Spectral estimators for finite non-commutative geometries

Barrett, John W.; Druce, Paul; Glaser, Lisa

Abstract

A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addreses the question of how to extract information about these geometries from the spectrum of the Dirac operator. Since the Dirac operator is a finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no sense and is replaced by measurements of the spectrum at a finite energy scale. The spectral dimension of the square of the Dirac operator is improved to provide a new spectral measure of the dimension of a space called the spectral variance. Similarly, the volume of a space can be computed from the spectrum once the dimension is known. Two methods of doing this are investigated: the well-known Dixmier trace and a recent improvement due to Abel Stern. Finally, the distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these techniques are tested on the explicit examples of the fuzzy spheres and fuzzy tori, which can be regarded as approximations of the usual Riemannian sphere and flat tori. Then they are applied to characterise some random fuzzy spaces using data generated by a Monte Carlo simulation.

Citation

Barrett, J. W., Druce, P., & Glaser, L. (2019). Spectral estimators for finite non-commutative geometries. Journal of Physics A: Mathematical and Theoretical, 52(27), https://doi.org/10.1088/1751-8121/ab22f8

• Journal Article Type: Article
• Acceptance Date: May 20, 2019
• Online Publication Date: May 20, 2019
• Publication Date: May 20, 2019
• Deposit Date: May 29, 2019
• Publicly Available Date: May 21, 2020
• Journal: Journal of Physics A: Mathematical and Theoretical
• Print ISSN: 1751-8113
• Electronic ISSN: 1751-8121
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 52
• Issue: 27
• DOI: https://doi.org/10.1088/1751-8121/ab22f8
• Keywords: Modelling and simulation; Statistics and probability; Mathematical physics; General physics and astronomy; Statistical and nonlinear physics
• Public URL: https://nottingham-repository.worktribe.com/output/2104173
• Publisher URL: https://iopscience.iop.org/article/10.1088/1751-8121/ab22f8
• Additional Information: This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://iopscience.iop.org/article/10.1088/1751-8121/ab22f8
• Contract Date: May 30, 2019