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2.5.Reduction of Spacetime to 4 Dimensions.

  One mechanism for dimensional reduction of the 8-dimensional base manifold of the F4 model is to restrict the physical spacetime to a 4-dimensional submanifold defined by a Cayley calibration 4-form (Lawson and Michelsohn20 and Harvey21) arising from the cohomology of the space a/g‚ of physical gauge orbit connections (Nash54).

There is also another way to see that the F4 model 8-dimensional base manifold must be reduced to a 4-dimensional spacetime.

Define (as will be discussed in Section 6.5.) the Planck energy to be the sum total energy of all the possible virtual (8-dimensional) particle-antiparticle fermion pairs permitted by the Pauli exclusion principle.

The Planck energy temperature is the temperature above which the instanton will evaporate (because there are as many virtual gauge bosons and fermion particle-antiparticle pairs outside the boundary as inside, so that the boundary has no physical significance), and below which it is stable.

Below the Planck energy vertices of spacetime cannot be spacelike closer to each other than the Planck length, the spacelike distance derived from the Planck energy. Also, the timelike separation must be at least the Planck time. The result is that, at and below the temperature of the Planck energy, the F4 model effectively has a Planck length lattice spacetime. Since all gauge bosons and fermions in the 8-dimensional F4 model are massless, the lattice must either have a light-cone structure or be reduced to a lower dimensional lattice that has a light-cone structure.

For the F4 model 8-dimensional spacetime to be a lattice with octonionic structure, the lattice must be one of seven similar but inequivalent E8 lattices (a lattice of integral octonions as described by Coxeter18). There are 240 paths from an E8 lattice vertex to a nearest neighbor vertex:

±1 (2 ±time axis paths);

±i, ±j, ±k, ±e, ±ie, ±je, ±ke (16 ±space axis paths); and

224 paths of the form (±a±b±c±d)/2.

None of the 240 paths to nearest neighbor vertices are light-cone paths in the 8-dimensional spacetime of the F4 model. (The nearest vertices on a light-cone are in the next layer out (at radius 2), which layer contains 2160 vertices.) The 8-dimensional F4 model has only massless particles, which must travel on light-cones. Therefore, to get a spacetime lattice with light-cones, the F4 model spacetime must be reduced from E8 8-dimensional octonionic lattice to 4-dimensional quaternionic lattice.

All E8 lattices project (by (e, ie, je, ke) -> (1, i, j, k)) to a D4 lattice of integral quaternions having 24 paths from a D4 vertex to a nearest neighbor vertex:

±1 (2 ±time axis paths);

±i, ±j, ±k (6 ±space axis paths); and

(±1±i±j±k)/2 (16 ±lightcone paths).

D4 and E8 lattices are described by Coxeter18, by Conway and Sloane19, and in Appendix 2.

The 4-dimensional F4 model has fundamentally a lattice spacetime with a D4 lattice. The quantum theory is defined by path integral sum over histories. From this point of view, a continuum spacetime base manifold is only an approximation.

 

 


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