal in SL(2,C): any larger semigroup must be the whole of SL(2,C).[3][4][5][6][7]

Using computations motivated by theoretical physics, Ferrara et al. (1973) introduced the semigroup  , defined through a set of inequalities. Without identification   as a compression semigroup, they established the maximality of  . Using the definition as a compression semigroup, maximality reduces to checking what happens when adding a new fractional transformation   to  . The idea of the proof depends on considering the positions of the two discs   and  . In the key cases, either one disc contains the other or they are disjoint. In the simplest cases,   is the inverse of a scaling transformation or  . In either case   and   generate an open neigbourhood of 1 and hence the whole of SL(2,C)

Later Lawson (1998) gave another more direct way to prove maximality by first showing that there is a g in S sending D onto the disk Dc, |z| > 1. In fact if   then there is a small disk D1 in D such that xD1 lies in Dc. Then for some h in H, D1 = hD. Similarly yxD1 = Dc for some y in H. So g = yxh lies in S and sends D onto Dc. It follows that g2 fixes the unit disc D so lies in SU(1,1). So g−1 lies in S. If t lies in H then tgD contains gD. Hence   So t−1 lies in S and therefore S contains an open neighbourhood of 1. Hence S = SL(2,C).

Exactly the same argument works for Möbius transformations on Rn and the open semigroup taking the closed unit sphere ||x|| ≤ 1 into the open unit sphere ||x|| < 1. The closure is a maximal proper semigroup in the group of all Möbius transformations. When n = 1, the closure corresponds to Möbius transformations of the real line taking the closed interval [–1,1] into itself.[8]

The semigroup H and its closure have a further piece of structure inherited from G, namely inversion on G extends to an antiautomorphism of H and its closure, which fixes the elements in exp C and its closure. For

 

the antiautomorphism is given by

 

and extends to an antiautomorphism of SL(2,C).

Similarly the antiautomorphism

 

leaves G1 invariant and fixes the elements in exp C1 and its closure, so it has analogous properties for the semigroup in G1.

Commutation relations of Heisenberg and WeylEdit

Let   be the space of Schwartz functions on R. It is dense in the Hilbert space L2(R) of square-integrable functions on R. Following the terminology of quantum mechanics, the "momentum" operator P and "position" operator Q are defined on   by

 

There operators satisfy the Heisenberg commutation relation

 

Both P and Q are self-adjoint for the inner product on   inherited from L2(R).

Two one parameter unitary groups U(s) and V(t) can be defined on   and L2(R) by

 

By definition

 

for  , so that formally

 

It is immediate from the definition that the one parameter groups U and V satisfy the Weyl commutation relation

 

The realization of U and V on L2(R) is called the Schrödinger representation.

Fourier transformEdit

The Fourier transform is defined on   by[9]

 

It defines a continuous map of   into itself for its natural topology.

Contour integration shows that the function

 

is its own Fourier transform.

On the other hand, integrating by parts or differentiating under the integral,

 

It follows that the operator on   defined by

 

commutes with both Q (and P). On the other hand,

 

and since

 

lies in  , it follows that

 

and hence

 

This implies the Fourier inversion formula:

 

and shows that the Fourier transform is an isomorphism of   onto itself.

By Fubini's theorem

 

When combined with the inversion formula this implies that the Fourier transform preserves the inner product

 

so defines an isometry of   onto itself.

By density it extends to a unitary operator on L2(R), as asserted by Plancherel's theorem.

Stone–von Neumann theoremEdit

Suppose U(s) and V(t) are one parameter unitary groups on a Hilbert space   satisfying the Weyl commutation relations

 

For   let[10][11]

 

and define a bounded operator on   by

 

Then

 

where

 

The operators T(F) have an important non-degeneracy property: the linear span of all vectors T(F)ξ is dense in  .

Indeed, if fds and gdt define probability measures with compact support, then the smeared operators

 

satisfy

 

and converge in the strong operator topology to the identity operator if the supports of the measures decrease to 0.

Since U(f)V(g) has the form T(F), non-degeneracy follows.

When   is the Schrödinger representation on L2(R), the operator T(F) is given by

 

It follows from this formula that U and V jointly act irreducibly on the Schrödinger representation since this is true for the operators given by kernels that are Schwartz functions.

Conversely given a representation of the Weyl commutation relations on  , it gives rise to a non-degenerate representation of the *-algebra of kernel operators. But all such representations are on an orthogonal direct sum of copies of L2(R) with the action on each copy as above. This is a straightforward generalisation of the elementary fact that the representations of the N × N matrices are on direct sums of the standard representation on CN. The proof using matrix units works equally well in infinite dimensions.

The one parameter unitary groups U and V leave each component invariant, inducing the standard action on the Schrödinger representation.

In particular this implies the Stone–von Neumann theorem: the Schrödinger representation is the unique irreducible representation of the Weyl commutation relations on a Hilbert space.

