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 overview.
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.
Semigroups in SL(2,C).
The group:
is a subgroup of "G""c" = SL(2,C), the group of complex 2 × 2 matrices with determinant 1. If "G"1 = 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 "G"1 lies in inside this subgroup.
The Lie algebra formula_6 of SU(1,1) consists of matrices
The period 2 automorphism σ of "G""c"
with
has fixed point subgroup "G" since
Similarly the same formula defines a period two automorphism σ of the Lie algebra formula_11 of "Gc", the complex matrices with trace zero. A standard basis of formula_11 over C is given by
Thus for −1 ≤ "m", "n" ≤ 1
There is a direct sum decomposition
where formula_6 is the +1 eigenspace of σ and formula_17 the –1 eigenspace.
The matrices "X" in formula_17 have the form
Note that
The cone "C" in formula_17 is defined by two conditions. The first is formula_22 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 formula_23
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 , can be defined by the geometric condition: 
The semigroup can be described explicitly in terms of the cone "C":
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"−2"y""z", 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 formula_28 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 "H"0 of "H".
Since
every matrix in "H"0 is conjugate to a diagonal matrix by a matrix "M" in "H"0.
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 "H"0.
It follows that there is a matrix "M" in "H"0 such that
with "S"0("t") diagonal. Similarly there is a matrix "N" in "H"0 such that
The semigroup "H"0 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 "H"0 for all "t" > 0 if and only if formula_40 and formula_41
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 "H"0. Take α with |α| < 1 and
If "a" = α2 and
with
for all α and Ψ"m" consists of all operators ψ("a") for such "a".
If "a" is in "S""m" 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:
The proof of boundedness of is particularly simple: if
then
where the bracketed operator has norm less than formula_52. So if "F" is supported in |"z"| ≤ "R", then
The property of "D"−1 is proved by taking
with
Then "R" = "I" – "DS" lies in Ψ−1, so that
lies in Ψ−2 and "T" = "DA" – "I" is smoothing. Hence
lies in Ψ−2 since "D"−1 "T" is smoothing.
The property for "D"1/2 is established similarly by constructing "B" in Ψ1/2 with real symbol such that "D" – "B"4 is a smoothing operator. Using the holomorphic functional calculus it can be checked that "D"1/2 – "B"2 is a smoothing operator.
The boundedness result above was used by 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 . He showed that such operators can be expressed as integrals over the oscillator semigroup and then estimated using the Cotlar-Stein lemma.
Applications and generalizations.
Theory for finite abelian groups.
 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.
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 formula_59 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 functions.
The metaplectic group was defined as the group
The coherent state
defines a holomorphic map of H into "L"2(R) satisfying
This is in fact a holomorphic map into each Sobolev space "Hk" and hence also formula_77.
On the other hand, in formula_78 (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 formula_79 where "A" = Z/"N"Z.
In fact let "m" > 0 and set "N" = 2"m". 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 "M"1, where
The sum defining Ψ"b" converges in formula_83 and depends only on the class of "b" in "M"1/"M". On the other hand, the operators "U"("x") and "V"("y") with "'x", "y" in "M"1 commute with all the corresponding operators for "M". So "M"1 leaves the subspace "V"0 spanned by the Ψ"b" invariant. Hence the group "A" = "M"1 acts on "V"0. 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 "M"1, it follows that they leave "V"0 invariant and on "V"0 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.
For "a" > 0 and "f" in formula_48 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/2"m"Z, with
assigned to an integer "n" modulo 2"m", the theta functions can be defined directly as matrix coefficients:
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 "f"τ and the distribution Ψ"b" lead immediately to the following transformation laws:
Derivation of law of quadratic reciprocity.
Because the operators π("S"), π ("R") and π("J") on "L"2(R) restrict to the corresponding operators on "V"0 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 "L"2(R) can be checked directly on "V"0.
But in this case
The relation can also be checked directly by applying both sides to the ground state exp -"x"2/2.
Consequently, it follows that for "m" ≥ 1 the Gauss sum can be evaluated:
For "m " odd, define
If "m" is odd, then, splitting the previous sum up into two parts, it follows that "G"(1,"m") equals "m"1/2 if "m" is congruent to 1 mod 4 and equals "i" "m"1/2 otherwise. If "p" is an odd prime and "c" is not divisible by "p", this implies
where formula_103 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 dimensions.
The theory of the oscillator representation can be extended from R to R"n" 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 or by using the fact that the "n"-dimensional case is a tensor product of "n" one-dimensional cases, reflecting the decomposition:
Let formula_107 be the space of Schwartz functions on R"n", a dense subspace of "L"2(R"n"). For "s", "t" in R"n", define "U"("s") and "V"("t") on formula_107 and "L"2(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 formula_107 by
satisfy the covariance relations above. This gives an ordinary unitary representation of the metaplectic group, a double cover of Sp(2"n",R). Indeed, Sp("n",R) acts by Möbius transformations on the generalised Siegel upper half plane H"n" 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(2"n",R) is defined as the group
and is a connected double covering group of Sp(2"n",R).
If formula_118, then it defines a coherent state
in "L"2, lying in a single orbit of Sp(2"n") 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(2"n",R) takes only values ±1.
Holomorphic Fock space is the Hilbert space formula_122 of holomorphic functions "f"("z") on C"n" with finite norm
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 formula_133. The embedding of Sp(2"n") in Sp(2("n"+1)) and the compatible identification
lead to an action on the whole of formula_122. It can be verified directly that it is compatible with the action of the operators "W"("z").
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 operators.
