Abstract
We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes.
The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint.
The Lorentz invariant spacetime volume operator, representing the 4-volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin.
The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.
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Acknowledgements
We are pleased to thank Joachim Cuntz and Jochen Zahn for helpful discussions and comments. S.D. gratefully acknowledges the support received from the Alexander von Humboldt Foundation in various stages of this research. D.B. was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
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Communicated by M. Salmhofer
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bahns, D., Doplicher, S., Fredenhagen, K. et al. Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators. Commun. Math. Phys. 308, 567–589 (2011). https://doi.org/10.1007/s00220-011-1358-y
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DOI: https://doi.org/10.1007/s00220-011-1358-y