Mariano A. del Olmo

Coherent states on the circle

A careful study of the physical properties of a family of coherent states on the circle, introduced some years ago by de Bievre and Gonzalez (in 1992 Semiclassical behaviour of the Weyl correspondence on the circle Group Theoretical Methods in Physics vol I (Madrid: Ciemat)), is carried out. They were obtained from the Weyl-Heisenberg coherent states in by means of the Weil-Brezin-Zak transformation, they are labelled by the points of the cylinder , and they provide a realization of by entire functions (similar to the well known Fock-Bargmann construction). In particular, we compute the expectation values of the position and momentum operators on the circle and we discuss the Heisenberg uncertainty relation.

Chains of twists for classical Lie algebras

For chains of regular injections of Hopf algebras the sets of maximal extended Jordanian twists {k} are considered. We prove that under certain conditions there exists for 0 the twist composed of the factors k. The general construction of a chain of twists is applied to the universal envelopings U( ) of classical Lie algebras . We study the chains for the infinite series An,Bn and Dn. The properties of the deformation produced by a chain U( ) are explicitly demonstrated for the case of = so(9).

The Stratonovich–Weyl correspondence for one‐dimensional kinematical groups

The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self‐adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one‐dimensional Galilei, Poincare, and Newton–Hooke groups as well as the two‐dimensional Euclidean group.

On Hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry

Abstract We show that for positive integer values l of the parameter in the conformal mechanics model the system possesses a hidden nonlinear superconformal symmetry, in which reflection plays a role of the grading operator. In addition to the even so ( 1 , 2 ) ⊕ u ( 1 ) -generators, the superalgebra includes 2 l + 1 odd integrals, which form the pair of spin- ( l + 1 2 ) representations of the bosonic subalgebra and anticommute for order 2 l + 1 polynomials of the even generators. This hidden symmetry, however, is broken at the level of the states in such a way that the action of the odd generators violates the boundary condition at the origin. In the earlier observed double nonlinear superconformal symmetry, arising in the superconformal mechanics for certain values of the boson–fermion coupling constant, the higher order symmetry is of the same, broken nature.

Electric Chern–Simons term, enlarged exotic Galilei symmetry and noncommutative plane

Abstract The extended exotic planar model for a charged particle is constructed. It includes a Chern–Simons-like term for a dynamical electric field, but produces usual equations of motion for the particle in background constant uniform electric and magnetic fields. The electric Chern–Simons term is responsible for the noncommutativity of the boost generators in the 10-dimensional enlarged exotic Galilei symmetry algebra of the extended system. The model admits two reduction schemes by the integrals of motion, one of which reproduces the usual formulation for the charged particle in external constant electric and magnetic fields with associated field-deformed Galilei symmetry, whose commuting boost generators are identified with the nonlocal in time Noether charges reduced on-shell. Another reduction scheme, in which electric field transmutes into the commuting space translation generators, extracts from the model a free particle on the noncommutative plane described by the twofold centrally extended Galilei group of the nonrelativistic anyons.

Classical superintegrable SO (p, q) Hamiltonian systems

Abstract A family of superintegrable real Hamiltonian systems exhibiting SO ( p , q ) symmetry is obtained by symmetry reduction from free SU ( p , q ) integrable Hamiltonian systems. Among them we find Poschl-Teller potentials. The Hamilton-Jacobi equation is solved in a separable coordinate system in a generic way for the whole family. We also study the projection of the geodesic flow from the complex to the real systems.

Locally operating realizations of transformation Lie groups

Using the Mackey theory of induced representations, a systematic study of the locally operating multiplier realizations of a connected Lie group G that acts transitively on a space‐time manifold is presented. We obtain a mathematical characterization of the locally operating multiplier realizations and a reduction of the problem of multiplier locally operating realizations to linear ones via a splitting group G‘;m for G. In this way the locally operating multiplier realizations are obtained by induction from finite‐dimensional linear representations of a well‐determined subgroup of G. Some examples, such as the two‐dimensional Euclidean group, the Galilei group, and the one‐dimensional Newton–Hooke group, are given.

Local realizations of kinematical groups with a constant electromagnetic field. I. The relativistic case

This paper is devoted to the study of the description of elementary physical systems interacting with an external constant electromagnetic field and the construction of their differential wave equations from a group‐theoretical point of view. In this context certain local realizations of the Poincare group are studied. The linearization of this problem is carried out by building the associated representation group that turns out to be the well‐known Maxwell group. In this way the usual method (concerning local realizations) that has been employed in studying free systems to the interacting case is extended.

Central extensions and realizations of one-dimensional Galilean systems and quantization

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.

Contraction of superintegrable Hamiltonian systems

We investigate the contraction of a class of superintegrable Hamiltonians by implementing the contraction of the underlying Lie groups. We also discuss the behavior of the coordinate systems that separate their equations of motion, the motion constants, as well as the corresponding solutions along such a process.