Abstract
The paper is intended to develop intuition about noncommutative spaces in general, in particular to explain notions of continuity, differential calculus and geometry applied to noncommutative algebras. To this end we discuss the simplest and the most prominent examples of fuzzy spaces. A specific variant of noncommutative differential geometry, the noncommutative frame formalism, is introduced in the same elementary way, through examples.
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Notes
We place \(m=1\) and \(\omega =1\) which leaves only \(\hbar\) as a dimensionful quantity. In fact, one often suppresses \(\hbar\) as well to simplify normalization.
In the following, a pair of the corresponding commutative and noncommutative quantities is denoted by the same letter with and without a tilde, e.g. \({\tilde{x}}\) and \(x\,\).
The letter \(k\!\!\!^{-}\) is chosen to denote the constant of noncommutativity by analogy with \(\hbar\); its dimension is length squared. If we assume that noncommutative geometry is an exact aspect of quantum gravity, it is reasonable to relate \(k\!\!\!^{-}\) to the Planck length, \(\,k\!\!\!^{-}\sim \ell _{Pl}^2\,\); if noncommutativity is only effective then \(\,k\!\!\!^{-}>\ell _{Pl}^2\,\) or \(k\!\!\!^{-}\gg \ell _{Pl}^2\).
We give just an outline of the reasoning; proofs can be found in [1].
References
J. Madore, An introduction to noncommutative differential geometry and its physical applications, in London Mathematical Society Lecture Note series, vol. 257, (CUP, 2000)
M. Nakahara, Geometry, Topology and Physics (IoP, 2003)
J. Madore, Class. Quantum Gravity 9, 69 (1992)
A. Perelomov, Generalized Coherent States and Their Applications (Springer, 1986)
H. Grosse, R. Wulkenhaar, JHEP 12, 019 (2003)
M. Buric, M. Wohlgenannt, JHEP 03, 053 (2010)
M. Buric, J. Madore, Eur. Phys. J. C 75, 502 (2015)
M. Buric, D. Latas, L. Nenadovic, Eur. Phys. J. C 78, 953 (2018)
A.P. Balachandran, S. Kurkcuoglu, S. Vaidya, Lectures on Fuzzy and Fuzzy SUSY Physics (World Scientific, 2007)
B.P. Dolan, D. O’Connor, P. Presnajder, JHEP 03, 013 (2002)
H. Grosse, J. Madore, Phys. Lett. B 283, 218 (1992)
U. Carow-Watamura, S. Watamura, Commun. Math. Phys. 183, 365 (1997)
H. Grosse, J. Madore, H. Steinacker, J. Geom. Phys. 38, 308 (2001)
R.J. Szabo, Phys. Rep. 378, 207 (2003)
C.S. Chu, J. Madore, H. Steinacker, JHEP 08, 038 (2001)
A. Connes, J. Lott, Nucl. Phys. B 18, 29 (1991)
P. Aschieri, J. Madore, P. Manousselis, G. Zoupanos, JHEP 04, 034 (2004)
M. Buric, T. Grammatikopoulos, J. Madore, G. Zoupanos, JHEP 04, 054 (2006)
M. Buric, J. Madore, G. Zoupanos, Eur. Phys. J. C 55, 489 (2008)
Acknowledgements
This work is supported by MPNTR Serbia grant 451-03-68/2022-14/200162.
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Noncommutativity and Physics. Guest editors: George Zoupanos, Konstantinos Anagnostopoulos, and Peter Schupp.
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Burić, M. A road to fuzzy physics. Eur. Phys. J. Spec. Top. 232, 3597–3606 (2023). https://doi.org/10.1140/epjs/s11734-023-00838-0
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DOI: https://doi.org/10.1140/epjs/s11734-023-00838-0