322 research outputs found
Representation Theory Approach to the Polynomial Solutions of q - Difference Equations : U_q(sl(3)) and Beyond,
A new approach to the theory of polynomial solutions of q - difference
equations is proposed. The approach is based on the representation theory of
simple Lie algebras and their q - deformations and is presented here for
U_q(sl(n)). First a q - difference realization of U_q(sl(n)) in terms of
n(n-1)/2 commuting variables and depending on n-1 complex representation
parameters r_i, is constructed. From this realization lowest weight modules
(LWM) are obtained which are studied in detail for the case n=3 (the well known
n=2 case is also recovered). All reducible LWM are found and the polynomial
bases of their invariant irreducible subrepresentations are explicitly given.
This also gives a classification of the quasi-exactly solvable operators in the
present setting. The invariant subspaces are obtained as solutions of certain
invariant q - difference equations, i.e., these are kernels of invariant q -
difference operators, which are also explicitly given. Such operators were not
used until now in the theory of polynomial solutions. Finally the states in all
subrepresentations are depicted graphically via the so called Newton diagrams.Comment: uuencoded Z-compressed .tar file containing two ps files
Classification of quantum relativistic orientable objects
Started from our work "Fields on the Poincare Group and Quantum Description
of Orientable Objects" (EPJC,2009), we consider here a classification of
orientable relativistic quantum objects in 3+1 dimensions. In such a
classification, one uses a maximal set of 10 commuting operators (generators of
left and right transformations) in the space of functions on the Poincare
group. In addition to usual 6 quantum numbers related to external symmetries
(given by left generators), there appear additional quantum numbers related to
internal symmetries (given by right generators). We believe that the proposed
approach can be useful for description of elementary spinning particles
considering as orientable objects. In particular, their classification in the
framework of the approach under consideration reproduces the usual
classification but is more comprehensive. This allows one to give a
group-theoretical interpretation to some facts of the existing phenomenological
classification of known spinning particles.Comment: 24 page
Representation Theory of Quantized Poincare Algebra. Tensor Operators and Their Application to One-Partical Systems
A representation theory of the quantized Poincar\'e (-Poincar\'e)
algebra (QPA) is developed. We show that the representations of this algebra
are closely connected with the representations of the non-deformed Poincar\'e
algebra. A theory of tensor operators for QPA is considered in detail.
Necessary and sufficient conditions are found in order for scalars to be
invariants. Covariant components of the four-momenta and the Pauli-Lubanski
vector are explicitly constructed.These results are used for the construction
of some q-relativistic equations. The Wigner-Eckart theorem for QPA is proven.Comment: 18 page
Structure Constants for New Infinite-Dimensional Lie Algebras of U(N+,N-) Tensor Operators and Applications
The structure constants for Moyal brackets of an infinite basis of functions
on the algebraic manifolds M of pseudo-unitary groups U(N_+,N_-) are provided.
They generalize the Virasoro and W_\infty algebras to higher dimensions. The
connection with volume-preserving diffeomorphisms on M, higher generalized-spin
and tensor operator algebras of U(N_+,N_-) is discussed. These
centrally-extended, infinite-dimensional Lie-algebras provide also the arena
for non-linear integrable field theories in higher dimensions, residual gauge
symmetries of higher-extended objects in the light-cone gauge and C^*-algebras
for tractable non-commutative versions of symmetric curved spaces.Comment: 8 pages, LaTeX, no figures; minor comments added; to appear in J.
Phys A (Math. Gen.
Hyperspherical harmonics with arbitrary arguments
The derivation scheme for hyperspherical harmonics (HSH) with arbitrary
arguments is proposed. It is demonstrated that HSH can be presented as the
product of HSH corresponding to spaces with lower dimensionality multiplied by
the orthogonal (Jacobi or Gegenbauer) polynomial. The relation of HSH to
quantum few-body problems is discussed. The explicit expressions for
orthonormal HSH in spaces with dimensions from 2 to 6 are given. The important
particular cases of four- and six-dimensional spaces are analyzed in detail and
explicit expressions for HSH are given for several choices of hyperangles. In
the six-dimensional space, HSH representing the kinetic energy operator
corresponding to i) the three-body problem in physical space and ii) four-body
planar problem are derived.Comment: 18 pages, 1 figur
Boson representations, non-standard quantum algebras and contractions
A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the
non-standard quantum deformation of which directly provides its
infinite and finite dimensional irreducible representations. Tensor product
decompositions are worked out for some examples. Relations between contraction
methods and boson realizations are also explored in several contexts. So, a
class of two-boson representations for the non-standard deformation of
is introduced and contracted to the non-standard quantum (1+1)
Poincar\'e representations. Likewise, a quantum extended Hopf
algebra is constructed and the Jordanian -oscillator algebra representations
are obtained from it by means of another contraction procedure.Comment: 21 pages, LaTeX; two new references adde
Non-Canonical Perturbation Theory of Non-Linear Sigma Models
We explore the O(N)-invariant Non-Linear Sigma Model (NLSM) in a different
perturbative regime from the usual relativistic-free-field one, by using
non-canonical basic commutation relations adapted to the underlying O(N)
symmetry of the system, which also account for the non-trivial (non-flat)
geometry and topology of the target manifold.Comment: 11 pages, 1 figure, LaTe
Q-Boson Representation of the Quantum Matrix Algebra
{Although q-oscillators have been used extensively for realization of quantum
universal enveloping algebras,such realization do not exist for quantum matrix
algebras ( deformation of the algebra of functions on the group ). In this
paper we first construct an infinite dimensional representation of the quantum
matrix algebra (the coordinate ring of and then use
this representation to realize by q-bosons.}Comment: pages 18 ,report # 93-00
Entanglement and statistics in Hong-Ou-Mandel interferometry
Hong-Ou-Mandel interferometry allows one to detect the presence of
entanglement in two-photon input states. The same result holds for
two-particles input states which obey to Fermionic statistics. In the latter
case however anti-bouncing introduces qualitative differences in the
interferometer response. This effect is analyzed in a Gedankenexperiment where
the particles entering the interferometer are assumed to belong to a
one-parameter family of quons which continuously interpolate between the
Bosonic and Fermionic statistics.Comment: 7 pages, 3 figures; minor editorial changes and new references adde
Photon-added coherent states as nonlinear coherent states
The states , defined as up to a
normalization constant and is a nonnegative integer, are shown to be the
eigenstates of where is a nonlinear
function of the number operator . The explicit form of
is constructed. The eigenstates of this operator for negative values of are
introduced. The properties of these states are discussed and compared with
those of the state .Comment: Rev Tex file with two figures as postscript files attached. Email:
[email protected]
- …