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

On the generating function of weight multiplicities for the representations of the Lie algebra C2C_2

We use the generating function of the characters of $C_2$ 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

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 $H$ giving the multiplicities of each weight in all the representations of a simple Lie algebra. The way to extract from $H$ explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple Lie algebras shown

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

Irreducible Characters and Clebsch-Gordan Series for the Exceptional Algebra E6: An Approach through the Quantum Calogero-Sutherland Model

Preprint[EN] We re-express the quantum Calogero-Sutherland model for the Lie algebra E_6 and the particular value of the coupling constant (\kappa=1) by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra. [ES]Hemos vuelto a expresar el modelo cuántico Calogero-Sutherland para el álgebra de Lie E_6 y el valor particular de la constante de acoplamiento (\kappa=1) utilizando los caracteres irreducibles fundamentales del álgebra como variables dinámicas. Para ello, es necesario desarrollar un procedimiento sistemático para obtener toda la serie de Clebsch-Gordan, necesario para realizar el cambio de variables. Se describe cómo el operador hamiltoniano cuántico resultante se puede utilizar para calcular más caracteres y series Clebsch-Gordan para este álgebra excepcional

A perturbative approach to the quantum elliptic Calogero-Sutherland model

Preprint[EN]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 ${\cal P}$ function is small. [ES]Resolvemos perturbativamente el modelo cuántico elíptico Calogero-Sutherland en el régimen en el que el cociente, entre los semiperiodos real e imaginario de la función Weierstrass ${\cal P}$, es pequeño