The Principle of the Fermionic Projector II, Derivation of the Effective Gauge Group

Felix Finster

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Submission date: 11. Feb. 2002
Pages: 84
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Abstract:
We study the principle of the fermionic projector for the two-point actioncorresponding to the Lagrangian

and a fermionic projector which in the vacuum is the direct sum of sevenidentical massive sectors and one masslessleft-handed sector, each of which is composed ofthree Dirac seas. It is shown under general assumptions and for aninteraction via general chiral and (pseudo)scalar potentials that thesectors spontaneously form pairs, which are referred toas blocks. The resulting so-called effective interaction can be describedby chiral potentials corresponding to the effective gauge group

This model has striking similarity to the standard model if the blockcontaining the left-handed sector is identified with the leptons and thethree other blocks with the quarks. Namely, the effective gauge fieldshave the following properties.

• The SU(3) corresponds to an unbroken gauge symmetry. TheSU(3) gauge fields couple to the quarks exactlyas the strong gauge fields in the standard model.
• The SU(2) potentials are left-handed and couple to the leptonsand quarks exactly as the weak gauge potentials in the standard model.Similar to the CKM mixing in the standard model,the off-diagonal components of these potentials must involve anon-trivial mixing of the generations.The SU(2) gauge symmetry is spontaneously broken.
• The U(1) of electrodynamics can be identified with an Abeliansubgroup of the effective gauge group.

The effective gauge group is larger than the gauge group of the standardmodel, but this is not inconsistent because a more detailed analysis ofour variational principle should give further constraints for theAbelian gauge potentials. Moreover, there are the following differencesto the standard model, which we derive mathematically without workingout their physical implications in detail.

• The SU(2) gauge field tensor F must be simple in the sensethat for a real 2-form and an su(2)-valuedfunction s.
• In the lepton block, the off-diagonal SU(2) gauge potentials areassociated with a new type of potential, called nil potential, which couplesto the right-handed component.

These results give a strong indication that the principle of the fermionicprojector is of physical significance.