Oscillator representation of SL(2,R)Edit

Given U and V satisfying the Weyl commutation relations, define

 

Then

 

so that W defines a projective unitary representation of R2 with cocycle given by

 

where   and B is the symplectic form on R2 given by

 

By the Stone–von Neumann theorem, there is a unique irreducible representation corresponding to this cocycle.

It follows that if g is an automorphism of R2 preserving the form B, i.e. an element of SL(2,R), then there is a unitary π(g) on L2(R) satisfying the covariance relation

 

By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with |ζ| = 1, so that π defines a projective unitary representation of SL(2,R).

This can be established directly using only the irreducibility of the Schrödinger representation. Irreducibility was a direct consequence of the fact the operators

 

with K a Schwartz function correspond exactly to operators given by kernels with Schwartz functions.

These are dense in the space of Hilbert–Schmidt operators, which, since it contains the finite rank operators, acts irreducibly.

The existence of π can be proved using only the irreducibility of the Schrödinger representation. The operators are unique up to a sign with

 

so that the 2-cocycle for the projective representation of SL(2,R) takes values ±1.

In fact the group SL(2,R) is generated by matrices of the form

 

and it can be verified directly that the following operators satisfy the covariance relations above:

 

The generators gi satisfy the following Bruhat relations, which uniquely specify the group SL(2,R):[12]

 

It can be verified by direct calculation that these relations are satisfied up to a sign by the corresponding operators, which establishes that the cocycle takes values ±1.

There is a more conceptual explanation using an explicit construction of the metaplectic group as a double cover of SL(2,R).[13] SL(2,R) acts by Möbius transformations on the upper half plane H. Moreover, if

 

then

 

The function

 

satisfies the 1-cocycle relation

 

For each g, the function m(g,z) is non-vanishing on H and therefore has two possible holomorphic square roots. The metaplectic group is defined as the group

 

By definition it is a double cover of SL(2,R) and is connected. Multiplication is given by

 

where

 

Thus for an element g of the metaplectic group there is a uniquely determined function m(g,z)1/2 satisfying the 1-cocycle relation.

If  , then

 

lies in L2 and is called a coherent state.

These functions lie in a single orbit of SL(2,R) generated by

 

since for g in SL(2,R)

 

More specifically if g lies in Mp(2,R) then

 

Indeed, if this holds for g and h, it also holds for their product. On the other hand, the formula is easily checked if gt has the form gi and these are generators.

This defines an ordinary unitary representation of the metaplectic group.

The element (1,–1) acts as multiplication by –1 on L2(R), from which it follows that the cocycle on SL(2,R) takes only values ±1.

Maslov indexEdit

As explained in Lion & Vergne (1980), the 2-cocycle on SL(2,R) associated with the metaplectic representation, taking values ±1, is determined by the Maslov index.

Given three non-zero vectors u, v, w in the plane, their Maslov index   is defined as the signature of the quadratic form on R3 defined by

 

Properties of the Maslov index:

  • it depends on the one-dimensional subpaces spanned by the vectors
  • it is invariant under SL(2,R)
  • it is alternating in its arguments, i.e. its sign changes if two of the arguments are interchanged
  • it vanishes if two of the subspaces coincide
  • it takes the values –1, 0 and +1: if u and v satisfy B(u,v) = 1 and w = au + bv, then the Maslov index is zero is if ab = 0 and is otherwise equal to minus the sign of ab
  •  

Picking a non-zero vector u0, it follows that the function

 

defines a 2-cocycle on SL(2,R) with values in the eighth roots of unity.

A modification of the 2-cocycle can be used to define a 2-cocycle with values in ±1 connected with the metaplectic cocycle.[14]

In fact given non-zero vectors u, v in the plane, define f(u,v) to be

  • i times the sign of B(u,v) if u and v are not proportional
  • the sign of λ if u = λv.

If

 

then

 

The representatives π(g) in the metaplectic representation can be chosen so that

 

where the 2-cocycle ω is given by

 

with

 

Holomorphic Fock spaceEdit

Holomorphic Fock space (also known as the Segal–Bargmann space) is defined to be the vector space   of holomorphic functions f(z) on C with

 

finite. It has inner product

 

  is a Hilbert space with orthonormal basis

 

Moreover, the power series expansion of a holomorphic function in   gives its expansion with respect to this basis.[15] Thus for z in C

 

so that evaluation at z is gives a continuous linear functional on   In fact

 

where[16]

 

Thus in particular   is a reproducing kernel Hilbert space.

For f in   and z in C define

 

Then

 

so this gives a unitary representation of the Weyl commutation relations.[17] Now

 

It follows that the representation   is irreducible.

Indeed, any function orthogonal to all the Ea must vanish, so that their linear span is dense in  .

If P is an orthogonal projection commuting with W(z), let f = PE0. Then

 

The only holomorphic function satisfying this condition is the constant function. So

 

with λ = 0 or 1. Since E0 is cyclic, it follows that P = 0 or I.

By the Stone–von Neumann theorem there is a unitary operator   from L2(R) onto  , unique up to multiplication by a scalar, intertwining the two representations of the Weyl commutation relations. By Schur's lemma and the Gelfand–Naimark construction, the matrix coefficient of any vector determines the vector up to a scalar multiple. Since the matrix coefficients of F = E0 and f = H0 are equal, it follows that the unitary   is uniquely determined by the properties

 

and

 

Hence for f in L2(R)

 

so that

 

where

 

The operator   is called the Segal–Bargmann transform[18] and B is called the Bargmann kernel.[19]

The adjoint of   is given by the formula:

 

Fock modelEdit

The action of SU(1,1) on holomorphic Fock space was described by Bargmann (1970) and Itzykson (1967).

The metaplectic double cover of SU(1,1) can be constructed explicitly as pairs (g, γ) with

 

and

 

If g = g1g2, then

 

using the power series expansion of (1 + z)1/2 for |z| < 1.

The metaplectic representation is a unitary representation π(g, γ) of this group satisfying the covariance relations

 

where

 

Since   is a reproducing kernel Hilbert space, any bounded operator T on it corresponds to a kernel given by a power series of its two arguments. In fact if

 

and F in  , then

 

The covariance relations and analyticity of the kernel imply that for S = π(g, γ),

 

for some constant C. Direct calculation shows that

 

leads to an ordinary representation of the double cover.[20]

Coherent states can again be defined as the orbit of E0 under the metaplectic group.

For w complex, set

 

Then   if and only if |w| < 1. In particular F0 = 1 = E0. Moreover,

 

where

 

Similarly the functions zFw lie in   and form an orbit of the metaplectic group:

 

Since (Fw, E0) = 1, the matrix coefficient of the function E0 = 1 is given by[21]

 

Disk modelEdit

The projective representation of SL(2,R) on L2(R) or on   break up as a direct sum of two irreducible representations, corresponding to even and odd functions of x or z. The two representations can be realized on Hilbert spaces of holomorphic functions on the unit disk; or, using the Cayley transform, on the upper half plane.[22][23]

The even functions correspond to holomorphic functions F+ for which

 

is finite; and the odd functions to holomorphic functions F for which

 

is finite. The polarized forms of these expressions define the inner products.

The action of the metaplectic group is given by

 

Irreducibility of these representations is established in a standard way.[24] Each representation breaks up as a direct sum of one dimensional eigenspaces of the rotation group each of which is generated by a C vector for the whole group. It follows that any closed invariant subspace is generated by the algebraic direct sum of eigenspaces it contains and that this sum is invariant under the infinitesimal action of the Lie algebra  . On the other hand, that action is irreducible.

The isomorphism with even and odd functions in   can be proved using the Gelfand–Naimark construction since the matrix coefficients associated to 1 and z in the corresponding representations are proportional. Itzykson (1967) gave another method starting from the maps

 
 

from the even and odd parts to functions on the unit disk. These maps intertwine the actions of the metaplectic group given above and send zn to a multiple of wn. Stipulating that U± should be unitary determines the inner products on functions on the disk, which can expressed in the form above.[25]

Although in these representations the operator L0 has positive spectrum—the feature that distinguishes the holomorphic discrete series representations of SU(1,1)—the representations do not lie in the discrete series of the metaplectic group. Indeed, Kashiwara & Vergne (1978) noted that the matrix coefficients are not square integrable, although their third power is.[26]

Harmonic oscillator and Hermite functionsEdit

Consider the following subspace of L2(R):

 

The operators

 

act on   X is called the annihilation operator and Y the creation operator. They satisfy

 

Define the functions

 

We claim they are the eigenfunctions of the harmonic oscillator, D. To prove this we use the commutation relations above:

 

Next we have:

 

This is known for n = 0 and the commutation relation above yields

 

The nth Hermite function is defined by

 

pn is called the nth Hermite polynomial.

Let

 

Thus

 

The operators P, Q or equivalently A, A* act irreducibly on   by a standard argument.[27][28]

Indeed, under the unitary isomorphism with holomorphic Fock space   can be identified with C[z], the space of polynomials in z, with

 

If a subspace invariant under A and A* contains a non-zero polynomial p(z), then, applying a power of A*, it contains a non-zero constant; applying then a power of A, it contains all zn.

Under the isomorphism Fn is sent to a multiple of zn and the operator D is given by

 

Let

 

so that

 

In the terminology of physics A, A* give a single boson and L0 is the energy operator. It is diagonalizable with eigenvalues 0, 1/2, 1, 3/2, ...., each of multiplicity one. Such a representation is called a positive energy representation.