Let "S" denote the unit sphere in C"n" and define the Hardy space H2("S") be the closure in "L"2("S") of the restriction of polynomials in the coordinates "z"1, ..., "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 and coincides up to a sign with the degree of "f" as a continuous mapping from "S" into the general linear group.
 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:
 noticed that there was a natural unitary isomorphism between H2("S") and "L"2(R"n") 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. This method of proving index theorems was generalised by Alain Connes within the framework of cyclic cohomology.
Theory in infinite dimensions.
The theory of the oscillator representation in infinite dimensions is due to Irving Segal and David Shale. 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 "L"2(R) and its extension to the Olshanskii semigroup.
The representation of commutation on Fock space is generalized to infinite dimensions by replacing C"n" (or its dual space) by an arbitrary complex Hilbert space "H". The symmetric group "Sk" acts on "H"⊗"k". "Sk"("H") is defined to be the fixed point subspace of "Sk" and the symmetric algebra is the algebraic direct sum
It has a natural inner product inherited from "H"⊗"k":
Taking the components "Sk"("H") to be mutually orthogonal, the symmetric Fock space "S"("H") is defined to be the Hilbert space completion of this direct sum.
For ξ in "H" define the coherent state "e"ξ by
It follows that their linear span is dense in "S"("H"), that the coherent states corresponding to "n" distinct vectors are linearly independent and that
When "H" is finite-dimensional, "S"("H") can naturally be identified with holomorphic Fock space for "H"*, since in the standard way "Sk"("H") are just homogeneous polynomials of degree "k" on "H"* and the inner products match up. Moreover, "S"("H") has functorial properties. Most importantly
A similar result hold for finite orthogonal direct sums and extends to infinite orthogonal direct sums, using von Neumman's definition of the infinite tensor product with 1 the reference unit vector in S0("Hi"). Any contraction operator between Hilbert spaces induces a contraction operator between the corresponding symmetric Fock spaces in a functorial way.
A unitary operator on "S"("H") is uniquely determined by it values on coherent states. Moreover, for any assignment "v"ξ such that
there is a unique unitary operator "U" on "S"("H") such that
As in the finite-dimensional case, this allows the unitary operators "W"("x") to be defined for "x" in "H":
It follows immediately from the finite-dimensional case that these operators are unitary and satisfy
In particular the Weyl commutation relations are satisfied:
Taking an orthonormal basis "e""n" of "H", "S"("H") can be written as an infinite tensor product of the "S"(C "e""n"). The irreducibility of "W" on each of these spaces implies the irreducibility of "W" on the whole of "S"("H"). W is called the complex wave representation.
To define the symplectic group in infinite dimensions let "H"R be the underlying real vector space of "H" with the symplectic form
and real inner product
The complex structure is then defined by the orthogonal operator
so that
A bounded invertible operator real linear operator "T" on "H"R lies in the symplectic group if it and its inverse preserve "B". This is equivalent to the conditions:
The operator "T" is said to be implementable on "S"("H") provided there is a unitary π("T") such that
The implementable operators form a subgroup of the symplectic group, the "restricted symplectic group". By Schur's lemma, π("T") is uniquely determined up to a scalar in T, so π gives a projective unitary representation of this subgroup.
The Segal-Shale quantization criterion states that "T" is implementable, i.e. lies in the restricted symplectic group, if and only if the commutator "TJ" – "JT" is a Hilbert–Schmidt operator.
Unlike the finite-dimensional case where a lifting π could be chosen so that it was multiplicative up to a sign, this is not possible in the infinite-dimensional case. (This can be seen directly using the example of the projective representation of the diffeomorphism group of the circle constructed below.)
The projective representation of the restricted symplectic group can be constructed directly on coherent states as in the finite-dimensional case.
In fact, choosing a real Hilbert subspace of "H" of which "H" is a complexification, for any operator "T" on "H" a complex conjugate of "T" is also defined. Then the infinite-dimensional analogue of SU(1,1) consists of invertible bounded operators
satisfying "gKg"* = "K" (or equivalently the same relations as in the finite-dimensional case). These belong to the restricted symplectic group if and only if "B" is a Hilbert–Schmidt operator. This group acts transitively on the infinite-dimensional analogue "D"≈ of the Seigel generalized unit disk consisting of Hilbert–Schmidt operators "W" that are symmetric with operator norm less than 1 via the formula
Again the stsblilizer subgroup of 0 consists of "g" with "A" unitary and "B" = 0. The metaplectic coherent states "f""W" can be defined as before and their inner product is given by the same formula, using the Fredholm determinant:
Define unit vectors by
and set
where μ(ζ) = ζ/|ζ|. As before this defines a projective representation and, if "g"3 = "g"1"g"2, the cocycle is given by
This representation extends by analytic continuation to define contraction operators for the complex semigroup by the same analytic continuation argument as in the finite-dimensional case. It can also be shown that they are strict contractions.
Example Let "H"R be the real Hilbert space consisting of real-valued functions on the circle with mean 0
and for which
The inner product is given by
An orthogonal basis is given by the function sin("n"θ) and cos("n"θ) for "n" > 0. The Hilbert transform on the circle defined by
defines a complex structure on "H"R. "J" can also be written
where sign "n" = ±1 denotes the sign of "n". The corresponding symplectic form is proportional to
In particular if φ is an orientation-preserving diffeomorphism of the circle and
then "T"φ is implementable.
The operators "W"("f") with "f" smooth correspond to a subgroup of the loop group LT invariant under the diffeomorphism group of the circle. The infinitesimal operators corresponding to the vector fields
can be computed explicitly. They satisfy the Virasoro relations
In particular they cannor be adjusted by addition of scalar operators to remove the second term on the right hand side. This shows that the cocycle on the restricted symplectic group is not equivalent to one taking only the values ±1.