Framework for liquid crystal based particle models

Abstract

Long-range e.g. Coulomb-like interactions for (quantized) topological charges are observed experimentally in liquid crystals, bringing open question this article is exploring: how far can we take this resemblance with particle physics? Uniaxial nematic liquid crystal of ellipsoid-like molecules can be represented using director field $\vec{n}(x)$ of unitary vectors. It has topological charge quantization: integrating field curvature over a closed surface $\mathcal{S}$, we get 3D winding number of $\mathcal{S}\to S^2$, which has to be integer - getting Gauss law with finally built-in missing charge quantization if interpreting field curvature as electric field. This article proposes a general mathematical framework \textit{LdGS}: combining Landau-de Gennes field with Skyrme kinetic term, to extend this similarity with particle physics to biaxial nematic, getting surprising agreement with the Standard Model. Specifically, recognising intrinsic twist of uniaxial nematic allows hedgehog configurations with one of 3 distinguishable axes: having the same topological charge, but different energy/mass - getting similarity with 3 leptons. Topological vortices correspond to quark strings building baryons and nuclei. Vacuum dynamics extends electromagnetism from 3D rotation dynamics, with Klein-Gordon-like equation for twists corresponding to quantum phase. Like in Einstein's teleparallelism we can add 4th time axis, extending vacuum dynamics to SO(1,3) Lorentz group by boosts, getting additional second set of Maxwell equations for GEM (gravitoelectromagnetism) approximation of general relativity.