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Dimensional Deception for the Noncommutative Torus

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Classical and Quantum Physics

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 229))

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Abstract

We study the dimensional aspect of the geometry of quantum spaces. Introducing a physically motivated notion of the scaling dimension, we study in detail the model based on a fuzzy torus. We show that for a natural choice of a deformed Laplace operator, this model demonstrates quite non-trivial behaviour: the scaling dimension flows from 2 in IR to 1 in UV. Unlike another model with the similar property, the so-called Horava-Lifshitz model, our construction does not have any preferred direction. The dimension flow is rather achieved by a rearrangement of the degrees of freedom. In this respect the number of dimensions is deceptive. Some physical consequences are discussed.

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Notes

  1. 1.

    There is one more case: \(\mu \ll 1\). It can be analysed in the complete analogy with the ones we have considered. We will not describe it as it does not offer anything new.

  2. 2.

    The construction can be made, with little changes, also for theta rational.

  3. 3.

    In the following, when there is no cause of confusion, we will sometime omit the subscript n from \(q:n, q'_n\) and the like. This will render some formulas more explicit.

  4. 4.

    It is also true in the zero-dimensional approximation described in [17]. This is because (13.5) is incompatible with \((\mathscr {C}_q )^q = \mathbbm {1}_q\).

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Acknowledgements

FL acknowledges the support of the COST action QSPACE, the INFN Iniziativa Specifica GeoSymQFT and Spanish MINECO under project MDM-2014-0369 of ICCUB (Unidad de Excelencia ‘Maria de Maeztu’).

Author information

Authors and Affiliations

  1. INFN, Sezione di Napoli & Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, Napoli, Italy

    Fedele Lizzi

  2. Departament de Física Quàntica i Astrofísica and Institut de Cíencies del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, Spain

    Fedele Lizzi

  3. Universidade de Brasília Instituto de Física & International Center of Physics, Brasilia, DF, Brazil

    Alexandr Pinzul

Authors
  1. Fedele Lizzi
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Corresponding author

Correspondence to Fedele Lizzi .

Editor information

Editors and Affiliations

  1. Università degli Studi di Napoli Federico II, Naples, Italy

    G. Marmo

  2. Institute of Mathematical Sciences (ICMAT), Madrid, Spain

    David Martín de Diego

  3. Universitat Politècnica de Catalunya, Barcelona, Spain

    Miguel Muñoz Lecanda

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Lizzi, F., Pinzul, A. (2019). Dimensional Deception for the Noncommutative Torus. In: Marmo, G., Martín de Diego, D., Muñoz Lecanda, M. (eds) Classical and Quantum Physics. Springer Proceedings in Physics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-030-24748-5_13

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