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.
- 1 Historical overview
- 2 Semigroups in SL(2,C)
- 3 Commutation relations of Heisenberg and Weyl
- 4 Fourier transform
- 5 Stone–von Neumann theorem
- 6 Oscillator representation of SL(2,R)
- 7 Maslov index
- 8 Holomorphic Fock space
- 9 Fock model
- 10 Disk model
- 11 Harmonic oscillator and Hermite functions
- 12 Sobolev spaces
- 13 Smooth vectors
- 14 Analytic vectors
- 15 Oscillator semigroup
- 16 Weyl calculus
- 17 Applications and generalizations
- 18 See also
- 19 Notes
- 20 References
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. 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 G = SU(1,1) is formed of matrices
It is a subgroup of Gc = SL(2,C), the group of complex 2 × 2 matrices with determinant 1.
If G1 = SL(2,R) and
The group SL(2,R) is generated as an abstract group by
and the subgroup of lower triangular matrices
with b real and a > 0. 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
with x real.
The period 2 automorphism σ of Gc
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
The cone C in is defined by two conditions. The first is that
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 that
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:
In fact the matrix X can be conjugated by an element of G to the matrix
Since the Möbius transformation corresponding to exp Y sends z to e−2y 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
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
If a matrix
is in H then so too are the matrices
since the latter is obtained by taking the transpose and conjugating by the diagonal matrix with entries ±1.
Hence H also contains the matrix
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 conjugating by 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.
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
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
then the matrix
has no square root in H0. For a square root would have the form
On the other hand,
In fact if S is a semigroup properly containing and g is in S but not then either:
- The image of the unit circle under g cuts the unit circle twice.
- The image of the unit disk under g is disjoint from the unit disk. In this case either the image of the unit circle touches the unit circle or is disjoint. Applying a scaling transformation in H reduces to case (1).
In case (1) if the circles cut at right angles, pre- and post-multiplying g by elements of SU(1,1) the image of the unit circle can be taken to be the real axis with the points ±1 fixed. But then g is the Cayley transform and its square k is the Möbius transformation z−1. Since k H k = H−1, S contains a neighbourhood of the identity and hence is the whole of SL(2,C). If the circles do not intersect at right angles, pre- and post-multiplying g by elements of SU(1,1) the points of intersection may be moved close together with 0 in the crescent formed by the obtuse angles and outside the second circle. Scaling transformations in H will transform the second circle into a touching interior circle. In between the angle will have been a right angle, for which the previous argument applies.
Lawson (1998) gives another 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 x is in S but not in , then there is a small disk D1 in D such that x D1 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 t g D contains g D. Hence g−1t−1 g lies in . 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.
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 Weyl
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
Formally both P and Q are formally 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
for f in , 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.
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,
lies in , it follows that
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 theorem
Now suppose that U(s) and V(t) are one parameter unitary groups on a Hilbert space satisfying the Weyl commutation relations
and define a bounded operator on by
The operators T(F) have an important non-degeneracy property: the linear span of all vectors T(F)ξ is dense in .
Indeed, if f ds and g dt define probability measures with compact support, then the smeared operators
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)
Given U and V satisfying the Weyl commutation relations, define
so that W defines a projective unitary representation of R2 with cocycle given by
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 operators
satisfy the covariance relations above.
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.
In fact SL(2,R) acts by Möbius transformations on the upper half plane H.
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
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.
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.
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.
The representatives π(g) in the metaplectic representation can be chosen so that
where the 2-cocycle ω is given by
Holomorphic Fock space
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
for n ≥ 0. Moreover, the power series expansion of a holomorphic function in gives its expansion with respect to this basis.
Thus for z in C
so that evaluation at z is gives a continuous linear functional on .
Thus in particular is a reproducing kernel Hilbert space.
For f in and z in C define
so this gives a unitary representation of the Weyl commutation relations. 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 = P E0. 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
Hence for f in L2(R)
The adjoint of is given by the formula:
The metaplectic double cover of SU(1,1) can be constructed explicitly as pairs (g, γ) with
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
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.
Coherent states can again be defined as the orbit of E0 under the metaplectic group.
For w complex, set
Then Fw lies in if and only if |w| < 1.
In particular F0 = 1 = E0.
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
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.
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
if g corresponds to the matrix
This gives the explicit formula
Irreducibility of these representations is established in a standard way. 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.
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.
Harmonic oscillator and Hermite functions
Let be the space of functions in L2(R) of the form
with p a polynomial.
act on and satisfy
X is called the annihilation operator and Y the creation operator.
Then the functions
Indeed, it follows by induction that
since XF0 = 0.
since 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.
They are thus eigenfunctions of the harmonic oscillator:
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
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.
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.
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.
on and let
be the harmonic oscillator.
The associated Sobolev spaces Hs, sometimes called Hermite-Sobolev spaces, are defined to
be the completions of with respect to the norms
is the expansion of f in Hermite functions.
The spaces are Hilbert spaces.
Moreover, Hs and H–s 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.
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
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.
