An Alternative Basis for the Wigner-Racah Algebra of the Group SU(2)

The Lie algebra of the classical group SU(2) is constructed from two quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder generators of the SU(2) Lie algebra and to (ii) an alternative to the (J,M) quantization scheme, viz., the (J,alpha) quantization scheme. The key ideas for developing the Wigner-Racah algebra of the group SU(2) in the (J,alpha) scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the (J,alpha) scheme are briefly discussed.Comment: 12 pages, Latex file. Submitted for publication to Turkish Journal of Physic

On the Wigner-Racah Algebra of the Group SU(2) in a Non-Standard Basis

The algebra su(2) is derived from two commuting quon algebras for which the parameter q is a root of unity. This leads to a polar decomposition of the shift operators of the group SU(2). The Wigner-Racah algebra of SU(2) is developed in a new basis arising from the simultanenous diagonalization of two commuting operators, viz., the Casimir of SU(2) and a unitary operator which takes its origin in the polar decomposition of the shift operators of SU(2).Comment: 13 pages, Latex file. Paper based on a lecture given to the Vth International School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" (Zaj\c aczkowo, Poland, 27 August - 2 September 1998

Angular Momentum and Mutually Unbiased Bases

The Lie algebra of the group SU(2) is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J2,Ur} scheme, an alternative to the familiar {J2,Jz} quantization scheme corresponding to common eigenvectors of the Casimir operator J2 and the Cartan operator Jz. A connection is established between the eigenvectors of the complete set of commuting operators {J2,Ur} and mutually unbiased bases in spaces of constant angular momentum.Comment: To be published in International Journal of Modern Physics

Representation theory and Wigner-Racah algebra of the SU(2) group in a noncanonical basis

The Lie algebra su(2) of the classical group SU(2) is built from two commuting quon algebras for which the deformation parameter is a common root of unity. This construction leads to (i) a not very well-known polar decomposition of the ladder generators of the SU(2) group, in terms of a unitary operator and a Hermitean operator, and (ii) a nonstandard quantization scheme, alternative to the usual quantization scheme correponding to the diagonalization of the Casimir of su(2) and of the z-generator. The representation theory of the SU(2) group can be developed in this nonstandard scheme. The key ideas for developing the Wigner-Racah algebra of the SU(2) group in the nonstandard scheme are given. In particular, some properties of the coupling and recoupling coefficients as well as the Wigner-Eckart theorem in the nonstandard scheme are examined in great detail.Comment: To be presented at ICSSUR'05 (9th International Conference on Squeezed States and Uncertainty Relations, France, 2-6 May 2005). Dedicated to Professor Josef Paldus on the occasion of his 70th birthday. To be published in Collection of Czechoslovak Chemical Communication

On Two Approaches to Fractional Supersymmetric Quantum Mechanics

Two complementary approaches of N = 2 fractional supersymmetric quantum mechanics of order k are studied in this article. The first one, based on a generalized Weyl-Heisenberg algebra W(k) (that comprizes the affine quantum algebra Uq(sl(2)) with q to k = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second one uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetric oscillator of order k.Comment: 25 pages, LaTex file, based on a talk given by M. Kibler at the "IX International Conference on Symmetry Methods in Physics" (Yerevan, Armenia, 3-8 July 2001) organized by the Joint Institute for Nuclear Research (Dubna, Russia) and the Yerevan State University (Yerevan, Armenia

Bases for qudits from a nonstandard approach to SU(2)

Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in one step the (1+p)p qupits (i.e., qudits with d = p a prime integer) of a complete set of 1+p mutually unbiased bases in C^p. Repeated application of the formula can be used for generating mutually unbiased bases in C^d with d = p^e (e > or = 2) a power of a prime integer. A connection between mutually unbiased bases and the unitary group SU(d) is briefly discussed in the case d = p^e.Comment: From a talk presented at the 13th International Conference on Symmetry Methods in Physics (Dubna, Russia, 6-9 July 2009) organized in memory of Prof. Yurii Fedorovich Smirnov by the Bogoliubov Laboratory of Theoretical Physics of the JINR and the ICAS at Yerevan State University