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2.6 Dimensional Reduction to 4 Dimensions - 3 Generations of Spinor Fermions.

  After reduction of the 8-dimensional spacetime of the F4 model to a 4-dimensional spacetime, the F4 model describes familiar physics.

The octonionic structure of S8, with basis {1,i,j,k,e,ie,je,ke}, is

reduced to a quaternionic structure of M4 with basis {1,i,j,k} by a projection : O -> H of octonions to quaternions taking e -> 1.

Consider the sum-over-histories path integral quantum structure of the F4 model. A path originating at a fermion vertex is a gauge boson link leading to the next fermion vertex in 8-dimensional spacetime, and so on, as in Fig. 1:

If both the origin and destination fermions were in the

4-dimensional hyperspacetime into which the 8-dimensional spacetime is reduced, then the fermions just remain fixed,

as in Fig. 2:

Fig. 2

The origin fermion is then a first generation fermion. It is identified with a basis element {1,i,j,k,e,ie,je,ke} of O.

If the origin fermion were not in the 4-dimensional hyperspacetime but the destination fermion were in the 4-dimensional hyperspacetime, the origin fermion would appear as the superposition of two fermions, as in Fig. 3:

Similarly, if the origin fermion were in the 4-dimensional hyperspacetime but the destination fermion were not in the 4-dimensional hyperspacetime, the origin fermion would appear as the superposition of two fermions,

as in Fig. 4:

In both the cases of Fig. 3 and Fig. 4, the origin fermion is a second generation fermion, as it is effectively a superposition of two fermions. It is identified with an element of O x O, or pairs of basis elements {1,i,j,k,e,ie,je,ke} of O.

If both the origin fermion and the destination fermion were not in the 4-dimensional hyperspacetime, the origin fermion would appear as the superposition of three fermions as in Fig. 5:

The origin fermion is then a third generation fermion, as it is effectively a superposition of three fermions. It is identified with an element of O x O x O, or triples of basis elements {1,i,j,k,e,ie,je,ke}.

Therefore dimensional reduction produces three generations of fermions.

The process does not give charged gauge bosons a three generation structure, even though a charged gauge boson can be a vertex from which a gauge boson link propagates. This is because gauge bosons correspond to elements of the gauge group, and a pair or triple of gauge bosons is equivalent to a single gauge boson that is the gauge group product of the pair or triple.

 

 

 


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