Framework for liquid crystal based particle models

TL;DR

The paper develops a unified, nonperturbative framework (LdGS) that combines Landau-de Gennes order with a Skyrme-like kinetic term to model electromagnetism, quantum phase dynamics, and gravity-like effects within a 4-axis, 4D field described by $M=O D O^T$. Topological charges emerge as built-in quantized invariants via curvature constructions from rotations and boosts, enabling a Maxwell-/Klein–Gordon–/GEM-like spectrum for elementary excitations such as hedgehogs, fluxons, and vortex loops. The author outlines qualitative correspondences to the Standard Model, including leptons, quarks, baryons, and neutrinos, via string-hadronization and knot-like field configurations, and suggests a path toward deriving SM structure from a reduced parameter set by integrating the Hamiltonian of these field configurations. The work proposes concrete steps—defining potentials, refining moduli-space parametrizations, and performing numerical simulations—to test and extend the framework, with potential implications for gravity–EM unification and new insights into SM puzzles.

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.

Figures (15)

• Figure 1: Unitary vector $\vec{n}\equiv \vec{n}(x)$ "director" field has quantization of topological charge, for which in liquid crystal experiments there were obtained long-range interactions - interpreting curvature as electric field, we can get Maxwell equations for their dynamics. Living in 3D suggests to extend it to 3 distinguishable axes representing rotating object - realized e.g. in biaxial nematic, getting 3 types of hedgehog with the same charge, but different energy (like 3 leptons), also magnetic singularity due to the https://en.wikipedia.org/wiki/Hairy_ball_theorem. We model such ellipsoid field like stress-energy tensor: with field of real symmetric matrices $M(x)\equiv M=ODO^T$ (orthogonal $OO^T=I$, $D=\textrm{diag}(\lambda_1,\lambda_2,\lambda_3)$), which prefers some shape as set of eigenvalues due to Higgs-like potential e.g. $V(M)=\sum_i (\lambda_i-\Lambda_i)^2$ for some fixed model parameters: $(\Lambda_i)$ - allowing to regularize singularity (discontinuity in the center) to finite energy as in top diagrams. Reminding that we live in 4D spacetime, like in Einstein's teleparallelism, suggests to add 0th time axis as the longest: with $\Lambda_0 (\textrm{gravity}) >> \Lambda_1 (\textrm{EM}) >>\Lambda_1 (\textrm{QM}) > \Lambda_0 (0)$, having the strongest tendency to be aligned in parallel, acting as local time direction: central axis of light cone. Mass/energy should enforce tiny perturbations as boost curvature of this time axis, with dynamics given by second set of Maxwell equations for GEM approximation of the general relativity, for boosts in vacuum extension from SO(3) to SO(1,3) Lorentz group.
• Figure 2: EM-hydrodynamics analogy EMh (top) missing charge quantization in Gauss law, added as topological - below calculation of Coulomb effective potential with shown Mathematica source (extended in GitHub). For "+-" or "++" pair of topological charges, there was postulated ansatz faber15: cylindrically symmetric configuration of unitary vector field $\vec{n}\equiv \vec{n}(x)$ in agreement with electric field of two elementary charges (in GitHub verified satisfaction of variation equation). Such various distance configurations are shown with visualized $H$ density. Seen as uniaxial nematic it corresponds to $M=nn^T$ matrix field, as discussed here with static energy density $H=\sum_{1\leq i<j\leq 3} \|[\partial_i M,\partial_j M]\|^2$. Integrating this energy density with cutoff around two singularities, there was numerically obtained $E(d)\approx 1590 \pm 25/d$ distance-energy dependence as in Coulomb law (shown values and fit). Finally the two singularities are to be regularized with Higgs-like potential e.g. $(1-\|n\|^2)^2$, due to Lorentz invariance leading to $m_0 \to m_0/\sqrt{1-v^2}$ energy scaling. To avoid infinite energy of singularities in charges there was used cutoff above, which finally should be replaced with regularization by Higgs-like potential - as discussed and calculated in faber4, leading to deformations of Coulomb interaction in tiny distances in agreement with the running coupling effect.
• Figure 3: Example of simplified calculation of Newton effective potential (in GitHub) - analogously as in Fig. \ref{['coulomb']}, but this time instead of large spatial rotations, using tiny boosts of 0th time axis for gravity (no mass quantization). Spherically symmetric curvature sources would have increased energy with reduced distance, hence to get attraction there was used dipole ansatz for microscopic scenarios like pair creation, hopefully averaging to spherically symmetric gravity. Final calculations will need further work.
• Figure 4: Summary of interactions by SO(1,3) dynamics of 4 orthogonal axes like in https://en.wikipedia.org/wiki/Teleparallelism, but of different lengths of axes: $g \gg 1 \gg \delta >0$. The 0th axis of length $g\sim 10^{10}$ is local time direction requiring high energy to change, its boost dynamics recreates https://en.wikipedia.org/wiki/Gravitoelectromagnetism (GEM) Maxwell equations of gravity confirmed by https://en.wikipedia.org/wiki/Gravity_Probe_B (diagrams) - minimal Lorentz invariant extension of Newton force in analogy to Coulomb, used as approximation of General Relativity. $S^2$ dynamics of 1st axis having length 1 gives EM Maxwell equations, and electric charge quantization as topological. Nonzero 2nd axis of length $\delta\sim 10^{-10}$ allows for tiny energy contributions of U(1) twists of 1st axis, as quantum phase in QED Lagrangian. EM-GEM interaction e.g. slows down EM propagation in gravitational field - leading to gravitational time dilation, and light lensing through Fermat principle. Additionally, there are degrees of freedom deforming these $(g,1,\delta,0)$ eigenvalues preferred by Higgs-like potential, activated mainly near particles for regularization, which resemble e.g. https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory Lagrangian contributions.
• Figure 5: Top: parts of hedgehog of the longest $\vec{n}$main axis of biaxial-nematic-like field. This axis can tilt in two directions already in uniaxial nematic, in biaxial there are additionally recognized its twists - here tilts correspond to high energy EM dynamics, twists to low energy QM phase dynamics. Such local rotation is affine connection $\Gamma_\mu=O^T (\partial_\mu O)$: antisymmetric matrix we can interpret as local rotation vector $\vec{\Gamma}_\mu := \left((\Gamma_{\mu})_{32},(\Gamma_{\mu})_{13},(\Gamma_{\mu})_{21}\right)$. It corresponds to $A_\mu$ four-vector weighted with shape $\lambda_i\approx \Lambda_i$ (far from charge fixed by potential e.g. $V=\sum_i (\lambda_i -\Lambda_i)^2$) distinguishing high energy tilts from low energy twists. Curvatures $\vec{R}_{\mu\nu}=\vec{\Gamma}_\mu \times \vec{\Gamma}_\nu$ after weighting with shape become dual $F^*_{\mu\nu}$ tensor: containing high energy tilt-tilt component $R^1_{\mu\nu}$ corresponding to EM, and low energy tilt-twist $R^2_{\mu\nu},R^3_{\mu\nu}$ corresponding to QM phase like in QED-like Lagrangian. Bottom: due to Aharonov-Bohm-like arguments, there is belief that $A$ four-vector is more fundamental than $E,B$ fields, however, it leaves gauge freedom. It also allows for non-integer charges like half-electron - to prevent that, there is postulated more fundamental field $M$ which (quantized) topological charge is calculated in Gauss law. This deeper field can be seen as extended quantum phase: from low energetic evolution of U(1) quantum phase (twists), to SO(3) evolution (+tilts) including also electromagnetism with built-in (topological) charge quantization, getting natural EM+QM unification (+GEM with 4D field, SO(1,3) vacuum).