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To account for the fermions, use the remaining OP2 = F4/Spin(9) part of F4.

In terms of the Lie algebra splitting F4 = Spin(8) + V8 + S8+ + S8-, that means using the two mirror image 8-dimensional Spin(8) half- spinor representations S8+ and S8- to define fermion particles and antiparticles. To do this, use an octonionic basis for S8+ and S8-. Recall that a basis for O is {1, i, j, k, ke, je, ke}, where 1 is the basis element for the real axis. Of the seven imaginaries, i, j, and k are just the three imaginary quaternions, and e, ie, je, and ke are constructed from the four quaternionic basis elements 1, i, j, and k by introducing an octonionic imaginary e. The octonionic basis for S8+ is identified with fermion particles as follows:

1 - electron neutrino;

i - red up quark;

j - blue up quark;

k - green up quark;

e - electron;

ie - red down quark;

je - blue down quark; and

ke - green down quark.

The octonionic basis for S8- is identified with antiparticles as follows:

1 - electron antineutrino;

i, j, k - red, blue, and green up antiquarks;

e - positron

ie, je, ke - red, blue, and green down antiquarks.

In the F4 model, a spinor matter field S8±(x) assigns to each point x of the spacetime base manifold a point in the space of superpositions of fundamental fermion particle or antiparticle states. The octonionic basis elements for S8+ and S8- determine the nature of the fermion, such as neutrino (1) or red down quark (ie). For each fermion present in the spinor field S8±(x) at x, there is a complex propagator amplitude phase exp(iq).

In the F4 model, particle masses and force strengths are calculated by ratios of volumes of compact symmetric spaces. Therefore, the spinor fermions must be represented by an 8-dimensional compact "spinor space", denoted by Q8+ for particles and by Q8- for antiparticles.

Q8+ should be constructed such that it has an octonionic basis {1,i,j,k,e,ie,je,ke} and each 1-dimensional subspace spanned each basis element, such as j, e, etc., is topologically S1 = U(1) and can be denoted S1j, S1e, etc.

Then, for those elements for which S8±(x) is nonzero,

S8±(x) [ (S11, S1i, S1j, S1k, S1e, S1ie, S1je, S1ke) and the position on each S1 determines the complex propagator amplitude phase.

The compact manifold RP1 x S7 has the structure required for Q8+ and Q8-, and in the F4 model Q8+ = Q8- = RP1 x S7.

Q8+ is the Silov boundary of the bounded complex domain D8+. Hua11 uses the term characteristic manifold for the Silov boundary.

D8+ is isomorphic to the 16-dimensional symmetric space

D8+ = Spin(10)/Spin(8) x Spin(2) = Spin(10)/Spin(8) x U(1). A standard reference on symmetric spaces and bounded domains is Helgason12.

Therefore Q8+ has the desirable properties of local Spin(8) symmetry for the Spin(8) gauge group of the F4 model in 8 dimensions and local U(1) symmetry for the complex propagator amplitude phase. Further, Q8+ is parallelizable10 and has natural octonionic structure.

The RP1 in Q8+ corresponds to the octonionic real axis and is topologically S1. Physically, RP1 is the Weyl neutrino-type fermion and, at tree level, can only be left-handed for Q8+ and right-handed for Q8-, and so must be massless at tree level.

The S7 in Q8+ corresponds to the imaginary octonions and, since it is parallelizable, corresponds to (0, S1i, S1j, S1k, S1e, S1ie, S1je, S1ke). Since S7 double covers RP7, with each point of RP7 corresponding to two antipodal points of S7, the northern hemisphere of S7 can be considered to be left-handed fermions and the southern hemisphere can be considered to be right-handed fermions. Physically, S7 is the set of Dirac electron-type or quark-type fermions. They can acquire tree-level mass by the Higgs mechanism.

Since quarks and the electron are Dirac fermions with two helicity states, they are capable of getting mass from a Higgs mechanism.

Counting Dirac fermion helicity states separately, the spinor field S8±(x) can have values in 15 particle states and 15 antiparticle states, a total of 30 states.

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