20 research outputs found
A simple construction of basic polynomials invariant under the Weyl group of the simple finite-dimensional complex Lie algebra
summary:For every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree
A perturbative approach to the quantum elliptic Calogero-Sutherland model
We solve perturbatively the quantum elliptic Calogero-Sutherland model in the
regime in which the quotient between the real and imaginary semiperiods of the
Weierstrass function is smallComment: 6 pages, no figure
Generating functions and multiplicity formulas: the case of rank two simple Lie algebras
A procedure is described that makes use of the generating function of
characters to obtain a new generating function giving the multiplicities of
each weight in all the representations of a simple Lie algebra. The way to
extract from explicit multiplicity formulas for particular weights is
explained and the results corresponding to rank two simple Lie algebras shown
On the generating function of weight multiplicities for the representations of the Lie algebra
We use the generating function of the characters of to obtain a
generating function for the multiplicities of the weights entering in the
irreducible representations of that simple Lie algebra. From this generating
function we derive some recurrence relations among the multiplicities and a
simple graphical recipe to compute them.Comment: arXiv admin note: text overlap with arXiv:1304.720
Berry phase in homogeneous K\"ahler manifolds with linear Hamiltonians
We study the total (dynamical plus geometrical (Berry)) phase of cyclic
quantum motion for coherent states over homogeneous K\"ahler manifolds X=G/H,
which can be considered as the phase spaces of classical systems and which are,
in particular cases, coadjoint orbits of some Lie groups G. When the
Hamiltonian is linear in the generators of a Lie group, both phases can be
calculated exactly in terms of {\em classical} objects. In particular, the
geometric phase is given by the symplectic area enclosed by the (purely
classical) motion in the space of coherent states.Comment: LaTeX fil
A squeezed review on coherent states and nonclassicality for non-Hermitian systems with minimal length
It was at the dawn of the historical developments of quantum mechanics when Schrödinger, Kennard and Darwin proposed an interesting type of Gaussian wave packets, which do not spread out while evolving in time. Originally, these wave packets are the prototypes of the renowned discovery, which are familiar as “coherent states” today. Coherent states are inevitable in the study of almost all areas of modern science, and the rate of progress of the subject is astonishing nowadays. Nonclassical states constitute one of the distinguished branches of coherent states having applications in various subjects including quantum information processing, quantum optics, quantum superselection principles and mathematical physics. On the other hand, the compelling advancements of non-Hermitian systems and related areas have been appealing, which became popular with the seminal paper by Bender and Boettcher in 1998. The subject of non-Hermitian Hamiltonian systems possessing real eigenvalues are exploding day by day and combining with almost all other subjects rapidly, in particular, in the areas of quantum optics, lasers and condensed matter systems, where one finds ample successful experiments for the proposed theory. For this reason, the study of coherent states for non-Hermitian systems have been very important. In this article, we review the recent developments of coherent and nonclassical states for such systems and discuss their applications and usefulness in different contexts of physics. In addition, since the systems considered here originated from the broader context of the study of minimal uncertainty relations, our review is also of interest to the mathematical physics communit
A simple construction of basic polynomials invariant under the Weyl group of the simple finite-dimensional complex Lie algebra
For every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree