191 research outputs found
A Natural Basis for Spinor and Vector Fields on the Noncommutative sphere
The product of two Heisenberg-Weil algebras contains the Jordan-Schwinger
representation of su(2). This Algebra is quotiented by the square-root of the
Casimir to produce a non-associative algebra denoted by . This algebra
may be viewed as the right-module over one of its associative subalgebras which
corresponds to the algebra of scalar fields on the noncommutative sphere. It is
now possible to interpret other subspaces as the space of spinor or vector
fields on the noncommutative sphere. A natural basis of is given which
may be interpreted as the deformed entries in the rotation matrices of SU(2).Comment: 21 pages Latex, No figures. Submitted to Journal of Mathematical
Physic
"Wick Rotations": The Noncommutative Hyperboloids, and other surfaces of rotations
A ``Wick rotation'' is applied to the noncommutative sphere to produce a
noncommutative version of the hyperboloids. A harmonic basis of the associated
algebra is given. It is noted that, for the one sheeted hyperboloid, the vector
space for the noncommutative algebra can be completed to a Hilbert space, where
multiplication is not continuous. A method of constructing noncommutative
analogues of surfaces of rotation, examples of which include the paraboloid and
the -deformed sphere, is given. Also given are mappings between
noncommutative surfaces, stereographic projections to the complex plane and
unitary representations. A relationship with one dimensional crystals is
highlighted.Comment: Latex, 12 pages, 0 figures, submitted to Lett. Math. Phy
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket
An infinite dimensional algebra which is a non-decomposable reducible
representation of is given. This algebra is defined with respect to two
real parameters. If one of these parameters is zero the algebra is the
commutative algebra of functions on the sphere, otherwise it is a
noncommutative analogue. This is an extension of the algebra normally refered
to as the (Berezin) quantum sphere or ``fuzzy'' sphere. A natural indefinite
``inner'' product and a basis of the algebra orthogonal with respect to it are
given. The basis elements are homogenious polynomials, eigenvectors of a
Laplacian, and related to the Hahn polynomials. It is shown that these elements
tend to the spherical harmonics for the sphere. A Moyal bracket is constructed
and shown to be the standard Moyal bracket for the sphere.Comment: 18 pages Latex, No figures, Submitted to Journal of Mathematical
Physics, March 199
Inhomogeneous Spatially Dispersive Electromagnetic Media
Two key types of inhomogeneous spatially dispersive media are described, both
based on a spatially dispersive generalisation of the single resonance model of
permittivity. The boundary conditions for two such media with different
properties are investigated using Lagrangian and distributional methods. Wave
packet solutions to Maxwell's equations, where the permittivity varies and is
periodic in the medium, are then found.Comment: Conference: Progress In Electromagnetics Research Symposium
Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 Published version available
at
http://piers.org/piersproceedings/piers2013StockholmProc.php?searchname=gratu
Classical and Quantum Implications of the Causality Structure of Two-Dimensional Spacetimes with Degenerate Metrics
The causality structure of two-dimensional manifolds with degenerate metrics
is analysed in terms of global solutions of the massless wave equation. Certain
novel features emerge. Despite the absence of a traditional Lorentzian Cauchy
surface on manifolds with a Euclidean domain it is possible to uniquely
determine a global solution (if it exists), satisfying well defined matching
conditions at the degeneracy curve, from Cauchy data on certain spacelike
curves in the Lorentzian region. In general, however, no global solution
satisfying such matching conditions will be consistent with this data.
Attention is drawn to a number of obstructions that arise prohibiting the
construction of a bounded operator connecting asymptotic single particle
states. The implications of these results for the existence of a unitary
quantum field theory are discussed.Comment: 27 pages LaTex (6 Figures), Journal of Mathematical Physics
(Accepted
Spatially Dispersive Inhomogeneous Electromagnetic Media with Periodic Structure
Spatially dispersive (also known as non-local) electromagnetic media are
considered where the parameters defining the permittivity relation vary
periodically. Maxwell's equations give rise to a difference equation
corresponding to the Floquet modes. A complete set of approximate solutions is
calculated which are valid when the inhomogeneity is small. This is applied to
inhomogeneous wire media. A new feature arises when considering spatially
dispersive media, that is the existence of coupled modes.Comment: Full Paper available from Journal of optics.
http://iopscience.iop.org/2040-8986/17/2/025105 17 Pages 7 Figure
- …