1,460 research outputs found
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
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