A Condensed Matter Interpretation of Sm Fermions and Gauge Fields

We present the bundle (Aff(3) ⊗ C ⊗ Λ)(R3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (C⊗ Λ)(R3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗C)(Z3). This space allows a simple physical interpretation as a phase space of a lattice of cells in R3. We find the SM SU(3)c ×SU(2)L ×U(1)Y action on (Aff(3)⊗C⊗Λ)(R3) 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 Z2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting Z2-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.