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[edit]

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:

${\displaystyle G=\operatorname {SU} (1,1)=\left\{\left.{\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\right||\alpha |^{2}-|\beta |^{2}=1\right\},}$

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

${\displaystyle G=CG_{1}C^{-1},\qquad C={\begin{pmatrix}1&i\\i&1\end{pmatrix}}.}$

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

${\displaystyle J={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$

and the subgroup of lower triangular matrices

${\displaystyle \left\{\left.{\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}}\right|a,b\in \mathbf {R} ,a>0\right\}.}$

Indeed, the orbit of the vector

${\displaystyle v={\begin{pmatrix}0\\1\end{pmatrix}}}$

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 ${\displaystyle {\mathfrak {g}}}$ of SU(1,1) consists of matrices

${\displaystyle {\begin{pmatrix}ix&w\\{\overline {w}}&-ix\end{pmatrix}},\quad x\in \mathbf {R} .}$

The period 2 automorphism σ of G_c

${\displaystyle \sigma (g)=M{\overline {g}}M^{-1},}$

with

${\displaystyle M={\begin{pmatrix}0&1\\1&0\end{pmatrix}},}$

has fixed point subgroup G since

${\displaystyle \sigma {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{pmatrix}{\overline {d}}&{\overline {c}}\\{\overline {b}}&{\overline {a}}\end{pmatrix}}.}$

Similarly the same formula defines a period two automorphism σ of the Lie algebra ${\displaystyle {\mathfrak {g}}_{c}}$ of G_c, the complex matrices with trace zero. A standard basis of ${\displaystyle {\mathfrak {g}}_{c}}$ over C is given by

${\displaystyle L_{0}={\begin{pmatrix}{1 \over 2}&0\\0&-{1 \over 2}\end{pmatrix}},\quad L_{-1}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad L_{1}={\begin{pmatrix}0&0\\-1&0\end{pmatrix}}.}$

Thus for −1 ≤ m, n ≤ 1

${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}.}$

There is a direct sum decomposition

${\displaystyle {\mathfrak {g}}_{c}={\mathfrak {g}}\oplus i{\mathfrak {g}},}$

where ${\displaystyle {\mathfrak {g}}}$ is the +1 eigenspace of σ and ${\displaystyle i{\mathfrak {g}}}$ the –1 eigenspace.

The matrices X in ${\displaystyle i{\mathfrak {g}}}$ have the form

${\displaystyle X={\begin{pmatrix}x&w\\-{\overline {w}}&-x\end{pmatrix}}.}$

Note that

${\displaystyle -\det X=x^{2}-|w|^{2}.}$

The cone C in ${\displaystyle i{\mathfrak {g}}}$ is defined by two conditions. The first is ${\displaystyle \det X<0.}$ 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 ${\displaystyle x<0.}$

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

${\displaystyle g({\overline {D}})\subset D.}$

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

${\displaystyle H=G\cdot \exp(C)=\exp(C)\cdot G.}$

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

${\displaystyle Y={\begin{pmatrix}-y&0\\0&y\end{pmatrix}}}$

with

${\displaystyle y={\sqrt {x^{2}-|w|^{2}}}>0.}$

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 ${\displaystyle e^{-Y}g}$ carries D onto itself so lies in G.

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

${\displaystyle {\overline {H}}=\{g\in \operatorname {SL} (2,\mathbf {C} )|gD\subseteq D\}=G\cdot \exp {\overline {C}}=\exp {\overline {C}}\cdot G.}$

From the above statement on conjugacy, it follows that

${\displaystyle H=GA_{+}G,}$

where

${\displaystyle A_{+}=\left\{\left.{\begin{pmatrix}e^{-y}&0\\0&e^{y}\end{pmatrix}}\right|y>0\right\}.}$

If

${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in H}$

then

${\displaystyle {\begin{pmatrix}{\overline {a}}&{\overline {b}}\\{\overline {c}}&{\overline {d}}\end{pmatrix}},\quad {\begin{pmatrix}a&-c\\-b&d\end{pmatrix}}\in H,}$

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

${\displaystyle {\begin{pmatrix}{\overline {a}}&-{\overline {c}}\\-{\overline {b}}&{\overline {d}}\end{pmatrix}}.}$

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

${\displaystyle M={\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}},\qquad |a|<1\quad {\text{and}}\quad |b|<|a|^{-1}-|a|.}$

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

Since

${\displaystyle M{\begin{pmatrix}x&0\\0&x^{-1}\end{pmatrix}}={\begin{pmatrix}x&0\\ba^{-1}(x-x^{-1})&x^{-1}\end{pmatrix}}M,}$

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

${\displaystyle MS(t)=S_{0}(t)M,}$

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

${\displaystyle S(t)N=NS_{0}(t),}$

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

${\displaystyle Z={\begin{pmatrix}z&0\\w&-z\end{pmatrix}}.}$

In particular the one parameter semigroup exp tZ lies in H₀ for all t > 0 if and only if ${\displaystyle \Re z<0}$ and ${\displaystyle |\Re z|>{\tfrac {1}{2}}|w|.}$

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

${\displaystyle \exp Z={\begin{pmatrix}e^{z}&0\\f(z)w&e^{-z}\end{pmatrix}},\qquad f(z)={\sinh z \over z}.}$

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

${\displaystyle \left|\alpha +\alpha ^{-1}\right|<|\alpha |+|\alpha ^{-1}|.}$

If a = α² and

${\displaystyle b=(1-\delta )(|a|^{-1}-|a|),}$

with

${\displaystyle (1-\delta )^{2}={|\alpha +\alpha ^{-1}| \over |\alpha |+|\alpha ^{-1}|},}$

then the matrix

${\displaystyle {\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}}}$

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

${\displaystyle {\begin{pmatrix}\alpha &0\\\beta &\alpha ^{-1}\end{pmatrix}}.}$

On the other hand,

${\displaystyle |\beta |={b \over |\alpha +\alpha ^{-1}|}={|\alpha |^{-1}-|\alpha | \over 1-\delta }>|\alpha |^{-1}-|\alpha |.}$

The closed semigroup ${\displaystyle {\overline {H}}}$ 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 ${\displaystyle H}$, defined through a set of inequalities. Without identification ${\displaystyle H}$ as a compression semigroup, they established the maximality of ${\displaystyle {\overline {H}}}$. Using the definition as a compression semigroup, maximality reduces to checking what happens when adding a new fractional transformation ${\displaystyle g}$ to ${\displaystyle {\overline {H}}}$. The idea of the proof depends on considering the positions of the two discs ${\displaystyle g(D)}$ and ${\displaystyle D}$. In the key cases, either one disc contains the other or they are disjoint. In the simplest cases, ${\displaystyle g}$ is the inverse of a scaling transformation or ${\displaystyle g(z)=-1/z}$. In either case ${\displaystyle g}$ and ${\displaystyle H}$ generate an open neighbourhood 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 D^c, |z| > 1. In fact if ${\displaystyle x\in S\setminus {\overline {H}},}$ then there is a small disk D₁ in D such that xD₁ lies in D^c. Then for some h in H, D₁ = hD. Similarly yxD₁ = D^c for some y in H. So g = yxh lies in S and sends D onto D^c. 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 ${\displaystyle g^{-1}t^{-1}g\in {\overline {H}}.}$ 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

${\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}},}$

the antiautomorphism is given by

${\displaystyle g^{+}={\begin{pmatrix}{\overline {a}}&-{\overline {c}}\\-{\overline {b}}&{\overline {d}}\end{pmatrix}}}$

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

Similarly the antiautomorphism

${\displaystyle g^{\dagger }={\begin{pmatrix}{\overline {d}}&-{\overline {b}}\\-{\overline {c}}&{\overline {a}}\end{pmatrix}}}$

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[edit]

Let ${\displaystyle {\mathcal {S}}}$ 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 ${\displaystyle {\mathcal {S}}}$ by

${\displaystyle Pf(x)=if'(x),\qquad Qf(x)=xf(x).}$

There operators satisfy the Heisenberg commutation relation

${\displaystyle PQ-QP=iI.}$

Both P and Q are self-adjoint for the inner product on ${\displaystyle {\mathcal {S}}}$ inherited from L²(R).

Two one parameter unitary groups U(s) and V(t) can be defined on ${\displaystyle {\mathcal {S}}}$ and L²(R) by

${\displaystyle U(s)f(x)=f(x-s),\qquad V(t)f(x)=e^{ixt}f(x).}$

By definition

${\displaystyle {d \over ds}U(s)f=iPU(s)f,\qquad {d \over dt}V(t)f=iQV(t)f}$

for ${\displaystyle f\in {\mathcal {S}}}$, so that formally

${\displaystyle U(s)=e^{iPs},\qquad V(t)=e^{iQt}.}$

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

${\displaystyle U(s)V(t)=e^{-ist}V(t)U(s).}$

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

Fourier transform[edit]

The Fourier transform is defined on ${\displaystyle {\mathcal {S}}}$ by^[9]

${\displaystyle {\widehat {f}}(\xi )={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)e^{-ix\xi }\,dx.}$

It defines a continuous map of ${\displaystyle {\mathcal {S}}}$ into itself for its natural topology.

Contour integration shows that the function

${\displaystyle H_{0}(x)={e^{-x^{2}/2} \over {\sqrt {2\pi }}}}$

is its own Fourier transform.

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

${\displaystyle {\widehat {Pf}}=-Q{\widehat {f}},\qquad {\widehat {Qf}}=P{\widehat {f}}.}$

It follows that the operator on ${\displaystyle {\mathcal {S}}}$ defined by

${\displaystyle Tf(x)={\widehat {\widehat {f}}}(-x)}$

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

${\displaystyle TH_{0}=H_{0}}$

and since

${\displaystyle g(x)={f(x)-f(a)H_{0}(x)/H_{0}(a) \over x-a}}$

lies in ${\displaystyle {\mathcal {S}}}$, it follows that

${\displaystyle T(x-a)g|_{x=a}=(x-a)Tg|_{x=a}=0}$

and hence

${\displaystyle Tf(a)=f(a).}$

This implies the Fourier inversion formula:

${\displaystyle f(x)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f}}(\xi )e^{ix\xi }\,d\xi }$

and shows that the Fourier transform is an isomorphism of ${\displaystyle {\mathcal {S}}}$ onto itself.

By Fubini's theorem

${\displaystyle \int _{-\infty }^{\infty }f(x){\widehat {g}}(x)\,dx={1 \over {\sqrt {2\pi }}}\iint f(x)g(\xi )e^{-ix\xi }\,dxd\xi =\int _{-\infty }^{\infty }{\widehat {f}}(\xi )g(\xi )\,d\xi .}$

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

${\displaystyle \left({\widehat {f}},{\widehat {g}}\right)=(f,g)}$

so defines an isometry of ${\displaystyle {\mathcal {S}}}$ onto itself.

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

Stone–von Neumann theorem[edit]

Suppose U(s) and V(t) are one parameter unitary groups on a Hilbert space ${\displaystyle {\mathcal {H}}}$ satisfying the Weyl commutation relations

${\displaystyle U(s)V(t)=e^{-ist}V(t)U(s).}$

For ${\displaystyle F(s,t)\in {\mathcal {S}}(\mathbf {R} \times \mathbf {R} ),}$ let^[10][11]

${\displaystyle F^{\vee }(x,y)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(t,y)e^{-itx}\,dt}$

and define a bounded operator on ${\displaystyle {\mathcal {H}}}$ by

${\displaystyle T(F)=\iint F^{\vee }(x,x+y)U(x)V(y)\,dxdy.}$

Then

${\displaystyle {\begin{aligned}T(F)T(G)&=T(F\star G)\\T(F)^{*}&=T(F^{*})\end{aligned}}}$

where

${\displaystyle {\begin{aligned}(F\star G)(x,y)&=\int F(x,z)G(z,y)\,dz\\F^{*}(x,y)&={\overline {F(y,x)}}\end{aligned}}}$

The operators T(F) have an important non-degeneracy property: the linear span of all vectors T(F)ξ is dense in ${\displaystyle {\mathcal {H}}}$.

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

${\displaystyle U(f)=\int U(s)f(s)\,ds,\qquad V(g)=\int V(t)g(t)\,dt}$

satisfy

${\displaystyle \|U(f)\|,\|V(g)\|\leq 1}$

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 ${\displaystyle {\mathcal {H}}}$ is the Schrödinger representation on L²(R), the operator T(F) is given by

${\displaystyle T(F)f(x)=\int F(x,y)f(y)\,dy.}$

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. A concrete description is provided by Linear canonical transformations.

Conversely given a representation of the Weyl commutation relations on ${\displaystyle {\mathcal {H}}}$, 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 C^N. 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

${\displaystyle W(x,y)=e^{\frac {ixy}{2}}U(x)V(y).}$

Then

${\displaystyle W(x_{1},y_{1})W(x_{2},y_{2})=e^{i(x_{1}y_{2}-y_{1}x_{2})}W(x_{1}+x_{2},y_{1}+y_{2}),}$

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

${\displaystyle \omega (z_{1},z_{2})=e^{iB(z_{1},z_{2})},}$

where ${\displaystyle z=x+iy=(x,y)}$ and B is the symplectic form on R² given by

${\displaystyle B(z_{1},z_{2})=x_{1}y_{2}-y_{1}x_{2}=\Im z_{1}{\overline {z_{2}}}.}$

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

${\displaystyle \pi (g)W(z)\pi (g)^{*}=W(g(z)).}$

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

${\displaystyle \iint K(x,y)U(x)V(y)\,dxdy,}$

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

${\displaystyle \pi (gh)=\pm \pi (g)\pi (h),}$

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

${\displaystyle g_{1}={\begin{pmatrix}a&0\\0&a^{-1}\end{pmatrix}},\,\,g_{2}={\begin{pmatrix}1&0\\b&1\end{pmatrix}},\,\,g_{3}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}},}$

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

${\displaystyle \pi (g_{1})f(x)=\pm a^{-{\frac {1}{2}}}f(a^{-1}x),\,\,\pi (g_{2})f(x)=\pm e^{-ibx^{2}}f(x),\,\,\pi (g_{3})f(x)=\pm e^{\frac {i\pi }{8}}{\widehat {f}}(x).}$

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

${\displaystyle g_{3}^{2}=g_{1}(-1),\,\,g_{3}g_{1}(a)g_{3}^{-1}=g_{1}(a^{-1}),\,\,g_{1}(a)g_{2}(b)g_{1}(a)^{-1}=g_{2}(a^{-2}b),\,\,g_{1}(a)=g_{3}g_{2}(a^{-1})g_{3}g_{2}(a)g_{3}g_{2}(a^{-1}).}$

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

${\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}},}$

then

${\displaystyle {dg(z) \over dz}={1 \over (cz+d)^{2}}.}$

The function

${\displaystyle m(g,z)=cz+d}$

satisfies the 1-cocycle relation

${\displaystyle m(gh,z)=m(g,hz)m(h,z).}$

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

${\displaystyle \operatorname {Mp} (2,\mathbf {R} )=\{(g,G)|G(z)^{2}=m(g,z)\}.}$

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

${\displaystyle (g,G)\cdot (h,H)=(gh,K),}$

where

${\displaystyle K(z)=G(hz)H(z).}$

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 ${\displaystyle \Im z>0}$, then

${\displaystyle f_{z}(x)=e^{izx^{2}/2}}$

lies in L² and is called a coherent state.

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

${\displaystyle f_{i}(x)=e^{-{\frac {x^{2}}{2}}},}$

since for g in SL(2,R)

${\displaystyle \pi ((g^{t})^{-1})f_{z}(x)=\pm m(g,z)^{-1/2}f_{gz}(x).}$

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

${\displaystyle \pi ((g^{t})^{-1})f_{z}(x)=m(g,z)^{-1/2}f_{gz}(x).}$

Indeed, if this holds for g and h, it also holds for their product. On the other hand, the formula is easily checked if g^t has the form g_i 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 index[edit]

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 ${\displaystyle \tau (u,v,w)}$ is defined as the signature of the quadratic form on R³ defined by

${\displaystyle Q(a,b,c)=abB(u,v)+bcB(v,w)+caB(w,u).}$

Properties of the Maslov index:

• it depends on the one-dimensional subspaces 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
• ${\displaystyle \displaystyle {\tau (v,w,z)-\tau (u,w,z)+\tau (u,v,z)-\tau (u,v,w)=0}}$

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

${\displaystyle \Omega (g,h)=\exp -{\pi i \over 4}\tau (u_{0},gu_{0},ghu_{0})}$

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

${\displaystyle b(g)=f(u_{0},gu_{0}),}$

then

${\displaystyle \Omega (g,h)^{2}=b(gh)b(g)^{-1}b(h)^{-1}.}$

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

${\displaystyle \pi (gh)=\omega (g,h)\pi (g)\pi (h)}$

where the 2-cocycle ω is given by

${\displaystyle \omega (g,h)=\Omega (g,h)\beta (gh)^{-1}\beta (g)\beta (h),}$

with

${\displaystyle \beta (g)^{2}=b(g).}$

Holomorphic Fock space[edit]

Holomorphic Fock space (also known as the Segal–Bargmann space) is defined to be the vector space ${\displaystyle {\mathcal {F}}}$ of holomorphic functions f(z) on C with

${\displaystyle {1 \over \pi }\iint _{\mathbf {C} }|f(z)|^{2}e^{-|z|^{2}}\,dxdy}$

finite. It has inner product

${\displaystyle (f_{1},f_{2})={1 \over \pi }\iint _{\mathbf {C} }f_{1}(z){\overline {f_{2}(z)}}e^{-|z|^{2}}\,dxdy.}$

${\displaystyle {\mathcal {F}}}$ is a Hilbert space with orthonormal basis

${\displaystyle e_{n}(z)={z^{n} \over {\sqrt {n!}}},\quad n\geq 0.}$

Moreover, the power series expansion of a holomorphic function in ${\displaystyle {\mathcal {F}}}$ gives its expansion with respect to this basis.^[15] Thus for z in C

${\displaystyle |f(z)|=\left|\sum _{n\geq 0}a_{n}z^{n}\right|\leq \|f\|e^{|z|^{2}/2},}$

so that evaluation at z is gives a continuous linear functional on ${\displaystyle {\mathcal {F}}.}$ In fact

${\displaystyle f(a)=(f,E_{a})}$

where^[16]

${\displaystyle E_{a}(z)=\sum _{n\geq 0}{(E_{a},e_{n})z^{n} \over {\sqrt {n!}}}=\sum _{n\geq 0}{z^{n}{\overline {a}}^{n} \over n!}=e^{z{\overline {a}}}.}$

Thus in particular ${\displaystyle {\mathcal {F}}}$ is a reproducing kernel Hilbert space.

For f in ${\displaystyle {\mathcal {F}}}$ and z in C define

${\displaystyle W_{\mathcal {F}}(z)f(w)=e^{-|z|^{2}/2}e^{w{\overline {z}}}f(w-z).}$

Then

${\displaystyle W_{\mathcal {F}}(z_{1})W_{\mathcal {F}}(z_{2})=e^{-i\Im z_{1}{\overline {z_{2}}}}W_{\mathcal {F}}(z_{1}+z_{2}),}$

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

${\displaystyle W_{\mathcal {F}}(a)E_{0}=e^{-|a|^{2}/2}E_{a}.}$

It follows that the representation ${\displaystyle W_{\mathcal {F}}}$ is irreducible.

Indeed, any function orthogonal to all the E_a must vanish, so that their linear span is dense in ${\displaystyle {\mathcal {F}}}$.

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

${\displaystyle f(z)=(PE_{0},E_{z})=e^{|z|^{2}}(PE_{0},W_{\mathcal {F}}(z)E_{0})=(PE_{-z},E_{0})={\overline {f(-z)}}.}$

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

${\displaystyle PE_{0}=\lambda E_{0},}$

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 ${\displaystyle {\mathcal {U}}}$ from L²(R) onto ${\displaystyle {\mathcal {F}}}$, 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 ${\displaystyle {\mathcal {U}}}$ is uniquely determined by the properties

${\displaystyle W_{\mathcal {F}}(a){\mathcal {U}}={\mathcal {U}}W(a)}$

and

${\displaystyle {\mathcal {U}}H_{0}=E_{0}.}$

Hence for f in L²(R)

${\displaystyle {\mathcal {U}}f(z)=({\mathcal {U}}f,E_{z})=(f,{\mathcal {U}}^{*}E_{z})=e^{-|z|^{2}}(f,{\mathcal {U}}^{*}W_{\mathcal {F}}(z)E_{0})=e^{-|z|^{2}}(W(-z)f,H_{0}),}$

so that

${\displaystyle {\mathcal {U}}f(z)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }e^{-(x^{2}+y^{2})}e^{-2ixy}f(t+x)e^{-t^{2}/2}\,dt={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }B(z,t)f(t)\,dt,}$

where

${\displaystyle B(z,t)=\exp[-z^{2}-t^{2}/2+zt].}$

The operator ${\displaystyle {\mathcal {U}}}$ is called the Segal–Bargmann transform^[18] and B is called the Bargmann kernel.^[19]

The adjoint of ${\displaystyle {\mathcal {U}}}$ is given by the formula:

${\displaystyle {\mathcal {U}}^{*}F(t)={1 \over \pi }\iint _{\mathbf {C} }B({\overline {z}},t)F(z)\,dxdy.}$

Fock model[edit]

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

${\displaystyle g={\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}}$

and

${\displaystyle \gamma ^{2}=\alpha .}$

If g = g₁g₂, then

${\displaystyle \gamma =\gamma _{1}\gamma _{2}\left(1+{\beta _{1}{\overline {\beta _{2}}} \over \alpha _{1}\alpha _{2}}\right)^{1/2},}$

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

${\displaystyle \pi (g,\gamma )W_{\mathcal {F}}(z)\pi (g,\gamma )^{*}=W_{\mathcal {F}}(g\cdot z),}$

where

${\displaystyle g\cdot z=\alpha z+\beta {\overline {z}}.}$

Since ${\displaystyle {\mathcal {F}}}$ 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

${\displaystyle K_{T}(a,b)=(TE_{\overline {b}},E_{a}),}$

and F in ${\displaystyle {\mathcal {F}}}$, then

${\displaystyle TF(a)=(TF,E_{a})=(F,T^{*}E_{a})={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z){\overline {(T^{*}E_{a},E_{z})}}e^{-|z|^{2}}\,dxdy={\frac {1}{\pi }}\iint _{\mathbf {C} }K_{T}(a,{\overline {z}})F(z)e^{-|z|^{2}}\,dxdy.}$

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

${\displaystyle K_{S}(a,z)=C\cdot \exp \,{1 \over 2\alpha }({\overline {\beta }}z^{2}+2az-\beta a^{2})}$

for some constant C. Direct calculation shows that

${\displaystyle C=\gamma ^{-1}}$

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

${\displaystyle F_{w}(z)=e^{wz^{2}/2}.}$

Then ${\displaystyle F_{w}\in {\mathcal {F}}}$ if and only if |w| < 1. In particular F₀ = 1 = E₀. Moreover,

${\displaystyle \pi (g,\gamma )F_{w}=({\overline {\alpha }}+{\overline {\beta }}w)^{-{\frac {1}{2}}}F_{gw}={\frac {1}{\overline {\gamma }}}\left(1+{{\overline {\beta }} \over {\overline {\alpha }}}w\right)^{-1/2}F_{gw},}$

where

${\displaystyle gw={\alpha w+\beta \over {\overline {\beta }}w+{\overline {\alpha }}}.}$

Similarly the functions zF_w lie in ${\displaystyle {\mathcal {F}}}$ and form an orbit of the metaplectic group:

${\displaystyle \pi (g,\gamma )[zF_{w}](z)=({\overline {\alpha }}+{\overline {\beta }}w)^{-3/2}zF_{gw}(z).}$

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

${\displaystyle (\pi (g,\gamma )1,1)=\gamma ^{-1}.}$

Disk model[edit]

The projective representation of SL(2,R) on L²(R) or on ${\displaystyle {\mathcal {F}}}$ 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

${\displaystyle {1 \over 2\pi }\iint |F_{+}(z)|^{2}(1-|z|^{2})^{-1/2}\,dxdy+{2 \over \pi }\iint |F'_{+}(z)|^{2}(1-|z|^{2})^{\frac {1}{2}}\,dxdy}$

is finite; and the odd functions to holomorphic functions F_– for which

${\displaystyle {1 \over 2\pi }\iint |F_{-}(z)|^{2}(1-|z|^{2})^{-1/2}\,dxdy}$

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

The action of the metaplectic group is given by

${\displaystyle {\begin{aligned}\pi _{\pm }(g^{-1})F_{\pm }(z)&=\left({\overline {\beta }}z+{\overline {\alpha }}\right)^{-1\pm {\frac {1}{2}}}F_{\pm }(gz)\\&=\left(-{\overline {\beta }}z+\alpha \right)^{-1\pm {\frac {1}{2}}}F_{\pm }\left({{\overline {\alpha }}z-\beta \over -{\overline {\beta }}z+\alpha }\right)&&g={\begin{pmatrix}\alpha &\beta \\{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\end{aligned}}}$

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 ${\displaystyle {\mathfrak {g}}}$. On the other hand, that action is irreducible.

The isomorphism with even and odd functions in ${\displaystyle {\mathcal {F}}}$ 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

${\displaystyle U_{+}(F)(w)={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z)e^{{\frac {1}{2}}w{\overline {z}}^{2}}e^{-|z|^{2}}\,dxdy,}$

${\displaystyle U_{-}(F)(w)={\frac {1}{\pi }}\iint _{\mathbf {C} }F(z){\overline {z}}e^{{\frac {1}{2}}w{\overline {z}}^{2}}e^{-|z|^{2}}\,dxdy,}$

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 functions[edit]

Consider the following subspace of L²(R):

${\displaystyle {\mathcal {H}}=\left\{f\in L^{2}(\mathbf {R} )\left|f(x)=p(x)e^{-{\frac {x^{2}}{2}}},p(x)\in \mathbf {R} [x]\right.\right\}.}$

The operators

${\displaystyle {\begin{aligned}X&=Q-iP={d \over dx}+x\\Y&=Q+iP=-{d \over dx}+x\end{aligned}}}$

act on ${\displaystyle {\mathcal {H}}.}$ X is called the annihilation operator and Y the creation operator. They satisfy

${\displaystyle {\begin{aligned}X&=Y^{*}\\XY&=D+I&&D=-{d^{2} \over dx^{2}}+x^{2}\\XY-YX&=2I\\XY^{n}-Y^{n}X&=2nY^{n-1}&&{\text{by induction}}\end{aligned}}}$

Define the functions

${\displaystyle F_{n}(x)=Y^{n}e^{-{\frac {x^{2}}{2}}}}$

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

${\displaystyle {\begin{aligned}DF_{n}&=DY^{n}F_{0}\\&=(XY-I)Y^{n}F_{0}\\&=\left(XY^{n+1}-Y^{n}\right)F_{0}\\&=\left(\left((2n+2)Y^{n}+Y^{n+1}X\right)-Y^{n}\right)F_{0}\\&=\left((2n+1)Y^{n}+Y^{n+1}X\right)F_{0}\\&=(2n+1)Y^{n}F_{0}+Y^{n+1}XF_{0}\\&=(2n+1)F_{n}&&XF_{0}=0\end{aligned}}}$

Next we have:

${\displaystyle \|F_{n}\|_{2}^{2}=2^{n}n!{\sqrt {\pi }}.}$

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

${\displaystyle (F_{n},F_{n})=\left(XY^{n}F_{0},Y^{n-1}F_{0}\right)=2n(F_{n-1},F_{n-1}).}$

The nth Hermite function is defined by

${\displaystyle H_{n}(x)=\|F_{n}\|^{-1}F_{n}(x)=p_{n}(x)e^{-{\frac {x^{2}}{2}}}.}$

p_n is called the nth Hermite polynomial.

Let

${\displaystyle {\begin{aligned}A&={1 \over {\sqrt {2}}}Y={1 \over {\sqrt {2}}}\left(-{d \over dx}+x\right)\\A^{*}&={1 \over {\sqrt {2}}}X={1 \over {\sqrt {2}}}\left({d \over dx}+x\right)\end{aligned}}}$

Thus

${\displaystyle AA^{*}-A^{*}A=I.}$

The operators P, Q or equivalently A, A* act irreducibly on ${\displaystyle {\mathcal {H}}}$ by a standard argument.^[27][28]

Indeed, under the unitary isomorphism with holomorphic Fock space ${\displaystyle {\mathcal {H}}}$ can be identified with C[z], the space of polynomials in z, with

