46 research outputs found
Spectral Quadruples
A set of data supposed to give possible axioms for spacetimes. It is hoped
that such a proposal can serve to become a testing ground on the way to a
general formulation. At the moment, the axioms are known to be sufficient for
cases with a sufficient number of symmetries, in particular for 1+1 de Sitter
spacetime.Comment: AMS-LaTeX, 7 pages, presented at the Euroconference "BRANE NEW WORLD
and Noncommutative Geometry", Torino, Villa Gualino (Italy), October 2 - 7,
200
Quantum Gravity coupled to Matter via Noncommutative Geometry
We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions
emerges in a semi-classical approximation from a construction which encodes the
kinematics of quantum gravity. The construction is a spectral triple over a
configuration space of connections. It involves an algebra of holonomy loops
represented as bounded operators on a separable Hilbert space and a Dirac type
operator. Semi-classical states, which involve an averaging over points at
which the product between loops is defined, are constructed and it is shown
that the Dirac Hamiltonian emerges as the expectation value of the Dirac type
operator on these states in a semi-classical approximation.Comment: 15 pages, 1 figur
Moduli spaces of discrete gravity I: A few points
Spectral triples describe and generalize Riemannian spin geometries by
converting the geometrical information into algebraic data, which consist of an
algebra , a Hilbert space carrying a representation of and the Dirac
operator (a selfadjoint operator acting on ). The gravitational action
is described by the trace of a suitable function of . In this paper we
examine the (path-integral-) quantization of such a system given by a finite
dimensional commutative algebra. It is then (in concrete examples) possible to
construct the moduli space of the theory, i.e. to divide the space of all Dirac
operators by the action of the diffeomorphism group, and to construct an
invariant measure on this space. We discuss expectation values of various
observables and demonstrate some interesting effects such as the effect of
coupling the system to Fermions (which renders finite quantities in cases,
where the Bosons alone would give infinite quantities) or the striking effect
of spontaneous breaking of spectral invariance.Comment: 35 pages, Latex, uses xy-package
Multiple noncommutative tori and Hopf algebras
We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite
fibrations of the quantum double torus and generalize the construction for
quantum multiple tori.Comment: 18 pages; AMSLaTeX (major revision, the construction of dual
rewritten using approach of multiplier Hopf algebras, references added
The Spectral Geometry of the Equatorial Podles Sphere
We propose a slight modification of the properties of a spectral geometry a
la Connes, which allows for some of the algebraic relations to be satisfied
only modulo compact operators. On the equatorial Podles sphere we construct
suq2-equivariant Dirac operator and real structure which satisfy these modified
properties.Comment: 6 pages. Latex. V2: Minor changes; to appear in Comptes Rendus
Mathematiqu