3,150 research outputs found
Quantum Gravity as Topological Quantum Field Theory
The physics of quantum gravity is discussed within the framework of
topological quantum field theory. Some of the principles are illustrated with
examples taken from theories in which space-time is three dimensional.Comment: 23 pages, amstex, JMP special issue (deadline permitting). (Text not
changed
State sum models for quantum gravity
This paper reviews the construction of quantum field theory on a
4-dimensional spacetime by combinatorial methods, and discusses the recent
developments in the direction of a combinatorial construction of quantum
gravity.Comment: amslatex, 7 pages, ICMP conference tal
A Lorentzian version of the non-commutative geometry of the standard model of particle physics
A formulation of the non-commutative geometry for the standard model of
particle physics with a Lorentzian signature metric is presented. The
elimination of the fermion doubling in the Lorentzian case is achieved by a
modification of Connes' internal space geometry so that it has signature 6 (mod
8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action
can be formulated, and it is shown that it allows an extension with
right-handed neutrinos and the correct mass terms for the see-saw mechanism of
neutrino mass generation.Comment: Approx. 14 pages. v2: minor corrections; conclusions unaffecte
Skein spaces and spin structures
This paper relates skein spaces based on the Kauffman bracket and spin
structures. A spin structure on an oriented 3-manifold provides an isomorphism
between the skein space for parameter A and the skein space for parameter -A.
There is an application to Penrose's binor calculus, which is related to the
tensor calculus of representations of SU(2). The perspective developed here is
that this tensor calculus is actually a calculus of spinors on the plane, and
the matrices a re determined by a type of spinor transport which generalises to
links in any 3-manifold.
A second application shows that there is a skein space which is the algebra
of functions on the set of spin structures for the 3-manifold.Comment: 9 pages, amstex, 15 figures. Revised by a substantial addition to
give a geometrical description of all the commutative skein algebras, for
A^6=1 (q^3=1
The equality of 3-manifold invariants
The invariants of 3-manifolds defined by Kuperberg for involutory Hopf
algebras and those defined by the authors for spherical Hopf algebras are the
same for Hopf algebras on which they are both defined.Comment: 8 pages, definition of state sum invariant improved for clarity, plus
minor typos corrected. With 3 postscript figures. further change: BoxedEPSF
macro now include
A Quasi-Variational Inequality Problem Arising in the Modeling of Growing Sandpiles
Existence of a solution to the quasi-variational inequality problem arising
in a model for sand surface evolution has been an open problem for a long time.
Another long-standing open problem concerns determining the dual variable, the
flux of sand pouring down the evolving sand surface, which is also of practical
interest in a variety of applications of this model. Previously, these problems
were solved for the special case in which the inequality is simply variational.
Here, we introduce a regularized mixed formulation involving both the primal
(sand surface) and dual (sand flux) variables. We derive, analyse and compare
two methods for the approximation, and numerical solution, of this mixed
problem. We prove subsequence convergence of both approximations, as the mesh
discretization parameters tend to zero; and hence prove existence of a solution
to this mixed model and the associated regularized quasi-variational inequality
problem. One of these numerical approximations, in which the flux is
approximated by the divergence-conforming lowest order Raviart-Thomas element,
leads to an efficient algorithm to compute not only the evolving pile surface,
but also the flux of pouring sand. Results of our numerical experiments confirm
the validity of the regularization employed.Comment: 51 p., low resolution fig
Relativistic spin networks and quantum gravity
Relativistic spin networks are defined by considering the spin covering of
the group SO(4), SU(2) times SU(2). Relativistic quantum spins are related to
the geometry of the 2-dimensional faces of a 4-simplex. This extends the idea
of Ponzano and Regge that SU(2) spins are related to the geometry of the edges
of a 3-simplex.
This leads us to suggest that there may be a 4-dimensional state sum model
for quantum gravity based on relativistic spin networks which parallels the
construction of 3-dimensional quantum gravity from ordinary spin networks.Comment: 10 pages, amstex, some errors corrected, more reference
Numerical approximation of corotational dumbbell models for dilute polymers
We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. In the case of a corotational drag term we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system
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