• DOI:10.1063/1.2204808
• Corpus ID: 117505922

Spectral geometry of ?-Minkowski space

@article{DAndrea2005SpectralGO,
  title={Spectral geometry of ?-Minkowski space},
  author={Francesco D’Andrea},
  journal={Journal of Mathematical Physics},
  year={2005}
}

• F. D’Andrea
• Published 1 March 2005
• Mathematics
• Journal of Mathematical Physics

After recalling Snyder’s idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as “coordinates on a noncommutative space,” we discuss a two-dimensional toy-model whose “dual” noncommutative coordinates form a Lie algebra: this is the well-known κ-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder’s idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of κ-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its “spectral properties,” and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbo...