4-dimensional lattice models for the quantum range
Authors: Gudrun Kalmbach H.E.  
Published: October 2021

Some finite subspace models L are presented for quantum structures which replace the use of countable infinite Hilbert space H dimensions. A maximal Boolean sublattice, called block, is 24, where its four atoms directly above 0εL, base vectors of H in 24 are drawn as four points on an interval. Blocks can overlap in one or two atoms. Different kinds of operators can map one block onto another and interpretations are given such that subspaces can carry on their base vector tuple real, complex or quaternionic numbers, energies, symmetries and generate coordinate lines. Describing states of physical systems is done using L and its applications for dynamical modelling. They don‘t need the infinte dimensional vectors of H. L has in the first model 11 blocks and 24 atoms (figure 1). They correspond to the 24 elements of the tetrahedral S4 symmetry. S4 arises from a spin-line rgb-graviton whirl operator with center at the tip of a tetrahedron and a nucleon triangle base with three quarks as vertices. The triangles factor group D3 of S4 is due to the CPT Klein normal subgroup Z2 x Z2 of S4 . It has a strong interaction SI rotor for the nucleons inner dynamics which is used for integrating functions, exchanging energies of nucleon with its environment and setting barycentrical coordinates in the triangle. At their intersection B as barycenter sets a Higgs boson or field the rescaled quark mass of a nucleon. Each factor class of one element from D3 assigns to it a color charge, a coordinate, an energy vector and a symmetry. Symmetries attached can be different according to interactions involved. Every atom of L has then a specific character with different properties.Three characters are added to octonian base vectors, listed by their indices as n = 0,1,…,7, and named for the atoms of L as na, nb, nc. The structure and element attributes of the finite subspace lattices L are desribed in many examples and models which technical constructed run macroscopically. Several models are described below. Example, the color charge whirl as rgb-graviton projection operator maps the block 2c3b5a6a to 0a1a2a3a. The symmetries change dimension from 3x3- to 2x2-matrices. From SU(3) are λ1 on 3b mapped to the SU(2) x-coordinate Pauli matrix σ1, from λ2 on 5a to σ2 y-coordinate and from λ3 on 6a to σ3 z-coordinate of real Euclidean space R³. The SU(3) matrices have complex w3 = z +ict, w2 = (iy,f), w1 = (x,m) coordinates. In figure 3 is shown how a rotation of two proton tetrahedrons for fusion changes the two linearly independent wj vectors to the 1-dimensional x,y,z base vectors. In deuteron then on one coordinate line sit with Cooper paire u-d-quarks at the ends the Heisenberg coupled energy or space vector rays 15 (x,m), m mass measured in kg, x in meter, 23 (iy,E(rot)), E(rot) rotational energy measured in Joule J, y in meter, 46 (ict,f), t time measured in seconds, f = 1/∆t frequency s inverse time interval measured in Hz. The six color charges are red r on +x as octonian coordinate 1, green g on +y as 2 , blue b on -z as 6, turquoise on -x as 5, magenta on -y as 3, yellow on +z as 4..