Moreover,

 

so that the Lie bracket with L0 defines a derivation of the Lie algebra spanned by A, A* and I. Adjoining L0 gives the semidirect product. The infinitesimal version of the Stone–von Neumann theorem states that the above representation on C[z] is the unique irreducible positive energy representation of this Lie algebra with L0 = A*A. For A lowers energy and A* raises energy. So any lowest energy vector v is annihilated by A and the module is exhausted by the powers of A* applied to v. It is thus a non-zero quotient of C[z] and hence can be identified with it by irreducibility.

Let

 

so that

 

These operators satisfy:

 

and act by derivations on the Lie algebra spanned by A, A* and I.

They are the infinitesimal operators corresponding to the metaplectic representation of SU(1,1).

The functions Fn are defined by

 

It follows that the Hermite functions are the orthonormal basis obtained by applying the Gram-Schmidt orthonormalization process to the basis xn exp -x2/2 of  .

The completeness of the Hermite functions follows from the fact that the Bargmann transform is unitary and carries the orthonormal basis en(z) of holomorphic Fock space onto the Hn(x).

The heat operator for the harmonic oscillator is the operator on L2(R) defined as the diagonal operator

 

It corresponds to the heat kernel given by Mehler's formula:

 

This follows from the formula

 

To prove this formula note that if s = σ2, then by Taylor's formula

 

Thus Fσ,x lies in holomorphic Fock space and

 

an inner product that can be computed directly.

Wiener (1933, pp. 51–67) establishes Mehler's formula directly and uses a classical argument to prove that

 

tends to f in L2(R) as t decreases to 0. This shows the completeness of the Hermite functions and also, since

 

can be used to derive the properties of the Fourier transform.

There are other elementary methods for proving the completeness of the Hermite functions, for example using Fourier series.[29]

Sobolev spacesEdit

The Sobolev spaces Hs, sometimes called Hermite-Sobolev spaces, are defined to be the completions of   with respect to the norms

 

where

 

is the expansion of f in Hermite functions.[30]

Thus

 

The Sobolev spaces are Hilbert spaces. Moreover, Hs and Hs are in duality under the pairing

 

For s ≥ 0,

 

for some positive constant Cs.

Indeed, such an inequality can be checked for creation and annihilation operators acting on Hermite functions Hn and this implies the general inequality.[31]

It follows for arbitrary s by duality.

Consequently, for a quadratic polynomial R in P and Q

 

The Sobolev inequality holds for f in Hs with s > 1/2:

 

for any k ≥ 0.

Indeed, the result for general k follows from the case k = 0 applied to Qkf.

For k = 0 the Fourier inversion formula

 

implies

 

If s < t, the diagonal form of D, shows that the inclusion of Ht in Hs is compact (Rellich's lemma).

It follows from Sobolev's inequality that the intersection of the spaces Hs is  . Functions in   are characterized by the rapid decay of their Hermite coefficients an.

Standard arguments show that each Sobolev space is invariant under the operators W(z) and the metaplectic group.[32] Indeed, it is enough to check invariance when g is sufficiently close to the identity. In that case

 

with D + A an isomorphism from   to  

It follows that

 

If   then

 

where the derivatives lie in  

Similarly the partial derivatives of total degree k of U(s)V(t)f lie in Sobolev spaces of order sk/2.

Consequently, a monomial in P and Q of order 2k applied to f lies in Hsk and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.

Smooth vectorsEdit

The smooth vectors for the Weyl commutation relations are those u in L2(R) such that the map

 

is smooth. By the uniform boundedness theorem, this is equivalent to the requirement that each matrix coefficient (W(z)u,v) be smooth.

A vector is smooth if and only it lies in  .[33] Sufficiency is clear. For necessity, smoothness implies that the partial derivatives of W(z)u lie in L2(R) and hence also Dku for all positive k. Hence u lies in the intersection of the Hk, so in  .

It follows that smooth vectors are also smooth for the metaplectic group.

Moreover, a vector is in   if and only if it is a smooth vector for the rotation subgroup of SU(1,1).

Analytic vectorsEdit

If Π(t) is a one parameter unitary group and for f in  

 

then the vectors Π(f)ξ form a dense set of smooth vectors for Π.

In fact taking

 

the vectors v = Π(fε)ξ converge to ξ as ε decreases to 0 and

 

is an analytic function of t that extends to an entire function on C.

The vector is called an entire vector for Π.

The wave operator associated to the harmonic oscillator is defined by

 

The operator is diagonal with the Hermite functions Hn as eigenfunctions:

 

Since it commutes with D, it preserves the Sobolev spaces.

The analytic vectors constructed above can be rewritten in terms of the Hermite semigroup as

 

The fact that v is an entire vector for Π is equivalent to the summability condition

 

for all r > 0.

Any such vector is also an entire vector for U(s)V(t), that is the map

 

defined on R2 extends to an analytic map on C2.

This reduces to the power series estimate

 

So these form a dense set of entire vectors for U(s)V(t); this can also be checked directly using Mehler's formula.

