Representations of the Schrödinger algebra and Appell systems
Abstract
We investigate the structure of the Schrödinger algebra and its representations in a Fock space realized in terms of canonical Appell systems. Generalized coherent states are used in the construction of a Hilbert space of functions on which certain commuting elements act as selfadjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space is found. Then Appell systems connected with certain evolution equations, analogs of the classical heat equation, on this algebra are computed.
Keywords: Lie algebras, Schrödinger algebra, HeisenbergWeyl algebra, Leibniz function, quantum probability, Appell systems
AMS classification: 17B81, 60BXX, 81R05
Contents
1 Introduction
The Schrödinger Lie algebra plays an important role in
mathematical physics and its applications
(see, e.g., [1, 2, 3, 8, 10]).
Using the technique of singular vectors, a classification of the
irreducible lowest weight representations of this algebra is given in
[5]. A main feature of the present paper is
a classification of the representations of the Schrödinger
algebra in an alternative way based on the semidirect product
structure of the algebra.
We begin in §2 with some interpretations of the notion of ‘Appell systems’.
Section 3 contains basic details of our approach to representations of the
Schrödinger algebra. In particular, we show how it is built and
determine a standard form. Some group calculations are done using a
matrix realization of the algebra.
In §4 we construct canonical Appell systems and find a family of probability
distributions associated to the
Schrödinger algebra that reflects its Lie algebraic structure.
In particular, we see that the results of [5] on
polynomial representations based on lowest weight modules fit into
our picture. The details of the associated
Hilbert space comprise §5. This starts with computing the Leibniz
function. We show how to recover the Lie algebra from the Leibniz
function and obtain an orthogonal basis for the Hilbert space.
In the final section,
we show how to construct Appell systems which provide solutions to generalized heat
equations on the Schrödinger algebra,
corresponding to classical twodimensional real Lévy processes.
2 Appell systems: some interpretations
There are three interpretations of the notion of ‘Appell systems’:

Appell systems in the classical sense. These are considered below from a more general viewpoint.

Canonical Appell systems associated to a Lie algebra. One uses the Lie algebra to construct a Hilbert space with the Appell system as basis. See §4.

General Appell systems on Lie groups. Here one uses the Lie algebra and group structure as a ‘black box’ into which a classical stochastic process goes in and produces a ‘Lie response’ — typically a process consisting of iterated stochastic integrals of the input process ([6, 7]). In this paper, for simplicity, we restrict to the abelian case. Appell systems provide solutions to evolution equations related to the input stochastic process. See §6.
Now we shall expand on the first point of view.
Considering the differentiation operator , we may think of the space of polynomials of degree not exceeding as the space of solutions, , to the equation . In this context an Appell system is defined to be a sequence of nonzero polynomials satisfying:

,

, for
(Note that this differs slightly from the usual definition, cf. [6], which has .) By analogy, for any operator , called the canonical lowering operator, we define a Appell system as follows. Set
for . Then the Appell space decomposition is the system of embeddings , and a Appell system is a sequence of nonzero functions satisfying:

