Classification of digital affine noncommutative geometries

Abstract

It is known that connected translation invariant n-dimensional noncommutative differentials dx i on the algebra k[x 1 , ⋯, x n ] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. This data also applies to construct differentials on the Heisenberg algebra 'spacetime' with relations [x µ , x ν ] = λΘ µν where Θ is an antisymmetric matrix as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k = F 2 of two elements, in which case translation invariant metrics (i.e. with constant coefficients) are equivalent to making V a Frobenius algebras. We classify all of these and their quantum Levi-Civita bimodule connections for n = 2, 3, with partial results for n = 4. For n = 2 we find 3 inequivalent differential structures admitting 1,2 and 3 invariant metrics respectively. For n = 3 we find 6 differential structures admitting 0, 1, 2, 3, 4, 7 invariant metrics respectively. We give some examples for n = 4 and general n. Surprisingly, not all our geometries for n ≥ 2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted 'sum' over all possible metrics but our results are a step towards a deeper approach in which we must also 'sum' over differential structures. Over F 2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of 'digital geometry'.2000 Mathematics Subject Classification. Primary 81R50, 58B32, 83C57.

Quantum geometry of Boolean algebras and de Morgan duality

“…Particularly, [2,18,19] already showed that quantum Riemannian geometry in this form specialises nontrivially over the field F 2 = {0, 1}. Here [19] classified such 'digital quantum geometries' for algebras up to dimension 3 while [2] constructed some first quantum geometries of algebra dimension 4.…”

“…While colourfully motivated as above, this article will be limited to some self-contained mathematics but which will include elements of gravity in the loose sense of a curved metric and an element of quantum theory in the minimal sense that differential forms on Boolean algebras do not commute with algebra elements. F 2 geometry is also interesting in its own right [2] and could have other applications, such as to the transfer of geometric ideas to digital electronics [18,19], providing another reason to consider the Boolean algebra case and within it de Morgan duality. We will also put the latter into a wider context beyond the Boolean case.…”

Digital finite quantum Riemannian geometries

“…Only two input wires are effective as 1 is in the kernel, and clearly Laplacians of practical interest would need to be somewhat more complicated. It is explained in [19] how to handle tensor products and the wiring diagrams for the algebra products of F 2 Z 2 , F 2 (Z 2 ), F 4 are given there. Such operations and their possible applications to engineering constitute another direction for further work.…”

“…Therefore the idea in two recent works [18,19] is that we can even work at the other extreme i.e. 'digitally' over the field F 2 = {0, 1} of two elements, and here we continue in this setting.…”

“…It is also possible, geometry being ubiquitous in science and engineering, that there could be applications of F 2 and F p d quantum geometries in theory own right. One of these could be to transfer geometric ideas into digital electronics as explained in [19]. Why exactly one would want to do this remains to be seen, but one area of application could be to build digital quantum computing gates as analogues of what we may wish to build in an actual quantum computer and with potentially some of the benefits, as well as being 'training wheels' for the real thing.…”