80 research outputs found
Graded Lie algebras, representation theory, integrable mappings and systems: nonabelian case
The exactly integrable systems connected with semisimple series for
arbitrary grading are presented in explicit form. Their general solutions are
expressed in terms of the matrix elements of various fundamental
representations of groups. The simplest example of such systems is the
generalized Toda chain with the matrices of arbitrary dimensions in each point
of the lattice.Comment: LaTeX, 19 page
The Solution of the N=2 Supersymmetric f-Toda Chain with Fixed Ends
The integrability of the recently introduced N=2 supersymmetric f-Toda chain,
under appropriate boundary conditions, is proven. The recurrent formulae for
its general solutions are derived. As an example, the solution for the simplest
case of boundary conditions is presented in explicit form.Comment: 15 pages, latex, no figure
Casimir interactions of an object inside a spherical metal shell
We investigate the electromagnetic Casimir interactions of an object
contained within an otherwise empty, perfectly conducting spherical shell. For
a small object we present analytical calculations of the force, which is
directed away from the center of the cavity, and the torque, which tends to
align the object opposite to the preferred alignment outside the cavity. For a
perfectly conducting sphere as the interior object, we compute the corrections
to the proximity force approximation (PFA) numerically. In both cases the
results for the interior configuration match smoothly onto those for the
corresponding exterior configuration.Comment: 4 pages, 3 figure
Fermionic flows and tau function of the N=(1|1) superconformal Toda lattice hierarchy
An infinite class of fermionic flows of the N=(1|1) superconformal Toda
lattice hierarchy is constructed and their algebraic structure is studied. We
completely solve the semi-infinite N=(1|1) Toda lattice and chain hierarchies
and derive their tau functions, which may be relevant for building
supersymmetric matrix models. Their bosonic limit is also discussed.Comment: 11 pages, no figures, revised version published in Nucl. Phys.
The Casimir Energy for a Hyperboloid Facing a Plate in the Optical Approximation
We study the Casimir energy of a massless scalar field that obeys Dirichlet
boundary conditions on a hyperboloid facing a plate. We use the optical
approximation including the first six reflections and compare the results with
the predictions of the proximity force approximation and the semi-classical
method. We also consider finite size effects by contrasting the infinite with a
finite plate. We find sizable and qualitative differences between the new
optical method and the more traditional approaches.Comment: v2: 14 pages, 11 eps figures; typo in eq. (21) removed, clarification
added, fig. 10 improved; version published in Phys. Rev.
Casimir potential of a compact object enclosed by a spherical cavity
We study the electromagnetic Casimir interaction of a compact object
contained inside a closed cavity of another compact object. We express the
interaction energy in terms of the objects' scattering matrices and translation
matrices that relate the coordinate systems appropriate to each object. When
the enclosing object is an otherwise empty metallic spherical shell, much
larger than the internal object, and the two are sufficiently separated, the
Casimir force can be expressed in terms of the static electric and magnetic
multipole polarizabilities of the internal object, which is analogous to the
Casimir-Polder result. Although it is not a simple power law, the dependence of
the force on the separation of the object from the containing sphere is a
universal function of its displacement from the center of the sphere,
independent of other details of the object's electromagnetic response.
Furthermore, we compute the exact Casimir force between two metallic spheres
contained one inside the other at arbitrary separations. Finally, we combine
our results with earlier work on the Casimir force between two spheres to
obtain data on the leading order correction to the Proximity Force
Approximation for two metallic spheres both outside and within one another.Comment: 12 pages, 6 figure
Casimir Effects: An Optical Approach I. Foundations and Examples
We present the foundations of a new approach to the Casimir effect based on
classical ray optics. We show that a very useful approximation to the Casimir
force between arbitrarily shaped smooth conductors can be obtained from
knowledge of the paths of light rays that originate at points between these
bodies and close on themselves. Although an approximation, the optical method
is exact for flat bodies, and is surprisingly accurate and versatile. In this
paper we present a self-contained derivation of our approximation, discuss its
range of validity and possible improvements, and work out three examples in
detail. The results are in excellent agreement with recent precise numerical
analysis for the experimentally interesting configuration of a sphere opposite
an infinite plane.Comment: Minor corrections. Submitted to Nucl. Phys.
Hamiltonian structures of fermionic two-dimensional Toda lattice hierarchies
By exhibiting the corresponding Lax pair representations we propose a wide
class of integrable two-dimensional (2D) fermionic Toda lattice (TL)
hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL
hierarchies as particular cases. We develop the generalized graded R-matrix
formalism using the generalized graded bracket on the space of graded operators
with involution generalizing the graded commutator in superalgebras, which
allows one to describe these hierarchies in the framework of the Hamiltonian
formalism and construct their first two Hamiltonian structures. The first
Hamiltonian structure is obtained for both bosonic and fermionic Lax operators
while the second Hamiltonian structure is established for bosonic Lax operators
only.Comment: 12 pages, LaTeX, the talks delivered at the International Workshop on
Classical and Quantum Integrable Systems (Dubna, January 24 - 28, 2005) and
International Conference on Theoretical Physics (Moscow, April 11 - 16, 2005
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