Framework for liquid crystal based particle models

Abstract: Long-range e.g. Coulomb-like interactions for (quantized) topological charges in liquid crystals are observed experimentally, 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 built-in missing charge quantization if interpreting field curvature as electric field. This article proposes a general mathematical framework, combining Landau-de Gennes and skyrmion models, to extend this similarity with particle physics to biaxial nematic, getting surprising agreement with the Standard Model. Specifically, recognising intrinsic twist of uniaxial nematic e.g. to propagate angular momentum also in this direction, allows hedgehog configurations with one of 3 distinguishable axes: having the same topological charge, but different energy/mass - getting similarity with 3 leptons (also neutrinos, quarks as living in 3D). Vacuum dynamics extends electromagnetism from 3D rotation dynamics, with Klein-Gordon-like equation for twists corresponding to quantum phase. If extending to 4D field with 0th axis, vacuum dynamics to SO(1,3) by boosts, we also get second set of Maxwell equations for GEM (gravitoelectromagnetism) approximation of general relativity.