The spaces of smooth and entire vectors for U(s)V(t) are each by definition invariant under the action of the metaplectic group as well as the Hermite semigroup.

Let

 

be the analytic continuation of the operators W(x,y) from R2 to C2 such that

 

Then W leaves the space of entire vectors invariant and satisfies

 

Moreover, for g in SL(2,R)

 

using the natural action of SL(2,R) on C2.

Formally

 

Oscillator semigroupEdit

There is a natural double cover of the Olshanski semigroup H, and its closure   that extends the double cover of SU(1,1) corresponding to the metaplectic group. It is given by pairs (g, γ) where g is an element of H or its closure

 

and γ is a square root of a.

Such a choice determines a unique branch of

 

for |z| < 1.

The unitary operators π(g) for g in SL(2,R) satisfy

 

for u in C2.

An element g of the complexification SL(2,C) is said to implementable if there is a bounded operator T such that it and its adjoint leave the space of entire vectors for W invariant, both have dense images and satisfy the covariance relations

 

for u in C2. The implementing operator T is uniquely determined up to multiplication by a non-zero scalar.

The implementable elements form a semigroup, containing SL(2,R). Since the representation has positive energy, the bounded compact self-adjoint operators

 

for t > 0 implement the group elements in exp C1.

It follows that all elements of the Olshanski semigroup and its closure are implemented.

Maximality of the Olshanki semigroup implies that no other elements of SL(2,C) are implemented. Indeed, otherwise every element of SL(2,C) would be implemented by a bounded operator, which would condradict the non-invertibility of the operators S0(t) for t > 0.

In the Schrödinger representation the operators S0(t) for t > 0 are given by Mehler's formula. They are contraction operators, positive and in every Schatten class. Moreover, they leave invariant each of the Sobolev spaces. The same formula is true for   by analytic continuation.

It can be seen directly in the Fock model that the implementing operators can be chosen so that they define an ordinary representation of the double cover of H constructed above. The corresponding semigroup of contraction operators is called the oscillator semigroup. The extended oscillator semigroup is obtained by taking the semidirect product with the operators W(u). These operators lie in every Schatten class and leave invariant the Sobolev spaces and the space of entire vectors for W.

The decomposition

 

corresponds at the operator level to the polar decomposition of bounded operators.

Moreover, since any matrix in H is conjugate to a diagonal matrix by elements in H or H−1, every operator in the oscillator semigroup is quasi-similar to an operator S0(t) with  . In particular it has the same spectrum consisting of simple eigenvalues.

In the Fock model, if the element g of the Olshanki semigroup H corresponds to the matrix

 

the corresponding operator is given by

 

where

 

and γ is a square root of a. Operators π(g,γ) for g in the semigroup H are exactly those that are Hilbert–Schmidt operators and correspond to kernels of the form

 

for which the complex symmetric matrix

 

has operator norm strictly less than one.

Operators in the extended oscillator semigroup are given by similar expressions with additional linear terms in z and w appearing in the exponential.

In the disk model for the two irreducible components of the metaplectic representation, the corresponding operators are given by

 

It is also possible to give an explicit formula for the contraction operators corresponding to g in H in the Schrödinger representation, It was by this formula that Howe (1988) introduced the oscillator semigroup as an explicit family of operators on L2(R).[34]

In fact consider the Siegel upper half plane consisting of symmetric complex 2x2 matrices with positive definite real part:

 

and define the kernel

 

with corresponding operator

 

for f in L2(R).

Then direct computation gives

 

where

 

Moreover,

 

where

 

By Mehler's formula for  

 

with

 

The oscillator semigroup is obtained by taking only matrices with B ≠ 0. From the above, this condition is closed under composition.

A normalized operator can be defined by

 

The choice of a square root determines a double cover.

In this case SZ corresponds to the element

 

of the Olshankii semigroup H.

Moreover, SZ is a strict contraction:

 

It follows also that

 

Weyl calculusEdit

For a function a(x,y) on R2 = C, let

 

So

 

where

 

Defining in general

 

the product of two such operators is given by the formula

 

where the twisted convolution or Moyal product is given by

 

The smoothing operators correspond to W(F) or ψ(a) with F or a Schwartz functions on R2. The corresponding operators T have kernels that are Schwartz functions. They carry each Sobolev space into the Schwartz functions. Moreover, every bounded operator on L2 (R) having this property has this form.

For the operators ψ(a) the Moyal product translates into the Weyl symbolic calculus. Indeed, if the Fourier transforms of a and b have compact support than

 

where

 

This follows because in this case b must extend to an entire function on C2 by the Paley-Wiener theorem.

This calculus can be extended to a broad class of symbols, but the simplest corresponds to convolution by a class of functions or distributions that all have the form T + S where T is a distribution of compact with singular support concentrated at 0 and where S is a Schwartz function. This class contains the operators P, Q as well as D1/2 and D−1/2 where D is the harmonic oscillator.