${\displaystyle A={\frac {\partial }{\partial z}},\qquad A^{*}=z.}$

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 F_n is sent to a multiple of zⁿ and the operator D is given by

${\displaystyle D=2A^{*}A+I.}$

Let

${\displaystyle L_{0}={1 \over 2}(A^{*}A+{1 \over 2})={1 \over 2}(z{\partial \over \partial z}+{1 \over 2})}$

so that

${\displaystyle L_{0}z^{n}={1 \over 2}(n+{1 \over 2})z^{n}.}$

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

Moreover,

${\displaystyle [L_{0},A]=-{1 \over 2}A,[L_{0},A^{*}]={1 \over 2}A^{*},}$

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 + 1/2. 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

${\displaystyle L_{-1}={1 \over 2}A^{2},L_{1}={1 \over 2}A^{*2},}$

so that

${\displaystyle [L_{-1},A]=0,\,\,\,[L_{-1},A^{*}]=A,\,\,\,[L_{1},A]=-A^{*},\,\,\,[L_{1},A^{*}]=0.}$

These operators satisfy:

${\displaystyle [L_{m},L_{n}]=(m-n)L_{m+n}}$

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 F_n are defined by

${\displaystyle F_{n}(x)=\left(x-{d \over dx}\right)^{n}e^{-{\frac {x^{2}}{2}}}=(-1)^{n}e^{\frac {x^{2}}{2}}{d^{n} \over dx^{n}}\left(e^{-x^{2}}\right)=\left(2^{n}x^{n}+\cdots \right)e^{-{\frac {x^{2}}{2}}}.}$

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 ${\displaystyle {\mathcal {H}}}$.

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

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

${\displaystyle e^{-Dt}H_{n}=e^{-(2n+1)t}H_{n}.}$

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

${\displaystyle K_{t}(x,y)\equiv \sum _{n\geq 0}e^{-(2n+1)t}H_{n}(x)H_{n}(y)=(4\pi t)^{-{1 \over 2}}\left({2t \over \sinh 2t}\right)^{1 \over 2}\exp \left(-{1 \over 4t}\left[{2t \over \tanh 2t}(x^{2}+y^{2})-{2t \over \sinh 2t}(2xy)\right]\right).}$

This follows from the formula

${\displaystyle \sum _{n\geq 0}s^{n}H_{n}(x)H_{n}(y)={1 \over {\sqrt {\pi (1-s^{2})}}}\exp {4xys-(1+s^{2})(x^{2}+y^{2}) \over 2(1-s^{2})}.}$

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

${\displaystyle F_{\sigma ,x}(z)\equiv \sum _{n\geq 0}\sigma ^{n}e_{n}(z)H_{n}(x)=\pi ^{-{1 \over 4}}e^{-{\frac {x^{2}}{2}}}\sum _{n\geq 0}{(-z)^{n}\sigma ^{n} \over 2^{n}n!}{d^{n}e^{x^{2}} \over dx^{n}}=\pi ^{-{\frac {1}{4}}}\exp \left(-{x^{2} \over 2}+{\sqrt {2}}xz\sigma -{z^{2}\sigma ^{2} \over 2}\right).}$

Thus F_σ,x lies in holomorphic Fock space and

${\displaystyle \sum _{n\geq 0}s^{n}H_{n}(x)H_{n}(y)=(F_{\sigma ,x},F_{\sigma ,y})_{\mathcal {F}},}$

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

${\displaystyle \int K_{t}(x,y)f(y)\,dy}$

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

${\displaystyle {\widehat {H_{n}}}=(-i)^{n}H_{n},}$

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 spaces[edit]

The Sobolev spaces H_s, sometimes called Hermite-Sobolev spaces, are defined to be the completions of ${\displaystyle {\mathcal {S}}}$ with respect to the norms

${\displaystyle \|f\|_{(s)}^{2}=\sum _{n\geq 0}|a_{n}|^{2}(1+2n)^{s},}$

where

${\displaystyle f=\sum a_{n}H_{n}}$

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

Thus

${\displaystyle \|f\|_{(s)}^{2}=(D^{s}f,f),\qquad (f_{1},f_{2})_{(s)}=(D^{s}f_{1},f_{2}).}$

The Sobolev spaces are Hilbert spaces. Moreover, H_s and H_–s are in duality under the pairing

${\displaystyle \langle f_{1},f_{2}\rangle =\int f_{1}f_{2}\,dx.}$

For s ≥ 0,

${\displaystyle \|(aP+bQ)f\|_{(s)}\leq (|a|+|b|)C_{s}\|f\|_{\left(s+{1 \over 2}\right)}}$

for some positive constant C_s.

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

It follows for arbitrary s by duality.

Consequently, for a quadratic polynomial R in P and Q

${\displaystyle \|Rf\|_{(s)}\leq C'_{s}\|f\|_{(s+1)}.}$

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

${\displaystyle |f(x)|\leq C_{s,k}\|f\|_{(s+k)}(1+x^{2})^{-k}}$

for any k ≥ 0.

Indeed, the result for general k follows from the case k = 0 applied to Q^kf.

For k = 0 the Fourier inversion formula

${\displaystyle f(x)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\widehat {f}}(t)e^{itx}\,dt}$

implies

${\displaystyle |f(x)|\leq C\left(\int \left|{\widehat {f}}(t)\right|^{2}(1+t^{2})^{s}\,dt\right)^{1 \over 2}=C\left(\left(I+Q^{2}\right)^{s}{\widehat {f}},{\widehat {f}}\right)^{1 \over 2}\leq C'\left\|{\widehat {f}}\right\|_{(s)}=C'\|f\|_{(s)}.}$

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

It follows from Sobolev's inequality that the intersection of the spaces H_s is ${\displaystyle {\mathcal {S}}}$. Functions in ${\displaystyle {\mathcal {S}}}$ are characterized by the rapid decay of their Hermite coefficients a_n.

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

${\displaystyle gDg^{-1}=D+A}$

with D + A an isomorphism from ${\displaystyle H_{t+2}}$ to ${\displaystyle H_{t}.}$

It follows that

${\displaystyle \|\pi (g)f\|_{(s)}^{2}=\left|((D+A)^{s}f,f)\right|\leq \left\|(D+A)^{s}f\right\|_{(-s)}\cdot \|f\|_{(s)}\leq C\|f\|_{(s)}^{2}.}$

If ${\displaystyle f\in H_{s},}$ then

${\displaystyle {d \over ds}U(s)f=iPU(s)f,\qquad {d \over dt}V(t)f=iQV(t)f,}$

where the derivatives lie in ${\displaystyle H_{s-1/2}.}$

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 H_s–k and can be expressed as a linear combination of partial derivatives of U(s)V(t)f of degree ≤ 2k evaluated at 0.

Smooth vectors[edit]

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

${\displaystyle \Phi (z)=W(z)u}$

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 ${\displaystyle {\mathcal {S}}}$.^[33] Sufficiency is clear. For necessity, smoothness implies that the partial derivatives of W(z)u lie in L²(R) and hence also D^ku for all positive k. Hence u lies in the intersection of the H_k, so in ${\displaystyle {\mathcal {S}}}$.

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

Moreover, a vector is in ${\displaystyle {\mathcal {S}}}$ if and only if it is a smooth vector for the rotation subgroup of SU(1,1).

Analytic vectors[edit]

If Π(t) is a one parameter unitary group and for f in ${\displaystyle {\mathcal {S}}}$

${\displaystyle \Pi (f)=\int _{-\infty }^{\infty }f(t)\Pi (t)\,dt,}$

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

In fact taking

${\displaystyle f_{\varepsilon }(x)={1 \over {\sqrt {2\pi \varepsilon }}}e^{-x^{2}/2\varepsilon }}$

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

${\displaystyle \Phi (t)=\Pi (t)v}$

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

${\displaystyle \Pi (t)=e^{it{\sqrt {D}}}.}$

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

${\displaystyle \Pi (t)H_{n}=e^{i(2n+1)^{1 \over 2}t}H_{n}.}$

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

${\displaystyle v=e^{-\varepsilon D}\xi .}$

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

${\displaystyle \sum _{n\geq 0}{r^{n}\|D^{n \over 2}v\| \over n!}<\infty }$

for all r > 0.

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

${\displaystyle F(s,t)=U(s)V(t)v}$

defined on R² extends to an analytic map on C².

This reduces to the power series estimate

${\displaystyle \left\|\sum _{m,n\geq 0}{1 \over m!n!}z^{m}w^{n}P^{m}Q^{n}v\right\|\leq C\sum _{k\geq 0}{(|z|+|w|)^{k} \over k!}\|D^{k \over 2}v\|<\infty .}$

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

${\displaystyle W(z,w)=e^{-izw/2}U(z)V(w)}$

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

${\displaystyle e^{-izw/2}F(z,w)=W(z,w)v.}$

Then W leaves the space of entire vectors invariant and satisfies

${\displaystyle W(z_{1},w_{1})W(z_{2},w_{2})=e^{i(z_{1}w_{2}-w_{1}z_{2})}W(z_{1}+z_{2},w_{1}+w_{2}).}$

Moreover, for g in SL(2,R)

${\displaystyle \pi (g)W(u)\pi (g)^{*}=W(gu),}$

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

Formally

${\displaystyle W(z,w)^{*}=W(-{\overline {z}},-{\overline {w}}).}$

Oscillator semigroup[edit]

There is a natural double cover of the Olshanski semigroup H, and its closure ${\displaystyle {\overline {H}}}$ 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

${\displaystyle g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$

and γ is a square root of a.

Such a choice determines a unique branch of

${\displaystyle \left(-{\overline {b}}z+{\overline {d}}\right)^{1 \over 2}}$

for |z| < 1.

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