The mathematical and computational aspects of quantum--mechanical angular momentum theory, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, play nowadays a prominent role also in quantum computing, quantum information and quantum gravity applications. Their modern formulations emphasize the underlying combinational aspects, and involve Wigner 3nj symbols, and the related problems of their calculations, their general properties, and their asymptotic limits. We refer to the ingredients of this theory --and of its extension to other Lie and quantum group-- by using the collective term of "spin networks" [1]. The workshop will discuss recent progress about the already established connections with the mathematical theory of discrete orthogonal polynomials (the so--called Askey Scheme), providing powerful tools based on asymptotic expansions, which correspond on the physical side to various levels of semi—classical limits [2]. These results are useful not only in theoretical molecular physics but also in motivating algorithms for the computationally demanding problems of molecular dynamics and chemical reaction theory, where large angular momenta are typically involved. Regarding quantum chemistry, applications of these techniques include selection and classification of complete orthogonal basis sets in atomic and molecular problems, either in configuration space (Sturmian orbitals) or in momentum space. Promising are relationships with Slater type orbitals. We will list and discuss some aspects of these developments--such as for instance the hyperquantization algorithm-- and studies of vector correlation in reaction stereodynamics and photodissociation, thus providing evidence of a unifying background structure. 

The Racah-Wigner tensor algebra

The coupling theory of quantum angular momenta is the most exhaustive framework in dealing with interacting systems that can be modeled by means of conservative polylocal two--body interactions [3]. The essential features of this algebra can be encoded, for each fixed number N=(n+1) of angular momentum variables, into a combinatorial object, the spin network graph. Vertices are associated with finite--dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. This combinatorial setting can indeed be promoted to (families of) computational quantum graphs, to be thought of as a sort of "abacus" encoding diagrammatical rules encountered in quantum collision events [4] or, more in general, as computational spaces of simulators able to support quantum algorithms for computational problems arising in mathematics and theoretical physics [5]. Semiclassical versions of this construction turn out to play a crucial role in many different contexts, ranging from molecular physics and quantum computing and discrete quantum gravity models. 

Mathematical and computational tools

Important ingredients of modern theoretical physics and chemistry are intimately linked to the mathematical theory of classical (as well as to q--deformed) orthogonal polynomials, in particular of hypergeometric families. The recently proposed formalization of the latter within the Askey and Nikiforov schemes [6,7] places Racah polynomials --and their q-deformed analogs [8]-- at the top of a hierarchy from which all the most relevant families are obtained as particular cases or via suitable limiting procedures. The study of such morphology may be considered as basic in the theory of the special functions of use in modern science: relationships among families, addition formulas, linearization formulas and sum rules look like obscure manipulations of abstract quantities unless a coherent interpretation arising from physical applications could be disclosed. Physical applications, on the other hand, can lead to new insights into previously unnoticed properties and relationships. The computational implementations should make soluble otherwise intractable problems. Dealing with the quantum theory of angular momentum, Racah found a finite sum expression for the basic "recoupling" coefficients or 6j symbols, representing matrix elements for the orthogonal basis transformation between two alternative binary coupling schemes of three angular momenta. Extensive developments of graphical techniques, initiated by the Yutsis school [9], allow one to deal efficiently with recouplings of (n+1) angular momenta, involving the so--called 3nj symbols (see the handbook [10] and the review [3] (Topic 12) also for a complete list of references). 

Spin networks in molecular physics and quantum chemistry

An introduction and some nomenclature are exhibited in [4], where the "spin networks", as well as the underlying "moves", are shown to be closely related to the tree-like graphical representation of hyperspherical coordinates and harmonics, as originally suggested in Ref. [11]. 

Explicit formulas and basic relationships among recoupling coefficients and harmonic superpositions (or "transplant" coefficients) and Racah polynomials can be found in [12] and in references therein. The close relationship of both vector recoupling coefficients and superposition matrix elements between alternative hyperspherical harmonics with orthogonal polynomials of a discrete variable made it possible to develop a common classification scheme, from which a number of relations and properties can be derived [4]. The important case of S3 harmonics, discussed in [13] from the viewpoint of hydrogenoid orbitals in momentum space, is reviewed in [14], where further applications for quantum chemistry are outlined. 
A presentation of angular momentum theory from the viewpoint of the applications in atomic and molecular physics,is given by Zare [15]. The theory developed about thirty years ago in [16] dealt with five alternative representations for the quantum-mechanical close-coupling formulation [16-18] of the motion along the internuclear distance of a vibrating diatomic molecule or colliding atoms having internal (spin and-or electronic) angular momenta. This unified frame transformation approaches of atomic collision theory and concepts of molecular spectroscopy, is originally due to Hund. The physical picture and the relevant nomenclature are reviewed in Ref [19]. Explicitly [20], starting from a sum rule equivalent to the Biedenharn--Elliott relation one can obtain the definition of a 6j symbol as a sum of four 3j symbols by taking a proper limit, since one angular momentum is much larger than the others. 

The coupling schemes of four angular momenta are illustrated in Ref. [19] as the basic ingredients underlying the classification of the five Hund cases and the relationships among representations. It was shown that the transition from one coupling scheme to another is performed by an orthogonal transformation whose matrix elements can be written in terms of 6j symbols. In the pentagonal arrangement of five alternative coupling schemes for four angular momenta (represented by the tree-like graphs at the vertices) [21], connections (the sides of the pentagon) are realized by orthogonal matrices and are related to 6j symbols. 
Recent extensions of this approach have shown that in a general theory of interacting open shell atoms, 3nj symbols up to n=6 occur [20]. A systematic study of 3nj morphogenesis and disentangling can be found in [22]
An important point is the view that a continuous variable limit is obtained at high angular momenta from the discrete structure typical of quantum mechanics. The opposite viewpoint can be considered as well, namely the semiclassical limit may describe a continuous structure, and quantum angular momentum algebra provides discretization. The search for both alternative reference frames and angular momentum coupling schemes has been a major challenge in quantum mechanics since its origin, and tranformations among them are represented by vector--coupling and recoupling coefficients, respectively. The relevant equations can then be formulated in terms of quantum numbers, which approximately correspond to constants of motion of the systems under study. Fundamental advances have been achieved over the years within this framework: in the last Fifties Jacob and Wick introduced the helicity formalism, widely used for the theoretical treatment of a variety of collisional problems; extending Hund introduction of alternative coupling schemes for a diatomic molecule carrying electronic, spin, and rotational angular momenta. These developments fit into the frame transformation theory pioneered by Fano and coworkers in the Seventies. 

Reaction theory

Indeed, for general anisotropic interaction, discretization procedures can be introduced by exploiting Racah algebra, which fosters the introduction of alternative coupling schemes labeled by "artificial" quantum numbers. This method has been shown to provide an elegant and powerful tool for the solution of the reactive scattering Schroedinger equation and, at present, a considerable number of methods have been developed in this spirit, among which the "hyperquantization" algorithm, outlined in greater detail in a number of references [21,23,24]. The technique relies on the hyperspherical coordinate approach when used for few-body processes, including rearrangement. For instance, in a reactive triatomic process, the reaction coordinate is represented by the hyperradius, which is a measure of the total inertia of the system, and an adiabatic representation of the total eigenfunction with respect to this coordinate is adopted. In such a way, a quantization problem on the surface of the a hypersphere (in this case the sphere in a 6-dimensional Euclidean space) must be solved for fixed values of the hyperradius. Then coupled-channel equations are integrated applying a standard propagation procedure. The success of this approach is strictly dependent on the accuracy and the effectiveness of the method used to solve the fixed hyperradius problem. The computation of the adiabatic eigenvalues containing detailed information on the structure, rotations, and internal modes parametrically in the hyperradius is typically very demanding. The hyperquantization algorithm exploits the peculiar properties of the discrete analogues of hyperspherical harmonics, i.e. generalized 3j symbols or Hahn polynomials, orthogonal on a grid of points that span the interaction region [25]. The computationally advantageous aspect of this algorithm, besides the elegance of unifying under the language of angular momentum theory the dynamical treatment of a reaction, is the structure of the Hamiltonian matrix: its kinetic part is simple, universal, highly symmetric, and sparse, while the potential displays the diagonal form characteristic of the stereodirected representations of the previous section. The hyperquantization algorithm, when implemented for reactive scattering calculations [21], allows considerable savings in memory requirements for storage and in computing time for the building up and diagonalization of large basis sets, exploiting the sparseness and the symmetry properties of the Hamiltonian matrix. 

Quantum computing

In the past few years there has been a vigorous activity aimed at introducing novel conceptual schemes for quantum computing. The model of quantum simulator proposed in [5,26] and further discussed in [27, 28] relies on the recoupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on "qubits" and Boolean gates [29]. The computational space of the simulator complies with the architecture of (families of) "automata" (an automaton in computer science is a graph whose nodes encode "internal states" while a link between two nodes represents an admissibile "transition" between the corresponding states). According with this kind of interpretation, a computational process on the spin network can be associated with a "directed path", namely an ordered sequence of vertices and edges starting from an initial quantum state and ending in some set of final states. 

In a series of papers [30-34] families of automata arising from the q-deformed analog of the spin network simulator have been implemented in order to deal with classes of computationally-hard problems in geometric topology (topological invariants associated with knots and with closed 3-dimensional manifolds). From the point of view of classical complexity theory, computing such invariants is "hard", namely could be achieved by a classical computer only by resorting to an exponential amount of resources. 

In [30,31] efficient ( i. e. running in polynomial time) quantum algorithms for approximating, within an arbitrarily small range, SU(2)-q-colored polynomial invariants of knots have been explicitly worked out. In [32, 33] such algorithms have been generalized to deal with 3-manifold invariants, and connections among quantized geometries, topological quantum field theory and quantum computing issues are developed in [34, 35].