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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

M8 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 V8 - 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:

V8 `S8± g S8±

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

V8 [ -F8 /\*F8 + `S8± g S8± ] .

 


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