Jonathan Smith

Quantization of web geometry

The prime motivation for this talk is the ongoing search for a "pregeometry" [1] as a spacetime geometry that is quantized at the Planck length of $10^{-38}$ miles. Classical web geometry is well placed to provide a background for this search. It already underlies both quantum mechanics (compare, say, Wooters' use of webs [2] in his approach to finite state Wigner functions) and general relativity (compare, say, the expression of curvature [3] within web geometry). Thus, the problem of quantizing web geometry emerges as a potentially important part of the general problem of appropriately quantizing spacetime.

Classical web geometry uses ordered pairs and projections to factors. However, in the general quantum setting of a symmetric, monoidal structure, this approach is no longer available. An alternative is needed. The points of a classical web appear as categorical points (morphisms out of the terminal object) in a suitable category of semisymmetrizations of classical quasigroups [4]. Semisymmetrization of quantum quasigroups thus presents itself as one possible approach to the quantization of web geometry [5].

We initiate this program by examining the most amenable setting, Cartesian symmetric monoidal categories of modules over a commutative, unital ring. We determine linear quantum quasigroup structures that provide comultiplications to extend the semisymmetrization multiplication of a linear quasigroup. In particular, if the linear quasigroup structure comes from a real or complex affine plane, a complete classification of the quantum semisymmetric comultiplications is provided, based on the solution of a system of cubic equations.

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Goldberg, V.V., Local differentiable quasigroups and webs, pp. 263-311 in "Quasigroups and Loops: Theory and Applications" (Chein, O. et al., eds.), Heldermann, Berlin, 1990.
Smith, J.D.H., Quasigroup homotopies, semisymmetrization, and reversible automata, Internat. J. Algebra Comput. 18 (2008), 1203-1221.
Smith, J.D.H., Quantization of web geometry: semisymmetrization of linear quantum quasigroups, J. Geom. Phys. (2026), 105781.