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2.3. The F4 Model - Spinor Fermion Term for the Lagrangian in 8 Dimensions.

In 8 dimensions, the F4 model Lagrangian action is generally of the form

ºM8 GB + SpF

where M8 is the spacetime base manifold, SpF is a spinor fermion term, and GB is a gauge boson term. In Section 2.1. the gauge boson part of the Lagrangian action was constructed to be ºV8 - F8 /\*F8 , where F8 is the Spin(8) gauge boson curvature 2-form. In this Section 2.3, the spinor fermion part of the Lagrangian is constructed.

The spinor field S8± is acted upon naturally by the Dirac operator g¶ defined in terms of the Clifford algebra Cl(8) of R8.

The Cl(8) Clifford algebra and Spin(8) spinors are described in Appendix 1.

Cl(8) is generated by {G1,G2,G3,G4,G5,G6,G7,G8}, where Gi can be written in terms of three independent sets of Pauli matrices ri, si, and ti as

G1 = r1 G5 = r3s3t1

G2 = i r2 G6 = i r3s3t2

G3 = r3s1 G7 = r3s3t3

G4 = i r3s2 G8 =    1

Then the Dirac operator g¶ is defined in terms of the M8 covariant derivative S ¶m and the Clifford product * by g¶ = S Gm * ¶m

(where m=1,2,3,4,5,6,7,8).

Since its symbol is Clifford multiplication, the Dirac operator g¶ interchanges fermion particles and antiparticles. It may change fermion type by permuting the half-spinor basis elements {1,i,j,k,e,ie,je,ke}.

The spinor term of the Lagrangian action is then:

ºV8 `S8± g¶ S8±

The full 8-dimensional classical Lagrangian action for the F4 model is therefore

ºV8 [ -F8 /\*F8 + `S8± g¶ S8± ] .

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