19,973 research outputs found
Symbol correspondences for spin systems
The present monograph explores the correspondence between quantum and
classical mechanics in the particular context of spin systems, that is,
SU(2)-symmetric mechanical systems. Here, a detailed presentation of quantum
spin-j systems, with emphasis on the SO(3)-invariant decomposition of their
operator algebras, is followed by an introduction to the Poisson algebra of the
classical spin system and a similarly detailed presentation of its
SO(3)-invariant decomposition. Subsequently, this monograph proceeds with a
detailed and systematic study of general quantum-classical symbol
correspondences for spin-j systems and their induced twisted products of
functions on the 2-sphere. This original systematic presentation culminates
with the study of twisted products in the asymptotic limit of high spin
numbers. In the context of spin systems, it shows how classical mechanics may
or may not emerge as an asymptotic limit of quantum mechanics.Comment: Research Monograph, 171 pages (book format, preliminary version
Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology
A key symmetry of classical -branes is invariance under worldvolume
diffeomorphisms. Under the assumption that the worldvolume, at fixed values of
the time, is a compact, quantisable K\"ahler manifold, we prove that the Lie
algebra of volume-preserving diffeomorphisms of the worldvolume can be
approximated by , for . We also prove, under the same
assumptions regarding the worldvolume at fixed time, that classical Nambu
brackets on the worldvolume are quantised by the multibrackets corresponding to
cocycles in the cohomology of the Lie algebra .Comment: This is a contribution to the Special Issue on Deformation
Quantization, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Even Dimensional Improper Affine Spheres
There are exactly two different types of bi-dimensional improper affine
spheres: the non-convex ones can be modeled by the center-chord transform of a
pair of planar curves while the convex ones can be modeled by a holomorphic
map. In this paper, we show that both constructions can be generalized to
arbitrary even dimensions: the former class corresponds to the center-chord
transform of a pair of Lagrangian submanifolds while the latter is related to
special K\"ahler manifolds. Furthermore, we show that the improper affine
spheres obtained in this way are solutions of certain exterior differential
systems. Finally, we also discuss the problem of realization of simple stable
Legendrian singularities as singularities of these improper affine spheres.Comment: 26 page
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