Noncommutativity and physics: a non-technical review
Physics Department, American University of Beirut, Beirut, Lebanon
2 College de France, 3 rue Ulm, 75005, Paris, France
3 I.H.E.S., 91440, Bures-sur-Yvette, France
4 Ludwig Maximilian University, Theresienstrasse 37, 80333, Munich, Germany
5 Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
Accepted: 14 April 2023
Published online: 26 May 2023
We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati–Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy.
© The Author(s) 2023
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