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arXiv · 1807.08492

Digital finite quantum Riemannian geometries

Shahn Majid·Anna Pachol
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Abstract

We study bimodule quantum Riemannian geometries over the field $\Bbb F_2$ of two elements as the extreme case of a finite-field adaptation of noncommutative-geometric methods for physics. We classify all parallelisable such geometries for coordinate algebras up to vector space dimension $n\le 3$, finding a rich moduli of examples for $n=3$ and top form degree 2, including many that are not flat. Their coordinate algebras are commutative but their differentials are not. We also study the quantum Laplacian $\Delta=(\ ,\ )\nabla{\rm d}$ on our models and characterise when it has a massive eigenvector.

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Shahn Majid, Anna Pachol. 2018-07-23. Digital finite quantum Riemannian geometries. https://doi.org/10.1088/1751-8121/ab1cf2

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