24,042 research outputs found
Finite Size Scaling and ``perfect'' actions: the three dimensional Ising model
Using Finite-Size Scaling techniques, we numerically show that the first
irrelevant operator of the lattice theory in three dimensions
is (within errors) completely decoupled at . This interesting
result also holds in the Thermodynamical Limit, where the renormalized coupling
constant shows an extraordinary reduction of the scaling-corrections when
compared with the Ising model. It is argued that Finite-Size Scaling analysis
can be a competitive method for finding improved actions.Comment: 13 pages, 3 figure
New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
A generalized version of Bertrand's theorem on spherically symmetric curved
spaces is presented. This result is based on the classification of
(3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two
families of Hamiltonian systems defined on certain 3-dimensional (Riemannian)
spaces. These two systems are shown to be either the Kepler or the oscillator
potentials on the corresponding Bertrand spaces, and both of them are maximally
superintegrable. Afterwards, the relationship between such Bertrand
Hamiltonians and position-dependent mass systems is explicitly established.
These results are illustrated through the example of a superintegrable
(nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and
physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International
Colloquium on Group Theoretical Methods in Physics, Northumbria University
(U.K.), 26-30th July 201
(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
All Lie bialgebra structures for the (1+1)-dimensional centrally extended
Schrodinger algebra are explicitly derived and proved to be of the coboundary
type. Therefore, since all of them come from a classical r-matrix, the complete
family of Schrodinger Poisson-Lie groups can be deduced by means of the
Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended
Galilei and gl(2) Lie bialgebras within the Schrodinger classification are
studied. As an application, new quantum (Hopf algebra) deformations of the
Schrodinger algebra, including their corresponding quantum universal
R-matrices, are constructed.Comment: 25 pages, LaTeX. Possible applications in relation with integrable
systems are pointed; new references adde
Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations
The link between 3D spaces with (in general, non-constant) curvature and
quantum deformations is presented. It is shown how the non-standard deformation
of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that
represent geodesic motions on 3D manifolds with a non-constant curvature that
turns out to be a function of the deformation parameter z. A different
Hamiltonian defined on the same deformed coalgebra is also shown to generate a
maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D
relativistic spaces whose sectional curvatures are all constant and equal to z.
This approach can be generalized to arbitrary dimension.Comment: 7 pages. Communication presented at the 14th Int. Colloquium on
Integrable Systems 14-16 June 2005, Prague, Czech Republi
Null-plane Quantum Universal -matrix
A non-linear map is applied onto the (non-standard) null-plane deformation of
(3+1) Poincar\'e algebra giving rise to a simpler form of this triangular
quantization. A universal -matrix for the null plane quantum algebra is then
obtained from a universal -matrix corresponding to a Hopf subalgebra.
Finally, the associated Poincar\'e Poisson--Lie group is quantized by using the
FRT approach.Comment: 8 pages, LaTe
On the Universality Class of Monopole Percolation in Scalar QED
We study the critical properties of the monopole-percolation transition in
U(1) lattice gauge theory coupled to scalars at infinite () gauge
coupling. We find strong scaling corrections in the critical exponents that
must be considered by means of an infinite-volume extrapolation. After the
extrapolation, our results are as precise as the obtained for the four
dimensional site-percolation and, contrary to previously stated, fully
compatible with them.Comment: 11 pages, 3 figure
Bases in Lie and Quantum Algebras
Applications of algebras in physics are related to the connection of
measurable observables to relevant elements of the algebras, usually the
generators. However, in the determination of the generators in Lie algebras
there is place for some arbitrary conventions. The situation is much more
involved in the context of quantum algebras, where inside the quantum universal
enveloping algebra, we have not enough primitive elements that allow for a
privileged set of generators and all basic sets are equivalent. In this paper
we discuss how the Drinfeld double structure underlying every simple Lie
bialgebra characterizes uniquely a particular basis without any freedom,
completing the Cartan program on simple algebras. By means of a perturbative
construction, a distinguished deformed basis (we call it the analytical basis)
is obtained for every quantum group as the analytical prolongation of the above
defined Lie basis of the corresponding Lie bialgebra. It turns out that the
whole construction is unique, so to each quantum universal enveloping algebra
is associated one and only one bialgebra. In this way the problem of the
classification of quantum algebras is moved to the classification of
bialgebras. In order to make this procedure more clear, we discuss in detail
the simple cases of su(2) and su_q(2).Comment: 16 pages, Proceedings of the 5th International Symposium on Quantum
Theory and Symmetries QTS5 (July 22-28, 2007, Valladolid (Spain)
Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions
A general procedure to get the explicit solution of the equations of motion
for N-body classical Hamiltonian systems equipped with coalgebra symmetry is
introduced by defining a set of appropriate collective variables which are
based on the iterations of the coproduct map on the generators of the algebra.
In this way several examples of N-body dynamical systems obtained from
q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2)
Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of
Ruijsenaars type arising from the same (non co-boundary) q-deformation of the
(1+1) Poincare' algebra. Also, a unified interpretation of all these systems as
different Poisson-Lie dynamics on the same three dimensional solvable Lie group
is given.Comment: 19 Latex pages, No figure
Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
The full spectrum and eigenfunctions of the quantum version of a nonlinear
oscillator defined on an N-dimensional space with nonconstant curvature are
rigorously found. Since the underlying curved space generates a
position-dependent kinetic energy, three different quantization prescriptions
are worked out by imposing that the maximal superintegrability of the system
has to be preserved after quantization. The relationships among these three
Schroedinger problems are described in detail through appropriate similarity
transformations. These three approaches are used to illustrate different
features of the quantization problem on N-dimensional curved spaces or,
alternatively, of position-dependent mass quantum Hamiltonians. This quantum
oscillator is, to the best of our knowledge, the first example of a maximally
superintegrable quantum system on an N-dimensional space with nonconstant
curvature.Comment: 26 pages, 5 figure
Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and
classified into four multiparametric families. Their quantum deformations are
obtained, together with the corresponding deformed Casimir operators. For the
coboundary cases quantum universal R-matrices are also given. Applications of
the quantum extended Galilei algebras to classical integrable systems are
explicitly developed.Comment: 16 pages, LaTeX. A detailed description of the construction of
integrable systems is carried ou
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