The mth order symbols Sm are given by smooth functions a satisfying

 

for all α and Ψm consists of all operators ψ(a) for such a.

If a is in Sm and χ is a smooth function of compact support equal to 1 near 0, then

 

with T and S as above.

These operators preserve the Schwartz functions and satisfy;

 

The operators P and Q lie in Ψ1 and D lies in Ψ2.

Properties:

  • A zeroth order symbol defines a bounded operator on L2(R).
  • D−1 lies in Ψ−2
  • If R = R* is smoothing, then D + R has a complete set of eigenvectors fn in   with (D + R)fn = λnfn and λn tends to ≈ as n tends to ≈.
  • D1/2 lies in Ψ1 and hence D−1/2 lies in Ψ−1, since D−1/2 = D1/2 ·D−1
  • Ψ−1 consists of compact operators, Ψs consists of trace-class operators for s > 1 and Ψk carries Hm into Hmk.
  •  

The proof of boundedness of Howe (1980) is particularly simple: if

 

then

 

where the bracketed operator has norm less than  . So if F is supported in |z| ≤ R, then

 

The property of D−1 is proved by taking

 

with

 

Then R = IDS lies in Ψ−1, so that

 

lies in Ψ−2 and T = DAI is smoothing. Hence

 

lies in Ψ−2 since D−1 T is smoothing.

The property for D1/2 is established similarly by constructing B in Ψ1/2 with real symbol such that DB4 is a smoothing operator. Using the holomorphic functional calculus it can be checked that D1/2B2 is a smoothing operator.

The boundedness result above was used by Howe (1980) to establish the more general inequality of Alberto Calderón and Remi Vaillancourt for pseudodifferential operators. An alternative proof that applies more generally to Fourier integral operators was given by Howe (1988). He showed that such operators can be expressed as integrals over the oscillator semigroup and then estimated using the Cotlar-Stein lemma.[35]

Applications and generalizationsEdit

Theory for finite abelian groupsEdit

Weil (1964) noted that the formalism of the Stone–von Neumann theorem and the oscillator representation of the symplectic group extends from the real numbers R to any locally compact abelian group. A particularly simple example is provided by finite abelian groups, where the proofs are either elementary or simplifications of the proofs for R.[36][37]

Let A be a finite abelian group, written additively, and let Q be a non-degenerate quadratic form on A with values in T. Thus

 

is a symmetric bilinear form on A that is non-degenerate, so permits an identification between A and its dual group A* = Hom (A, T).

Let   be the space of complex-valued functions on A with inner product

 

Define operators on V by

 

for x, y in A. Then U(x) and V(y) are unitary representations of A on V satisfying the commutation relations

 

This action is irreducible and is the unique such irreducible representation of these relations.

Let G = A × A and for z = (x, y) in G set

 

Then

 

where

 

a non-degenerate alternating bilinear form on G. The uniqueness result above implies that if W'(z) is another family of unitaries giving a projective representation of G such that

 

then there is a unitary U, unique up to a phase, such that

 

for some λ(z) in T.

In particular if g is an automorphism of G preserving B, then there is an essentially unique unitary π(g) such that

 

The group of all such automorphisms is called the symplectic group for B and π gives a projective representation of G on V.

The group SL(2.Z) naturally acts on G = A x A by symplectic automorphisms. It is generated by the matrices

 

If Z = –I, then Z is central and

 

These automorphisms of G are implemented on V by the following operators:

 

It follows that

 

where μ lies in T. Direct calculation shows that μ is given by the Gauss sum

 

Transformation laws for theta functionsEdit

The metaplectic group was defined as the group

 

The coherent state

 

defines a holomorphic map of H into L2(R) satisfying

 

This is in fact a holomorphic map into each Sobolev space Hk and hence also  .

On the other hand, in   (in fact in H–1) there is a finite-dimensional space of distributions invariant under SL(2,Z) and isomorphic to the N-dimensional oscillator representation on   where A = Z/NZ.

In fact let m > 0 and set N = 2m. Let

 

The operators U(x), V(y) with x and y in M all commute and have a finite-dimensional subspace of fixed vectors formed by the distributions

 

with b in M1, where

 

The sum defining Ψb converges in   and depends only on the class of b in M1/M. On the other hand, the operators U(x) and V(y) with 'x, y in M1 commute with all the corresponding operators for M. So M1 leaves the subspace V0 spanned by the Ψb invariant. Hence the group A = M1 acts on V0. This action can immediately be identified with the action on V for the N-dimensional oscillator representation associated with A, since

 

Since the operators π(R) and π(S) normalise the two sets of operators U and V corresponding to M and M1, it follows that they leave V0 invariant and on V0 must be constant multiples of the operators associated with the oscillator representation of A. In fact they coincide. From R this is immediate from the definitions, which show that

 

For S it follows from the Poisson summation formula and the commutation properties with the operators U)x) and V(y). The Poisson summation is proved classically as follows.[38]

For a > 0 and f in   let

 

F is a smooth function on R with period a:

 

The theory of Fourier series shows that

 

with the sum absolutely convergent and the Fourier coefficients given by

 

Hence

 

the usual Poisson summation formula.

This formula shows that S acts as follows

 

and so agrees exactly with formula for the oscillator representation on A.

Identifying A with Z/2mZ, with

 

assigned to an integer n modulo 2m, the theta functions can be defined directly as matrix coefficients:[39]

 

For τ in H and z in C set

 

so that |q| < 1. The theta functions agree with the standard classical formulas for the Jacobi-Riemann theta functions:

 

By definition they define holomorphic functions on H × C. The covariance properties of the function fτ and the distribution Ψb lead immediately to the following transformation laws:

 

Derivation of law of quadratic reciprocityEdit

Because the operators π(S), π (R) and π(J) on L2(R) restrict to the corresponding operators on V0 for any choice of m, signs of cocycles can be determined by taking m = 1. In this case the representation is 2-dimensional and the relation

 

on L2(R) can be checked directly on V0.

But in this case

 

The relation can also be checked directly by applying both sides to the ground state exp -x2/2.

Consequently, it follows that for m ≥ 1 the Gauss sum can be evaluated:[40]

 

For m odd, define

 

If m is odd, then, splitting the previous sum up into two parts, it follows that G(1,m) equals m1/2 if m is congruent to 1 mod 4 and equals i m1/2 otherwise. If p is an odd prime and c is not divisible by p, this implies

 

where   is the Legendre symbol equal to 1 if c is a square mod p and –1 otherwise. Moreover, if p and q are distinct odd primes, then

 

From the formula for G(1,p) and this relation, the law of quadratic reciprocity follows:

 

Theory in higher dimensionsEdit

The theory of the oscillator representation can be extended from R to Rn with the group SL(2,R) replaced by the symplectic group Sp(2n,R). The results can be proved either by straightforward generalisations from the one-dimensional case as in Folland (1989) or by using the fact that the n-dimensional case is a tensor product of n one-dimensional cases, reflecting the decomposition:

 

Let   be the space of Schwartz functions on Rn, a dense subspace of L2(Rn). For s, t in Rn, define U(s) and V(t) on   and L2(R) by

 

From the definition U and V satisfy the Weyl commutation relation

 

As before this is called the Schrödinger representation.

The Fourier transform is defined on   by

 

The Fourier inversion formula

 

shows that the Fourier transform is an isomorphism of   onto itself extending to a unitary mapping of L2(Rn) onto itself (Plancherel's theorem).

The Stone–von Neumann theorem asserts that the Schrödinger representation is irreducible and is the unique irreducible representation of the commutation relations: any other representation is a direct sum of copies of this representation.

If U and V satisfying the Weyl commutation relations, define

 

Then

 

so that W defines a projective unitary representation of R2n with cocycle given by

 

where   and B is the symplectic form on R2n given by

 

The symplectic group Sp (2n,R) is defined to be group of automorphisms g of R2n preserving the form B. It follows from the Stone–von Neumann theorem that for each such g there is a unitary π(g) on L2(R) satisfying the covariance relation

 

By Schur's lemma the unitary π(g) is unique up to multiplication by a scalar ζ with |ζ| = 1, so that π defines a projective unitary representation of Sp(n). Representatives can be chosen for π(g), unique up to a sign, which show that the 2-cocycle for the projective representation of Sp(2n,R) takes values ±1. In fact elements of the group Sp(n,R) are given by 2n × 2n real matrices g satisfying

 

where

 

Sp(2n,R) is generated by matrices of the form

 

and the operators

 

satisfy the covariance relations above. This gives an ordinary unitary representation of the metaplectic group, a double cover of Sp(2n,R). Indeed, Sp(n,R) acts by Möbius transformations on the generalised Siegel upper half plane Hn consisting of symmetric complex n × n matrices Z with strictly imaginary part by

 

if

 

The function

 

satisfies the 1-cocycle relation

 

The metaplectic group Mp(2n,R) is defined as the group

 

and is a connected double covering group of Sp(2n,R).

If  , then it defines a coherent state

 

in L2, lying in a single orbit of Sp(2n) generated by

 

If g lies in Mp(2n,R) then

 

defines an ordinary unitary representation of the metaplectic group, from which it follows that the cocycle on Sp(2n,R) takes only values ±1.

Holomorphic Fock space is the Hilbert space   of holomorphic functions f(z) on Cn with finite norm

 

inner product

 

and orthonormal basis

 

for α a multinomial. For f in   and z in Cn, the operators

 

define an irreducible unitary representation of the Weyl commutation relations. By the Stone–von Neumann theorem there is a unitary operator   from L2(Rn) onto   intertwining the two representations. It is given by the Bargmann transform

 

where

 

Its adjoint   is given by the formula:

 

Sobolev spaces, smooth and analytic vectors can be defined as in the one-dimensional case using the sum of n copies of the harmonic oscillator

 

The Weyl calculus similarly extends to the n-dimensional case.