,

, for
Typically, one starts with a ‘standard Appell system’, such as
, for . Then Appell systems are generated from the
standard one via timeevolution. To accomplish this for Appell systems,
the symmetry algebra of comes into play.
If is an operator acting on a space of smooth functions, we define its unrestricted symmetry algebra to be the Lie algebra of vector fields, , such that there exists an operator in the center of with
If in every case
we require to be multiplication by a scalar function, we shall talk of
the restricted symmetry algebra, as in [5].
If we consider only those for which , we have the strict
symmetry algebra . Clearly, .
Also, it is clear that contains the center of
.
Proposition 2.1
The strict symmetry algebra contains the derived algebra of unrestricted symmetries: . That is, implies .
Proof: Let and . Then, by the Jacobi identity,
using the property that the operators are central.
The relevance for Appell systems is this.
Proposition 2.2
The unrestricted symmetry algebra of an operator preserves the Appell space decomposition , that is, for every .
Proof:
Write in the form . Fix
and let . Since
commutes with , we have
.
New Appell systems are generated from a given one by the adjoint action
of a group element generated by a ‘Hamiltonian’ — a
function of elements of the symmetry algebra. The structure of
the spaces is preserved, while the Appell systems provide
‘polynomial solutions’ to the evolution equation corresponding to the
Hamiltonian. Indeed, if is a function of operators in , with
, then will be an Appell system.
For each , the function satisfies , with .
In the simplest situation where is a function of and the initial
Appell
sequence is , different choices of yield many of the
classically important sequences of polynomials (with perhaps minor variations).
In the paper [5], a hierarchy of solutions to
is developed for the Schrödinger operator
.
The representations discussed there can be viewed as Appell
systems in the above sense. These correspond to
finitedimensional representations of sl(2) in the standard form of
the Schrödinger algebra given below.
3 Schrödinger algebra
Referring to [5] for details, recall that the (, centrallyextended) Schrödinger algebra is spanned by the following elements (commented by their physical origins):
mass  
special conformal transformation  
Galilei boost  
dilation (not differentiation!)  
spatial translation  
time translation 
which satisfy the following commutation relations, given here in the form of a matrix with rows and columns labelled by the corresponding operators
Note that elements span a HeisenbergWeyl subalgebra, while span an sl(2) subalgebra. This fact, that the Schrödinger algebra is a semidirect product
is the basis for analyzing the representations of the Schrödinger
algebra. We continue with the case and indicate how the general
case goes at the end of the discussion of the standard form, since
the rotation generators, , do not appear in the case .
3.1 Structural decomposition for Fock calculus
In general, in order to construct representations,
we first seek a generalized Cartan decomposition of the Schrödinger algebra
into a triple
where and are abelian subalgebras, and
is a subalgebra normalizing both and .
The main idea is that elements of and
act as raising and lowering operators, respectively.
The possibility of finding a scalar product in which each element of
has a corresponding adjoint in is important, since
we wish to construct a family of selfadjoint operators that provide
a family of commuting quantum observables or classical random variables in the
probabilistic interpretation. In many cases, this family arises
by conjugating elements of by a group element
with a generator from .
This technique may be viewed as an extension of the Cayley transform
for symmetric spaces. Notice that for this to work, the
subalgebras and must be in onetoone
correspondence — the Cartan involution in the theory of
symmetric spaces.
The Schrödinger algebra admits the following generalized Cartan decomposition:
(1) 
Note however that and cannot be put into 11
correspondence and therefore this is of no direct use for us.
We will use instead the following decomposition ( cf. [9, p. 31]):
(2) 
acts here as a scalar . We take and as raising
operators. Even though is not in “Cartan’s ”, as in
equation (1), we use it as
the lowering operator dual to , so take and .
Even though the decomposition (2) is not technically
a Cartan decomposition, it will lead to interesting results
for representations of the Schrödinger algebra.
3.2 A matrix representation and group calculations
A 4dimensional representation (see [4]) of the Schrödinger algebra () is given by embedding into su(4). Let denote a typical element of the Lie algebra. Set,
(3) 
We will denote a typical group element according to the basis we have chosen by
The variables are coordinates of the second kind. The group element corresponding to (3) is
From this we have
Proposition 3.1
Given in matrix form a group element , we can recover the secondkind coordinates according to
Referring to decomposition (2), we specialize variables, writing for respectively. Basic for our analysis is the partial group law:
We will get the required results using the matrix representation noted above. The general elements of and are:
As the square of each of these matrices is zero, the exponential of each reduces to simply adding the identity. We find the matrix product
Applying Proposition 3.1 to the matrix found above yields
Proposition 3.2
In coordinates of the second kind, we have the Leibniz formula,
In general, a Leibniz formula is the group law for commuting
the operators past the ’s, in analogy to the classical formula
of Leibniz for derivatives.
3.3 Standard form of the Schrödinger algebra
Now we show the internal structure of the Schrödinger algebra ().
Remark 3.3
Note that we work in enveloping algebras throughout, so our calculations are based on relations in an associative algebra. In particular, we often use
(4) 
Definition 3.4
Denote the basis for a standard HeisenbergWeyl (HW) algebra, , satisfying
A representation of HWalgebra such that acts as the scalar times the identity operator will be denoted as HW algebra.
Definition 3.5
Denote the basis for a standard sl(2) algebra, , by , satisfying
We write .
The following Lemma is wellknown. It follows readily from the equations in remark 3.3 (also see calculations below).
Lemma 3.6
Given an HW algebra, setting
yields a standard sl(2) algebra.
Now for our first main observation, which follows immediately from the commutation rules for the Schrödinger algebra.
Theorem 3.7
(HW form of the Schrödinger algebra) Given an HW algebra, setting
yields a representation of .
And the main theorem, which gives the standard form.
Theorem 3.8
(Standard form of the Schrödinger algebra) Any representation of the Schrödinger algebra contains a standard sl(2) algebra such that, with the HW algebra from the given representation of , the sl(2) subalgebra is of the form
where commutes with .
Conversely, given any HW representation, use it for . Now take any sl(2) algebra commuting with , and form the direct product with the standard sl(2) algebra constructed from by the Lemma. Then this yields a representation of .
Proof: The converse is clear by construction and our previous observations. What must be checked is that given a representation of , setting
yields an sl(2) algebra that commutes with . From equation (4), we have
and similar relations for and show that commutes with . Now, using remark 3.3, we note these relations
Thus, using the fact that , we have
while
and
which completes the proof.
Remark 3.9
The theorem, extended to include rotations, holds also for , where we use spanned by
and for the rotations,
with the rotations commuting with .
As an application of Theorem 3.7, consider the special realization, with scalar and denoting multiplication by the variable ,
(5) 
In this realization, acting on the function identically equal to 1, we have , and . Applying a group element to the function , we find
Clearly, can be identified with the function itself. Now apply the Leibniz formula, Proposition 3.2, to find
Corollary 3.10
The differential realization of the Schrödinger algebra has the following “partial group law”
4 Canonical Appell systems for the Schrödinger algebra
Now to construct the representation space and basis — the canonical Appell system. To start, define a vacuum state such that, for constants and ,
Notation
The standard form (cf. Theorem 3.8) gives , which shows that . Hence in the following we denote by .
The (commuting) elements and of can be used to form basis elements
of a Fock space
on which and act as raising operators,
while and act as lowering operators.
4.1 Adjoint operators and Appell systems
The goal is to find an abelian subalgebra spanned by some selfadjoint operators acting on the representation space just constructed. Such a twodimensional subalgebra can be obtained by an appropriate “turn” of the plane in the Lie algebra, namely via the adjoint action of the group element formed by exponentiating . The resulting plane, say, is abelian and is spanned by
(6)  
(7) 
Next we determine our canonical Appell systems. We want to compute . Setting , , , and in Proposition 3.2 yields
(8) 
To get the generating function for the basis , set in equation (4.1)
(9) 
Substituting throughout, we have
Proposition 4.1
The generating function for the canonical Appell system, is
where we identify and in the realization as functions of .
With , we recognize the generating function for the Laguerre
polynomials, while reduces to the generating function for
Hermite polynomials.
This corresponds to the results of Section 4 of [5].
From the exponentials , equation (4.1), we identify as operators and . Using script notation for the as operators, relations (9) take the form
To act on polynomials, expand in geometric series
So we have both a Appell system and a Appell system as in Section 2. The Appell space decompositions are, for and ,
respectively, where poly(), resp. poly, denote arbitrary polynomials in the indicated variable, resp. of degree a most in the variable. Now symmetries are generated by functions of and . We will see explicit examples in Section 6.
4.2 Probability distributions
Now we shall consider some probabilistic observations.
We introduce an inner product such that and
. The , which are formally symmetric,
extend to selfadjoint operators on appropriate domains.
Expectation values are taken in the state , i.e., for any operator ,
where the normalization is understood.
From follows that and moving and across in the inner product, that as well. Going back to equation (4.1), take the inner product on the left with . The exponential factors in and average to 1, yielding
This result has an interesting probabilistic interpretation
for positive values of and .
Observe that the marginal distribution of (i.e., for )
is gamma distribution, while the marginal distribution of
(now ) is Gaussian. Note, however, that these are not
independent random variables.
To recover the joint distribution of , let us first recall some probability integrals (Fourier transforms):
where denotes the usual Heaviside function, if , zero otherwise. Replacing by respectively and taking inverse Fourier transforms, we have
Proposition 4.2
The joint density of the random variables is given by
for , where .
In the first factor, writing shows
where the Gaussian factor comes in. The result says that the marginal
distribution of is Gaussian with mean 0 and variance
. Conditional on , is gamma with parameters
and taking values in the interval
. In the special case , i.e., ,
the gamma density collapses to a delta function:
.
5 Leibniz function and orthogonal basis
Once the Leibniz formula for our Lie algebra is known
(Proposition 3.2), we can proceed to define coherent
states, find the Leibniz function — inner product of coherent states —
and show that we have a Hilbert space with selfadjoint
commuting operators and
(here the in equations (6) is set equal to 1).
We recover the raising and lowering operators as elements of the
Lie algebra acting on the Hilbert space with basis consisting of the
canonical Appell system.
Proposition 5.1
With and , the Leibniz function is
Note that the Leibniz function is symmetric in and , which is equivalent
to the inner product being symmetric, and thus the Hilbert space
being welldefined.
It is remarkable that the Lie algebra can be reconstructed from
the Leibniz function .
The idea is that differentiation with respect to
brings down acting on ,
while differentiation with respect to brings down a
acting on which moves across the inner product as
acting on . Similarly for and . We thus introduce
canonical bosons, creation operators , and annihilation
(velocity) operators ,
satisfying .
We thus identify , .
Note, however, that is not the adjoint of , nor
that of .
In fact, our goal is to determine the boson
realization of and , the respective adjoints.
Here is a method to find the boson realization. First, one determines the partial differential equations for :
Then, one interprets each multiplication by as the operator and each differentiation by as the operator . This gives the following action of the operators and on polynomial functions of and :
This means that acts on as follows
and does similarly. The element is recovered via
Summarizing, we have
Theorem 5.2
The representation of the Schrödinger algebra on the Fock space with basis is given by