A condensed matter interpretation of SM fermions and gauge fields

TL;DR

This paper offers a geometric interpretation of Standard Model fermions and gauge fields using a lattice-based approach, introducing a Dirac equation on a specific bundle with a novel discretization and gauge action.

Contribution

It introduces a new geometric framework for SM fermions and gauge fields using the bundle Aff(3) x C x /(R^3), with a doubler-free lattice Dirac equation and a compatible gauge action.

Findings

A geometric Dirac equation on Aff(3) x C x /(R^3) models SM fermions.

A doubler-free staggered lattice discretization is developed.

The model includes a compatible metric theory of gravity.

Abstract

We present the bundle Aff(3) x C x /(R^3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each C x /(R^3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space Aff(3) x C (Z^3). This space allows a simple physical interpretation as a phase space of a lattice of cells in R^3. We find the SM SU(3)_c x SU(2)_L x U(1)_Y action on Aff(3) x C x /(R^3) to be a maximal anomaly-free special gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with Z_2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting Z_2-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.