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Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group

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Towards Quantum Gravity

Part of the book series: Lecture Notes in Physics ((LNP,volume 541))

Abstract

This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the ‘Planck scale quantum group’ ℂ[x]⋈ℂ[p] and its observable-state T-duality-like properties are explained. The general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. More general quantum groups ℂ(G*)⋈U(g) and U q(g) are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian geometry is outlined. The semiclassical limit of the latter is a theory with generalised non-symmetric metric g μv obeying ∇μgvp ∇vg μp 0.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, Silver Street, CB3 9EW, Cambridge

    Shahn Majid

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  1. Shahn Majid
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Editors and Affiliations

  1. Institute of Theoretical Physics, University of Wrocław, Pl.Maxa Borna 9, 50-204, Wrocław, Poland

    Jerzy Kowalski-Glikman

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© 2000 Springer-Verlag Berlin Heidelberg

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Majid, S. (2000). Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group. In: Kowalski-Glikman, J. (eds) Towards Quantum Gravity. Lecture Notes in Physics, vol 541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46634-7_10

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  • DOI: https://doi.org/10.1007/3-540-46634-7_10

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  • Print ISBN: 978-3-540-66910-4

  • Online ISBN: 978-3-540-46634-5

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