Indeed, it is enough to check invariance when g is sufficiently close to the identity.
In that case
with D + A an isomorphism from Ht+2 to Ht.
It follows that
If f is in Hs, then
where the derivatives lie in Hs–½.
Similarly the partial derivatives of total degree k of U(s)V(t)f lie in Sobolev spaces of order s–k/2.
Consequently, a monomial in P and Q of order 2k applied to f lies in Hs–k and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.
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 . Sufficiency is clear. For necessity, smoothness implies that the partial derivatives of W(z)u lie in L2(R) and hence also Dk u 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).
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.
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.
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.
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
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).
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
By Mehler's formula for
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
For a function a(x,y) on R2 = C, let
Defining in general
the product of two such operators is given by the formula
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
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.
- 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 = λn fn 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, Ψ−s consists of trace-class operators for s > 1 and Ψk carries Hm into Hm–k.
The proof of boundedness of Howe (1980) is particularly simple: if
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
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 D1/2 is established similarly by constructing B in Ψ1/2 with real symbol such that D – B4 is a smoothing operator. Using the holomorphic functional calculus it can be checked that D1/2 – B2 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.
Applications and generalizations
Theory for finite abelian groups
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.
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).
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 x A and for z = (x, y) in G set
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:
- (the Fourier transform for A),
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 L2(R) satisfying
This is in fact a holomorphic map into each Sobolev space Hk and hence also H≈ = .
On the other hand, in H–≈ = (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 H–1 ⊂ 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 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
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:
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 reciprocity
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:
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 dimensions
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
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
so that W defines a projective unitary representation of R2n with cocycle given by
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
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
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
and orthonormal basis
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
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.
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
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).
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 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
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:
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. 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 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 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 eξ 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 vξ 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 en of H, S(H) can be written as an infinite tensor product of the S(C en). 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 HR 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
A bounded invertible operator real linear operator T on HR 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 fW can be defined as before and their inner product is given by the same formula, using the Fredholm determinant:
Define unit vectors by
where μ(ζ) = ζ/|ζ|. As before this defines a projective representation and, if g3 = g1g2, 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 HR 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(nθ) and cos(nθ) for n > 0. The Hilbert transform on the circle defined by
defines a complex structure on HR. 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 Tφ 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.
- Folland 1989
- Hilgert & Neeb 1993, pp. 59–60
- Hilgert & Neeb 1993, pp. 250–253
- Lawson 1998, pp. 146–147
- Ferrara et al. 1973
- Hörmander 1983, pp. 160–163
- Folland 1989, pp. 35–36
- von Neumann 1929
- Lang 1985, p. 209
- Pressley & Segal 1986
- Lion & Vergne 1980, pp. 73–83
- Folland 1989
- Hall 2013, pp. 299–300
- Hall 2013, pp. 297–299
- Hall 2013, pp. 300–301
- Folland 1989
- Folland 1989, pp. 181–184
- He 2007
- Itzykson 1967
- Folland 1989
- Folland 1989, p. 94
- Folland (1989, pp. 210–215)
- He 2007
- Howe & Tan 1992
- Kac & Raina 1987
- Igusa 1972
- Sohrab 1981
- Goodman & Wallach 1984
- Goodman & Wallach 1984
- Goodman 1969
- Folland 1989, pp. 223–255
- Folland 1989, pp. 121–129
- Mumford, Nori & Norman 2006
- Igusa 1972
- Hörmander 1983, pp. 178–179
- Lion & Vergne 1980, pp. 149–161
