2005/03/31 by Francesco D’Andrea, Francesco D'Andrea · 21 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Algebraic structures and combinatorial models #Clifford algebra #Dirac operator #Gamma matrices #Hilbert space #Mathematical physics #Mathematics #Minkowski space #Noncommutative algebraic geometry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Operator algebra #Pure mathematics #Quantum differential calculus #Spectral triple #hep-th
paper · pdf · doi:10.1063/1.2204808
published in Journal of Mathematical Physics 47(6) (American Institute of Physics) · 23 pages, expanded version
arxiv created 2006/01/05 · openalex publication_date 2006/06/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 unbounded commutators with the algebra.