4,767 research outputs found
Duality Principle and Braided Geometry
We give an overview of a new kind symmetry in physics which exists between
observables and states and which is made possible by the language of Hopf
algebras and quantum geometry. It has been proposed by the author as a feature
of Planck scale physics. More recent work includes corresponding results at the
semiclassical level of Poisson-Lie groups and at the level of braided groups
and braided geometry.Comment: 24 page
Hopf quasigroups and the algebraic 7-sphere
We introduce the notions of Hopf quasigroup and Hopf coquasigroup
generalising the classical notion of an inverse property quasigroup
expressed respectively as a quasigroup algebra and an algebraic
quasigroup . We prove basic results as for Hopf algebras, such as
anti(co)multiplicativity of the antipode , that S^2=\id if is
commutative or cocommutative, and a theory of crossed (co)products. We also
introduce the notion of a Moufang Hopf (co)quasigroup and show that the
coordinate algebras of the parallelizable spheres are algebraic
quasigroups (commutative Hopf coquasigroups in our formulation) and Moufang. We
make use of the description of composition algebras such as the octonions via a
cochain introduced in \cite{Ma99}. We construct an example
of a Hopf coquasigroup which is noncommutative and
non-trivially Moufang. We use Hopf coquasigroup methods to study differential
geometry on including a short algebraic proof that is
parallelizable. Looking at combinations of left and right invariant vector
fields on we provide a new description of the structure constants of
the Lie algebra in terms of the structure constants of the octonions.
In the concluding section we give a new description of the -deformation
quantum group \C_q[S^3] regarded trivially as a Moufang Hopf coquasigroup
(trivially since it is in fact a Hopf algebra) but now in terms of built up
via the Cayley-Dickson process.Comment: 43 pages latex; added Maurer-Cartan equation (Prop 6.5) and
computation of it for S^7 (lemma 6.8). No other change aside typo
Quantum Deformations of Space-Time Symmetries with Mass-Like Deformation Parameter
The difficulties with the measurability of classical space-time distances are
considered. We outline the framework of quantum deformations of D=4 space-time
symmetries with dimensionfull deformation parameter, and present some recent
results.Comment: 4 pages, LaTeX, uses file stwol.sty, to be published in the
Proceedings of XXXII International Rochester Conference in High Energy
Physics (Warsaw, 24.07-31.07 1996
Almost commutative Riemannian geometry: wave operators
Associated to any (pseudo)-Riemannian manifold of dimension is an
-dimensional noncommutative differential structure (\Omega^1,\extd) on
the manifold, with the extra dimension encoding the classical Laplacian as a
noncommutative `vector field'. We use the classical connection, Ricci tensor
and Hodge Laplacian to construct (\Omega^2,\extd) and a natural
noncommutative torsion free connection on . We show
that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to
\Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only
when the original is flat, i.e their failure is governed by the Riemann
curvature, and that \sigma^2=\id only when is Einstein. We show that if
has a conformal Killing vector field then the cross product algebra
viewed as a noncommutative analogue of has a
natural -dimensional calculus extending and a natural spacetime
Laplacian now directly defined by the extra dimension. The case
recovers the Majid-Ruegg bicrossproduct flat spacetime model and the
wave-operator used in its variable speed of light preduction, but now as an
example of a general construction. As an application we construct the wave
operator on a noncommutative Schwarzschild black hole and take a first look at
its features. It appears that the infinite classical redshift/time dilation
factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a
separate paper (now ArXived) to meet referee length requirements.
Corresponding slight restructure but no change to remaining conten
Towards Spinfoam Cosmology
We compute the transition amplitude between coherent quantum-states of
geometry peaked on homogeneous isotropic metrics. We use the holomorphic
representations of loop quantum gravity and the
Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at
first order in the vertex expansion, second order in the graph (multipole)
expansion, and first order in 1/volume. We show that the resulting amplitude is
in the kernel of a differential operator whose classical limit is the canonical
hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an
indication that the dynamics of loop quantum gravity defined by the new vertex
yields the Friedmann equation in the appropriate limit.Comment: 8 page
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