731 research outputs found
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
A SU(2) recipe for mutually unbiased bases
A simple recipe for generating a complete set of mutually unbiased bases in
dimension (2j+1)**e, with 2j + 1 prime and e positive integer, is developed
from a single matrix acting on a space of constant angular momentum j and
defined in terms of the irreducible characters of the cyclic group C(2j+1). As
two pending results, this matrix is used in the derivation of a polar
decomposition of SU(2) and of a FFZ algebra.Comment: v2: abstract enlarged, a corollary added, acknowledgments added, one
reference added, presentation improved; v3: two misprints correcte
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
SU(2) nonstandard bases: the case of mutually unbiased bases
This paper deals with bases in a finite-dimensional Hilbert space. Such a
space can be realized as a subspace of the representation space of SU(2)
corresponding to an irreducible representation of SU(2). The representation
theory of SU(2) is reconsidered via the use of two truncated deformed
oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme
{j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the
enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting
set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1),
2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where
2j+1 is prime, the corresponding eigenvectors generate a complete set of
mutually unbiased bases. Some useful relations on generalized quadratic Gauss
sums are exposed in three appendices.Comment: 33 pages; version2: rescaling of generalized Hadamard matrices,
acknowledgment and references added, misprints corrected; version 3:
published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA/ (22 pages
An Rotor Model for Rotational Bands of Superdeformed Nuclei
A nonrigid rotor model is developed from the two-parameter quantum algebra
. [This model presents the symmetry and
shall be referred to as the qp-rotor model.] A rotational energy formula as
well as a qp-deformation of E2 reduced transition probabilities are derived.
The qp-rotor model is applied (through fitting procedures) to twenty rotational
bands of superdeformed nuclei in the , 150 and 190 mass regions.
Systematic comparisons between the qp-rotor model and the q-rotor model of
Raychev, Roussev and Smirnov, on one hand, and a basic three-parameter model,
on the other hand, are performed on energy spectra, on dynamical moments of
inertia and on B(E2) values. The physical signification of the deformation
parameters q and p is discussed.Comment: 24 pages, Latex File, to appear in IJMP
On -Deformations in Statistical Mechanics of Bosons in D Dimensions
The Bose distribution for a gas of nonrelativistic free bosons is derived in
the framework of -deformed second quantization. Some thermodynamical
functions for such a system in D dimensions are derived. Bose-Einstein
condensation is discussed in terms of the parameters q and p as well as a
parameter which characterizes the representation space of the
oscillator algebra.Comment: 15 pages, Latex File, to be published in Symmetry and Structural
Properties of Condensed Matter, Eds. T. Lulek, B. Lulek and W. Florek (World
Scientific, Singapore, 1997
Phase operators, phase states and vector phase states for SU(3) and SU(2,1)
This paper focuses on phase operators, phase states and vector phase states
for the sl(3) Lie algebra. We introduce a one-parameter generalized oscillator
algebra A(k,2) which provides a unified scheme for dealing with su(3) (for k <
0), su(2,1) (for k > 0) and h(4) x h(4) (for k = 0) symmetries. Finite- and
infinite-dimensional representations of A(k,2) are constructed for k < 0 and k
> 0 or = 0, respectively. Phase operators associated with A(k,2) are defined
and temporally stable phase states (as well as vector phase states) are
constructed as eigenstates of these operators. Finally, we discuss a relation
between quantized phase states and a quadratic discrete Fourier transform and
show how to use these states for constructing mutually unbiased bases
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