- Folland 1989
- Segal 1981, pp. 315–320
- Hörmander 1985, p. 188
- Connes 1990
- Segal 1981, pp. 315–320
- Baez, J. C.; Segal, I. E.; Zhou, Z.-F.; Kon, Mark A. (1992), "Introduction to algebraic and constructive quantum field theory", Physics Today, Princeton University Press, 46: 43, Bibcode:1993PhT....46l..43B, doi:10.1063/1.2809125, ISBN 0-691-08546-3
- Bargmann, V. (1970), Group representations on Hilbert spaces of analytic functions, Analytic methods in mathematical physics, Gordon and Breach, pp. 27–63
- Berg, M. C. (2000), The Fourier-analytic proof of quadratic reciprocity, Pure and Applied Mathematics, Wiley-Interscience, ISBN 0-471-35830-4
- Brunet, M.; Kramer, P. (1980), "Complex extension of the representation of the symplectic group associated with the canonical commutation relations", Reports on Mathematical Physics, 17: 205–215, Bibcode:1980RpMP...17..205B, doi:10.1016/0034-4877(80)90063-4
- Đokovic, D. Z.; Hofmann, K.-H. (1997), "The surjectivity question for the exponential function of real Lie groups: a status report", Journal of Lie Theory, 7: 171–199
- Coburn, L. A. (1973), "Singular integral operators and Toeplitz operators on odd spheres", Indiana University Mathematics Journal, 23: 433–439, doi:10.1512/iumj.1974.23.23036
- Connes, A. (1990), Géométrie non commutative, InterEditions, ISBN 2-7296-0284-4
- Ferrara, S.; Mattiolia, G.; Rossic, G.; Toller, M. (1973), "Semi-group approach to multiperipheral dynamics", Nuclear Physics B, 53: 366–394, Bibcode:1973NuPhB..53..366F, doi:10.1016/0550-3213(73)90451-3
- Folland, G. B. (1989), Harmonic analysis in phase space, Annals of Mathematics Studies, 122, Princeton University Press, ISBN 9780691085289
- Goodman, R. (1969), "Analytic and entire vectors for representations of Lie groups", Transactions of the American Mathematical Society, 143: 55–76, doi:10.1090/s0002-9947-1969-0248285-6
- Goodman, R.; Wallach, N. R. (1984), "Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle", Journal für die Reine und Angewandte Mathematik, 347: 69–133
- Hall, B. C. (2013), Quantum Theory for Mathematicians, Springer
- He, H. (2007), "Functions on symmetric spaces and oscillator representation", Journal of Functional Analysis, 244: 536–564, doi:10.1016/j.jfa.2006.11.008
- Helton, J. W.; Howe, R. E. (1975), "Traces of commutators of integral operators", Acta Mathematica, 135: 271–305, doi:10.1007/bf02392022
- Hilgert, J.; Neeb, K.-H. (1993), Lie semigroups and their applications, Lecture notes in mathematics, 1552, Springer-Verlag, ISBN 0387569545
- Hille, E. (1940), "Contributions to the theory of Hermitian series. II. The representation problem", Transactions of the American Mathematical Society, 47: 80–94, doi:10.1090/s0002-9947-1940-0000871-3
- Hörmander, Lars (1983), Analysis of Partial Differential Operators I, Springer-Verlag, ISBN 3-540-12104-8
- Hörmander, Lars (1985), Analysis of Partial Differential Operators III, Springer-Verlag, ISBN 3-540-13828-5
- Howe, R. (1980), "Quantum mechanics and partial differential equations", Journal of Functional Analysis, 38: 188–254, doi:10.1016/0022-1236(80)90064-6
- Howe, R. (1988), "The Oscillator Semigroup", Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 48: 61–132, doi:10.1090/pspum/048/974332
- Howe, R.; Tan, Eng-Chye (1992), Non-abelian harmonic analysis: applications of SL(2,R), Universitext, Springer-Verlag, ISBN 0387977686
- Igusa, J. (1972), Theta functions, Die Grundlehren der mathematischen Wissenschaften, 194, Springer-Verlag
- Itzykson, C. (1967), "Remarks on boson commutation rules", Communications in Mathematical Physics, 4: 92–122, Bibcode:1967CMaPh...4...92I, doi:10.1007/bf01645755
- Kashiwara, M.; Vergne, M. (1978), "On the Segal–Shale–Weil representations and harmonic polynomials", Inventiones Mathematicae, 44: 1–47, Bibcode:1978InMat..44....1K, doi:10.1007/bf01389900
- Kac, V.G.; Raina, A.K. (1987), Bombay lectures on highest weight representations, World Scientific, ISBN 9971503956
- Kac, V.G. (1990), Infinite dimensional Lie algebras (3rd ed.), Cambridge University Press, ISBN 0521466938
- Kramer, P.; Moshinsky, M.; Seligman, T. H. (1975), Complex extensions of canonical transformations and quantum mechanics, Group Theory and its Applications, 3, Academic Press
- Lang, S. (1985), SL2(R), Graduate Texts in Mathematics, 105, Springer-Verlag, ISBN 0-387-96198-4
- Lawson, J. D. (1998), "Semigroups in Möbius and Lorentzian geometry", Geometriae Dedicata, 70: 139–180
- Lion, G.; Vergne, M. (1980), The Weil representation, Maslov index and theta series, Progress in Mathematics, 6, Birkhäuser, ISBN 3-7643-3007-4
- Mackey, G. W. (1989), Unitary group representations in physics, probability, and number theory (2nd ed.), Addison-Wesley, ISBN 0-201-51009-X
- Mumford, D.; Nori, M.; Norman, P. (2006), Tata Lectures on Theta III, Progress in mathematics, Springer, ISBN 0817645705
- Neretin, Y. A. (1996), Categories of symmetries and infinite-dimensional groups, London Mathematical Society Monographs, 16, Oxford University Press, ISBN 0-19-851186-8
- von Neumann, J. (1932), "Ueber Einen Satz Von Herrn M. H. Stone", Annals of Mathematics, 33: 567–573, doi:10.2307/1968535, JSTOR 1968535
- Olshanskii, G. I. (1981), "Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series" (PDF), Functional Analysis and Its Applications, 15: 275–285, doi:10.1007/bf01106156
- Pressley, A.; Segal, G. B. (1986), Loop groups, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853535-X
- Segal, G. B. (1981), "Unitary representations of some infinite-dimensional groups", Communications in Mathematical Physics, 80: 301–342, Bibcode:1981CMaPh..80..301S, doi: