S. Lievens

On rotation distance between binary coupling trees and applications for 3nj-coefficients

Generalized recoupling coefficients or 3nj-coefficients for a Lie algebra (with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6j-coefficients). In general there exist many such expressions; we say that such an expression is optimal if the number of Racah coefficients in such a product (and, correlated, the number of summation indices) is minimal. The problem of finding an optimal expression for a given 3nj-coefficient is equivalent to finding a shortest path in a graph Gn. The vertices of this graph Gn consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3nj-coefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2n − 1)!!. As the order increases so rapidly, finding a shortest path is computationally achievable only for n < 11. We present some mathematical tools to compute or estimate the length of such shortest paths between binary coupling trees. The diameter of Gn is determined explicitly up to n < 11, and it is shown to grow like n log(n). Thus for n large enough, the number of Racah coefficients in the expansion of a 3nj-coefficient is of order nlog(n). We also show that this problem in Racah—Wigner theory is equivalent to a problem in mathematical biology, where one is concerned with the quantitative comparison of classifications or dendrograms. From this context, some algorithms for approximating the shortest path can be deduced.

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the pos...

On the diameter of the rotation graph of binary coupling trees

A binary coupling tree on n + 1 leaves is a binary tree in which the leaves have distinct labels. The rotation graph Gn is defined as the graph of all binary coupling trees on n + 1 leaves, with edges connecting trees that can be transformed into each other by a single rotation. In this paper, we study distance properties of the graph Gn. Exact results for the diameter of Gn for values up to n = 10 are obtained. For larger values of n, we prove upper and lower bounds for the diameter, which yield the result that the diameter of Gn grows like nlg(n).

Harmonic oscillator chains as Wigner quantum systems: Periodic and fixed wall boundary conditions in gl(1|n) solutions

We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the nth oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representa...

3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

In the tensor product of n+1 positive discrete series representations of su(1,1), a coupled basis vector can be described by a certain binary coupling tree. To every such binary coupling tree, polynomials Rl(k)(x) and Rl(k)(x) are associated. These polynomials are n-variable Jacobi and continuous Hahn polynomials, and are orthogonal with respect to a weight function. The connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj-coefficients of su(1,1).

Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

In a recent paper, Alisauskas deduced different triple sum expressions for the 9-j coefficient of su(2) and suq(2). For a singly stretched 9-j coefficient, these reduce to different double sum series. Using these distinct series, we deduce a set of new transformation formulas for double hypergeometric series of Kampe de Feriet type and their basic analogs. These transformation formulas are valid for rather general parameters of the series, although a common feature is that all the series appearing here are terminating. It is also shown that the transformation formulas deduced here generate a group of transformation formulas, thus yielding an invariance group or symmetry group of particular double series.

The Wigner function of a q-deformed harmonic oscillator model

The phase space representation for a q-deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q-oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h ? 0 (q ? 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q-deformed model shows that, when q 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n. The ground-state-like behaviour of the distribution functions for excited states in the q-deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.

The finite group of the Kummer solutions

In this short communication, which is self-contained, we show that the set of 24 Kummer solutions of the classical hypergeometric differential equation has an elegant, simple group theoretic structure associated with the symmetries of a cube; or, in other words, that the underlying symmetry group is the symmetric group S 4.

On the eigenvalue problem for arbitrary odd elements of the Lie superalgebra and applications

In a Wigner quantum mechanical model, with a solution in terms of the Lie superalgebra , one is faced with determining the eigenvalues and eigenvectors for an arbitrary self-adjoint odd element of in any unitary irreducible representation W. We show that the eigenvalue problem can be solved by the decomposition of W with respect to the branching . The eigenvector problem is much harder, since the Gelfand?Zetlin basis of W is involved, and the explicit actions of generators on this basis are fairly complicated. Using properties of the Gelfand?Zetlin basis, we manage to present a solution for this problem as well. Our solution is illustrated for two special classes of unitary representations: the so-called Fock representations and the ladder representations.

Realizations of coupled vectors in the tensor product of representations of su(1, 1) and su(2)

Using the realization of positive discrete series representations of su(1,1) in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of v + 1 representations as polynomials in v + 1 variables z1,.....,zv+1. These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch-Gordan coefficients of su(1,1). In general, these polynomials can be written as (terminating) multiple hypergeometric series. For v=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The cast of su(2) is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.