The complexification Sp(2n,C) of the symplectic group is defined by the same relation, but allowing the matrices A, B, C and D to be complex. The subsemigroup of group elements that take the Siegel upper half plane into itself has a natural double cover. The representations of Mp(2n,R) on L2(Rn) and   extend naturally to a representation of this semigroup by contraction operators defined by kernels, which generalise the one-dimensional case (taking determinants where necessary). The action of Mp(2n,R) on coherent states applies equally well to operators in this larger semigroup.[41]

As in the 1-dimensional case, where the group SL(2,R) has a counterpart SU(1,1) threough the Cayley transform with the upper half plane replaced by the unit disc, the symplectic group has a complex counterpart. Indeed, if C is the unitary matrix

 

then C Sp(2n) C−1 is the group of all matrices

 

such that

 

or equivalently

 

where

 

The Siegel generalized disk Dn is defined as the set of complex symmetric n x n matrices W with operator norm less than 1.

It consist precisely of Cayley transforms of points Z in the Siegel generalized upper half plane:

 

Elements g act on Dn

 

and, as in the one dimensional case this action is transitive. The stabilizer subgroup of 0 consists of matrices with A unitary and B = 0.

For W in Dn the metaplectic coherent states in holomorphic Fock space are defined by

 

The inner product of two such states is given by

 

Moreover, the metaplectic representation π satisfies

 

The closed linear span of these states gives the even part of holomorphic Fock space  . The embedding of Sp(2n) in Sp(2(n+1)) and the compatible identification

 

lead to an action on the whole of  . It can be verified directly that it is compatible with the action of the operators W(z).[42]

Since the complex semigroup has as Shilov boundary the symplectic group, the fact that this representation has a well-defined contractive extension to the semigroup follows from the maximum modulus principle and the fact that the semigroup operators are closed under adjoints. Indeed, it suffices to check, for two such operators S, T and vectors vi proportional to metaplectic coherent states, that

 

which follows because the sum depends holomorphically on S and T, which are unitary on the boundary.

Index theorems for Toeplitz operatorsEdit

Let S denote the unit sphere in Cn and define the Hardy space H2(S) be the closure in L2(S) of the restriction of polynomials in the coordinates z1, ..., zn. Let P be the projection onto Hardy space. It is known that if m(f) denotes multiplication by a continuous function f on S, then the commutator [P,m(f)] is compact. Consequently, defining the Toeplitz operator by

 

on Hardy space, it follows that T(fg) – T(f)T(g) is compact for continuous f and g. The same holds if f and g are matrix-valued functions (so that the corresponding Toeplitz operators are matrices of operators on H2(S)). In particular if f is a function on S taking values in invertible matrices, then

 

are compact and hence T(f) is a Fredholm operator with an index defined as

 

The index has been computed using the methods of K-theory by Coburn (1973) and coincides up to a sign with the degree of f as a continuous mapping from S into the general linear group.

Helton & Howe (1975) gave an analytic way to establish this index theorem, simplied later by Howe. Their proof relies on the fact if f is smooth then the index is given by the formula of McKean and Singer:[43]

 

Howe (1980) noticed that there was a natural unitary isomorphism between H2(S) and L2(Rn) carrying the Toeplitz operators

 

onto the operators

 

These are examples of zeroth order operators constructed within the Weyl calculus. The traces in the McKean-Singer formula can be computed directly using the Weyl calculus, leading to another proof of the index theorem.[44] This method of proving index theorems was generalised by Alain Connes within the framework of cyclic cohomology.[45]

Theory in infinite dimensionsEdit

The theory of the oscillator representation in infinite dimensions is due to Irving Segal and David Shale.[46] Graeme Segal used it to give a mathematically rigorous construction of projective representations of loop groups and the group of diffeomorphisms of the circle. At an infinitesimal level the construction of the representations of the Lie algebras, in this case the affine Kac–Moody algebra and the Virasoro algebra, was already known to physicists, through dual resonance theory and later string theory. Only the simplest case will be considered here, involving the loop group LU(1) of smooth maps of the circle into U(1) = T. The oscillator semigroup, developed independently by Neretin and Segal, allows contraction operators to be defined for the semigroup of univalent holomorphic maps of the unit disc into itself, extending the unitary operators corresponding to diffeomorphisms of the circle. When applied to the subgroup SU(1,1) of the diffeomorphism group, this gives a generalization of the oscillator representation on L2(R) and its extension to the Olshanskii semigroup.

The representation of commutation on Fock space is generalized to infinite dimensions by replacing Cn (or its dual space) by an arbitrary complex Hilbert space H. The symmetric group Sk acts on Hk. Sk(H) is d