arXiv subjects

Joris Van der Jeugt

Publications and source records attributed to Joris Van der Jeugt.

22 records · Page 2Linked to original sources

arXiv2010

Wigner quantization of some one-dimensional Hamiltonians

Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2).

math-ph↗

arXiv2009

Wigner Quantization of Hamiltonians Describing Harmonic Oscillators Coupled by a General Interaction Matrix

In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix. The quantization of a Hamiltonian describing such a system has been done in the canonical case. In this paper, we take a more general approach and look at the system as a Wigner quantum system. Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations. Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n). We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail. We also make the connection with the well-known canonical case.

quant-ph↗

arXiv1997

Representations and Clebsch-Gordan coefficients for the Jordanian quantum algebra U_h(sl(2))

Representation theory for the Jordanian quantum algebra $U=U_h(sl(2))$ is developed. Closed form expressions are given for the action of the generators of U on the basis vectors of finite dimensional irreducible representations. It is shown how representation theory of U has a close connection to combinatorial identities involving summation formulas. A general formula is obtained for the Clebsch-Gordan coefficients $C^{j_1,j_2,j}_{n_1,n_2,m}(h)$ of U. These Clebsch-Gordan coefficients are shown to coincide with those of su(2) for $n_1+n_2 \leq m$, but for $n_1+n_2 > m$ they are in general a nonzero monomial in $h^{n_1+n_2-m}$.

q-alg↗

arXiv1995

Transformation and summation formulas for Kampe de Feriet series.

The double hypergeometric Kampé de Fériet series $F^{0:3}_{1:1}(1,1)$ depends upon 9 complex parameters. We present three cases with 2 relations between those 9 parameters, and show that under these circumstances $F^{0:3}_{1:1}(1,1)$ can be written as a ${}_4F_3(1)$ series. Some limiting cases of these transformation formulas give rise to new summation results for special $F^{0:3}_{1:1}(1,1)$'s. The actual transformation results arose out of the study of 9-$j$ coefficients.

math.CA↗