Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials
Abstract
Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three angular momentum operators, can be analyzed on the basis of a discrete Schrödinger–like equation which provides a semiclassical Hamiltonian picture of the evolution of a ‘quantum of space’, as shown by the authors in [1]. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuous) variable. Key tool for this comparative analysis is the duality operation defined on the generators of the quadratic algebra and suitably extended to the various families of overlap functions (generalized recoupling coefficients). These families, recognized as lying at the top level of the Askey scheme, are classified and a few limiting cases are addressed.
1 Introduction and a brief review
In [1] a family of orthogonal polynomials has been introduced based on a three–term recursion relationship which plays the role of a discrete Schrödinger equation describing the action of a ‘volume’ operator. This operator occurs in the symmetric treatment of the quantum few–body problem as well as in spin–network modeling of a quantum of space, as pointed out originally in [2]. In this section a short introduction to the necessary mathematical background will be given, together with a summary of a few significant results found by the authors in [1].
Improved insights into algebraic and analytical aspects of the subject will be provided in the next sections.
The theory of (re)coupling of eigenstates of three angular momentum operators , , to states of sharp total angular momentum (with projection along the quantization axis) is usually carried out in the setting of ‘binary couplings’ (see [3], Topic 12 and original references therein). Referring to the ordered triple as above, the admissible schemes are and , respectively. The corresponding eigenvectors are denoted
(1) 
where small s are labelings of the eigenvalues associated with the angular momentum operators (e.g. , ) running over , in units, and () is the eigenvalue of with in integer steps. Thus the ket vectors above belong to Hilbert spaces representing simultaneous eigenspaces of the two, partially overlapping sets of commuting operators , , , , and (, respectively). The Racah–Wigner symbol is the unitary (actually orthogonal by CondonâShortley convention) transformation relating the two sets (1) according to
(2) 
where and the weights are the dimensions of the spin– representations of which provide the standard normalization of such a ‘recoupling coefficient’ as encoded in the shorthand notation in the left–hand side. Therefore a basis transform is simply written as while the inverse one is achieved by the transpose (all non–null matrix elements obey the selection rule by Wigner–Eckart theorem). Recall in passing that the symbol in (2) encodes naturally the symmetry of an Euclidean tetrahedron, a fact which is at the basis of the huge amount of literature about ‘spin–network’ models for 3–dimensional discretized quantum gravity and quantum computing flourished in the past two decades (see [4] and references therein).
The treatment of the ‘symmetric’ coupling scheme for the addition of three angular momenta to give (with projection ) is characterized in terms of a ‘volume’ operator . Unlike what happens with binary coupling schemes, the s appear now to be all on the same footing, indicating that the volume operator can be thought of as acting democratically on either a composite system of four objects with vanishing total angular momentum (a configuration that can be associated with a not necessarily planar quadrilateral vector diagram ), or a system of three objects with total angular momentum (see again [3], Topic 12, last section, and original references therein). The present scheme is characterized by the six commuting Hermitian operators , , , , and , so that eigenvectors and eigenvalues of are given formally (consistently with the notation used in (1)) as
(3) 
Eigenvalues and matrix elements of are naturally found within an imaginary antisymmetric representation based on a three–terms recursion relationship [2], which can be turned into a real, time–independent Schrödinger equation which governs the dynamics of a ‘quantum of space’ as a function of a discrete variable denoted (see below). This has been achieved in [1] (which can be referred to also for a complete list of previous and related papers) where the introduction of discrete, potential–like functions highlights the surprising crucial role of ‘hidden’ symmetries, first discovered by Regge [5] for the symbols. The Schrödinger equation is discretized with respect to a lattice variable given by the label of the operator which characterizes the first of the binary schemes in (1) and reads
(4) 
where the matrix elements are expressed in terms of geometric quantities, namely
(5) 
Here is the Heron’s formula for the area of a triangle with side lengths and . Thus is proportional to the product of the areas of the two triangles sharing the side of length and forming a quadrilateral of sides and . Such a parameter quadrilateral, together with its Regge–conjugate (see below), is the guiding tool of the combinatorial and geometric analysis, in the asymptotic limit, of the Hamiltonian dynamics governing both tetrahedral and ‘fluttery’ quadrilateral configurations, see sections 2 and 3 of [1] for more details.
2 Quadratic symmetry algebras
Following [6, 7], the quantum version of a classical dynamical algebra associated with a pair of ‘mutually integrable’ dynamical variables calls into play a triple of linear operators acting on a (suitably defined) Hilbert space with Hermitian and algebraically independent and anti–Hermitian. The request that these generators do fulfill the Jacobi identity constrains the fundamental commutation relations to be of the form ( is the anticommutator)
(6) 
where are real parameters (the structure constants) and the right–hand sides of the last two relations contain only Hermitian terms. Such a kind of algebraic structures was actually introduced by Sklyanin [8] and they are called ‘quadratic’ algebras for the obvious reason that the commutators (Poisson brackets in the classical cases) are combinations of quadratic (and linear) terms in each of the generators. Mutual
integrability is a sharper requirement with respect to the original formulation, and amounts to look at the symmetry algebra as a dynamical one –where is a constant of the motion for taken as the Hamiltonian operator, as well as the other way around.
Further improvements in the the study of classical, quantum and deformed symmetries along these lines have been provided over the past few decades by a number of authors. Often the admissible structures associated with (6)
and listed in the table below [6] are referred to as ‘Zhedanov’s algebras’ in the literature.
Note that for completeness the last line includes the two ‘standard’ Lie algebras on three generators (whose commutation relations are by definition linear).
Classification of quadratic algebras
AW(3) (Askey–Wilson)  *  *  *  * 
R(3) (Racah)  0  *  *  * 
H(3) (Hahn)  0  0  *  * 
J(3) (Jacobi)  0  0  *  0 
Lie algebras:  0  0  0  * 
, 
The denominations of the algebras, Askey–Wilson, Racah, …, are strongly reminiscent of the Askey–Wilson scheme of hypergeometric orthogonal polynomials of one (continuous or discrete) variable [9]. This is not accidental:
rather, this remark turns out to be crucial in order to recognize the deep connection between algebraic symmetries of (quantum) systems and special function theory in a quite straightforward way.
Indeed the ‘overlap functions’ stemming from the analysis of the eigenvalue problems for the operators which generate the quadratic algebras are, under mild conditions, orthogonal families of Wilson, Racah, Hanh, Jacobi, …, Hermite polynomials. In what follows an account of a few technical details is given for the case of the Racah algebra
R(3) which corresponds to set in (6).
Suppose that the Hermitian operators and –defined on a separable Hilbert space and possibly depending on a same (finite) set of real parameters– are both ladder operators, namely possess discrete, non–degenerate spectra, and start considering the eigenvalue problem for
(7) 
Then it can be easily shown that the operator is tridiagonal in this basis
(8) 
and, similarly, by exchanging the role of and , one would get
(9) 
and
(10) 
The (real) matrix coefficients can be evaluated explicitly in terms of the commutation relations (6) and contain also the parameters which the operators may depend on (such parameters are dropped in the present simplified treatment aimed to point out the overall structural properties). Once chosen suitable normalizations for the two sets of eigenbases (7) and (9), it is possible to introduce two families of overlap functions by resorting to the Dirac braket convention (in which for instance stands for the eigenfunction of a system in the position representation)
(11) 
which are both hypergeometric orthogonal polynomials of one discrete variable (the spectral parameter and respectively) to be identified, up to suitable rearrangements of the hidden parameters, with the Racah polynomial on the top of the Askey scheme [9].
In the –eigenbasis the operator satisfies
(12) 
where eigenvalues and matrix elements are iteratively evaluated from (7) and (8). has a discrete spectrum found as a solution of
(13) 
It is worth noting that in general the diagonalization of cannot be carried out analytically, except in a few cases in which at least the lowest eigenvalues turn out to be representable in closed algebraic forms. The associated families of (normalized) overlap functions are denoted
(14) 
and can be shown to be orthogonal (on different suitably defined lattices), each depending on one discrete variable, but in principle they might not be included into the Askey scheme.
Similarly, other two families of (normalized) overlap functions associated with the pair can be defined by notation consistency as
(15) 
A crucial feature of the Racah algebra R(3) and associated overlap functions is the duality property. It relies on the following transformation of the generators
(16) 
which can be easily shown to represent an automorphism of the Racah algebra R(3). The notion of duality is extended to (all of) the sets of overlap functions introduced above. More precisely

Under the automorphism (16) the discrete variables of the two hypergeometric families of overlap functions associated with given in (11) and their degrees as polynomials are interchanged. Since in the present case the operator is not called into play, the stronger property of ‘self–duality’ of these families holds true: both of them are recognized as Racah polynomials, as already mentioned above.

Referring to the families in (14), under the automorphism (16) the discrete spectral variable of the first family, which is orthogonal on the lattice , is turned into the second family, where the variable is and the polynomial degree is given in terms of the labels of the eigenvalues of .
A similar property is shared by the families associated with the pair given in (15).
More details on the nature of the automorphism group and on the statements about the overlap functions will be reported in the next section when dealing with a specific ‘realization’ of the Racah algebra.
3 Generalized recoupling theory, Regge symmetry and duality
The realization of the Racah algebra R(3) within the setting of generalized recoupling theory was actually the issue addressed originally in [7] which has inspired further work on quadratic algebras. Combining the definitions and notation of section 2 with those of section 1 it is straightforward to recognize the following correspondence
(17) 
between the abstract ordered set of operators and its realization as
.
The next step would consist in associating eigenvalue equations and three–term recursion relations of the abstract approach with their realizations in generalized quantum (re)coupling theory. Here we do not enter into much details about this matter since the translation of (8) based on the pair , represents the three–term recursion relation for the coefficient in disguise (see e.g. [10]). The analysis for the pair , which gives the abstract three–term relation as written in
(12) is examined in details in [2] (and references therein) while its symmetrized counterpart is nothing but the discretized Schrödinger–like equation displayed already in (4) [1].
Focusing on the specific issue regarding the families of solutions of such relationships, one would directly be lead to establish the correspondence
(18) 
where the arrow stands for the specific realization (3) of R(3). To achieve this goal in a transparent and consistent way a few more steps are needed, the first one of which consists in establishing suitable notations for all of the recoupling coefficients. The symbol in (2) and the functions in (4) are thus denoted and defined respectively as
(19) 
Actually this is not a mere question of notation, since in this way the objects may reveal their ‘double’ meaning as i) quantum mechanical transition amplitudes, namely the square modulus is the probability that a system, prepared in the state , be measured to be in the state ; ii) eigenfunctions of the operator whose quantum number is in in the representation labeled by the eigenvalue of the other operator, namely through the projection onto . The latter interpretation will be under focus in what follows and more details about the correspondence (18) can be worked out by introducing explicitly the (so far ignored) parameters of the problem. Upon replacement of the original (ordered) set of labeling of the four angular momenta forming a quadrilateral according to
(20) 
the functionals are rewritten as
(21) 
Recall that geometrically the first functional is associated with a tetrahedron ( and being a pair of opposite edges) and the second one to a quadrilateral (actually two triangles hinged by one of its diagonal, or ) bounding, so to speak, a portion of volume of amount , the eigenvalue of the volume operator given in (3). In order to select in a convenient way the Hilbert space on which the volume operator acts and all the functionals above can be defined consistently, the role of Regge symmetries, originally introduced for the [5], is crucial. Such symmetries in their original formulation are recognized as functional relations on the arguments (namely they cannot be derived by interchanging the arguments as happens for the so–called ‘classical’ or tetrahedral symmetries) and read
(22) 
where is the semi–perimeter of the parameter quadrilateral and in the last equality the new set is defined. It can be checked that the total number of classical and Regge symmetries is 144, which equals the order of the product permutation group .
Denoting by the smallest value among the eight parameters , it can be shown that a consistent ordering of the other parameters compatible with all the due inequalities is given by . This sort of gauge fixing implies that the whole problem becomes finite–dimensional and workable out for each fixed values of the parameters . Moreover: i) the tetrahedron can be chosen as the reference one, calling its Regge–conjugate; ii) the same thing holds for the quadrilateral denoted and its conjugate . More technical details about this specific parametrization and the denomination Regge–‘conjugate’ (as well as the proof that the volume operators and all quantities in its three–term recursion relation (4) are Regge–invariant) can be found in [11] and [1] respectively.
Coming back to the statement regarding the correspondence (18), the remarks above should have made clear that Regge symmetry is strictly related to the duality property of the Racah algebra discussed at the end of section 2. Note that in [7] it had been already recognized that (classical + Regge) symmetries do have the group structure given by , to be identified with the automorphism group of the Racah algebra.
4 Classification of discrete polynomial families
In this section the focus will be on interconnections among the families of discrete orthogonal polynomials in view of the formalization presented in section
2 and summarized there in items i) and ii). This analysis –not addressed elsewhere to our knowledge– is just sketched here, leaving aside a number of technical details that can be found in [12].
The various cases, together with the most significant properties of each family, are summarized in the following table.
Finite families of discrete orthogonal polynomials [ fixed ]
#  family  orthogonality on lattice  eigenvalue  degree 

(related to the variable)  related to  
I.A  
I.B  
II.A  
II.B  
III.A  
III.B 
Comparing the notations adopted here –the bar stands for complex conjugation or simply transposition in the real cases– with those of section 2, it is straightforward to recognized that the classes I, II and III are in correspondence with the overlap functions in (11), (14) and (15) (restricted to finite sets by suitable choices of the omitted parameters), respectively.
Looking at the family IA, observe that by the convention chosen for symbols in (2) (and similarly for IB). Thus ‘self–duality’ relations for class I read either way
(23) 
once fulfilled the completeness relations and for the binary coupled eigenbases introduced in (1). Note that the operators associated with class I ( and ) represent a ‘Leonard pair’ so that the associated overlap functions (recoupling coefficients) are necessarily hypergeometric of Racah type [13]. More generally, in connection with the analysis of the other classes, a stringent result holds true: any finite system of orthogonal polynomials whose dual is a finite system of orthogonal polynomials must be (possibly –deformed) Racah or one of its limiting cases which constitute finite systems (refer to [14] for a modern monograph on hypergeometric polynomials in the Askey–Wilson scheme). Indeed here all of the families are consistently defined, for fixed parameters , as finite sets (recall the choice on the ordering discussed in connection with Regge symmetry) but the recognition of classes II and III as belonging to the Askey scheme is certainly not straightforward. (More precisely, the reduction process to specific hypergeometric functions of type would require to find out a closed algebraic form for the sets of eigenvalues of the volume operator for given parameters, a task not yet accomplished.)
For what concerns duality within class II, a first remark is about the bar operation: is , but the latter, unlike what happens for the , is not necessarily equal to because this property actually depends on the volume operator being Hermitian (imaginary antisymmetric) [2] or real symmetric (see [1] also for plots of the family of eigenfunctions ). Anyway, both options can be included through a suitable notation into the duality relations
(24) 
according to the choiche of the representation of . Duality relations in class III are similar to (24), with taking the role of .
To conclude this general overview on duality relationships, a further remarkable property –transversal with respect to the classes– has to be mentioned, namely
(25) 
Such a ‘triangular relation’ (and the other ones that can be derived by using the properties of the single classes given above) closely resembles the Racah identity satisfied by three symbols and might be used also to explore a formalization of the whole subject within the general scheme of tensor categories.
5 Limiting cases
The issue of asymptotic (semiclassical) limits of angular momentum functions is of continuous interest in many fields, ranging from special function theory [14] to applied quantum mechanics [15]. Here just a few remarks concerning two limiting cases of families II.A and III.B are sketched.
The reference model of asymptotics is the well–know limit of the symbol for three large entries (see [16, 10]), , where the latter is the Wigner symbol, the symmetrized version of a Clebsch–Gordan coefficient. The counterpart of this operation in the Askey scheme is achieved by moving one step downwards from top, namely from (Racah) to (Hahn and dual Hahn) hypergeometric families.
A new change of notation is needed which consists in restoring the string for the parameters (see (20)) and in writing down as an array the functions in (21) (equivalently, in family II.A) according to
(26) 
where the vertical bars in front of the last column of this symbol indicate that not all of the entries are constrained by standard triangular inequalities, as happens for the . To address any limit in which (some of) the arguments of the symbols become large –a fact that implies that all of the arguments can be ‘running’– a convenient notation is to substitute capital to small letters. Thus the formal limiting process for the symbol in (26) when the arguments of the lower row become large can be displayed as a generalized coefficient, denoted , related in turn to a generalized dual Hahn polynomial; schematically
(27) 
On applying a similar procedure to family III.B, and denoting the previous generic argument (playing the role of ), the resulting correspondence would read
(28) 
A few comments on these results are in order, leaving aside a more careful analysis and most technical details reported in [12]. As already noticed, the symbols in round brackets on the right–hand sides of (27) and (28) are generalized counterparts of coefficients, the arguments in the lower row being interpreted as magnetic quantum numbers. They actually share with standard s a suitable formulation of Regge symmetry [17] and their properties as orthogonal families are inferred from three–term recursion relationships. The latter can in turn be derived as limits of the three–term recursions at the upper level (in particular, the relation for (27) can be quite easily worked out). The motivation for associating dual Hahn and Hahn families respectively is related with the specific lattices these three–term recursion relations are defined on. Thus it is found that the relation for (27) mimics the behavior of the relation of a on a quadratic lattice (), so that it is functionally similar to the standard dual Hahn polynomial family. Conversely, the relation for (28) mimics the behavior of the relation of a on a linear lattice (given by scaling the quantum number ) and thus these functions represent counterparts of the Hahn polynomial family.
6 Outlook
Further developments can be addressed in parallel, from algebraic–analytical and geometric viewpoints. A schematic list of ongoing works (and still open questions) follows:

convolution rules for overlap functions (specifically, symmetric recoupling coefficients) of Racah algebra;

composition rules of collections of quadrilaterals able to provide new classes of integrable quantum systems to be associated with extended quantum geometries;

–deformed extensions and limiting cases of the dual sets of orthogonal polynomials also in view of applications in quantum chemistry.
In particular, a systematic study of limiting procedures –to be carried out on recurrence relations, on families of polynomials and possibly directly on the defining relations (6) of the underlying quadratic algebras– seems particularly promising also in view of recent analytical and numerical work on strictly related issues [19, 20, 21, 22].
Acknowledgments
D M and A M acknowledge partial support from PRIN 20102011 Geometrical and analytical theories of finite and infinite dimensional Hamiltonian systems.
References
Footnotes
 J. Phys.: Conf. Ser. (2014), in press.
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