350 research outputs found
The spectral distance on the Moyal plane
We study the noncommutative geometry of the Moyal plane from a metric point
of view. Starting from a non compact spectral triple based on the Moyal
deformation A of the algebra of Schwartz functions on R^2, we explicitly
compute Connes' spectral distance between the pure states of A corresponding to
eigenfunctions of the quantum harmonic oscillator. For other pure states, we
provide a lower bound to the spectral distance, and show that the latest is not
always finite. As a consequence, we show that the spectral triple [20] is not a
spectral metric space in the sense of [5]. This motivates the study of
truncations of the spectral triple, based on M_n(C) with arbitrary integer n,
which turn out to be compact quantum metric spaces in the sense of Rieffel.
Finally the distance is explicitly computed for n=2.Comment: Published version. Misprints corrected and references updated;
Journal of Geometry and Physics (2011
A Lorentzian Signature Model for Quantum General Relativity
We give a relativistic spin network model for quantum gravity based on the
Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral
over suitable representations of this algebra. This generalises the state sum
models for the case of the four-dimensional rotation group previously studied
in gr-qc/9709028.
As a technical tool, formulae for the evaluation of relativistic spin
networks for the Lorentz group are developed, with some simple examples which
show that the evaluation is finite in interesting cases. We conjecture that the
`10J' symbol needed in our model has a finite value.Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation.
Version 2 is a major revision with explicit formulae included for the
evaluation of relativistic spin networks and the computation of examples
which have finite value
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