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MiS Preprint
11/2002
The Principle of the Fermionic Projector II, Derivation of the Effective Gauge Group
Felix Finster
Abstract
We study the principle of the fermionic projector for the two-point action corresponding to the Lagrangian \[ {\cal{L}}[A] \;=\; |A^2|^2 \:-\: \mu\: |A|^4 \;,\;\;\;\;\; \mu \in \mathbb{R} \] and a fermionic projector which in the vacuum is the direct sum of seven identical massive sectors and one massless left-handed sector, each of which is composed of three Dirac seas. It is shown under general assumptions and for an interaction via general chiral and (pseudo)scalar potentials that the sectors spontaneously form pairs, which are referred to as blocks. The resulting so-called effective interaction can be described by chiral potentials corresponding to the effective gauge group \[ SU(2) \otimes SU(3) \otimes U(1)^3 \;. \] This model has striking similarity to the standard model if the block containing the left-handed sector is identified with the leptons and the three other blocks with the quarks. Namely, the effective gauge fields have the following properties.
The $SU(3)$ corresponds to an unbroken gauge symmetry. The $SU(3)$ gauge fields couple to the quarks exactly as the strong gauge fields in the standard model.
The $SU(2)$ potentials are left-handed and couple to the leptons and 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 a non-trivial mixing of the generations. The $SU(2)$ gauge symmetry is spontaneously broken.
The $U(1)$ of electrodynamics can be identified with an Abelian subgroup of the effective gauge group.
The effective gauge group is larger than the gauge group of the standard model, but this is not inconsistent because a more detailed analysis of our variational principle should give further constraints for the Abelian gauge potentials. Moreover, there are the following differences to the standard model, which we derive mathematically without working out their physical implications in detail.
The $SU(2)$ gauge field tensor $F$ must be simple in the sense that $F=\Lambda \:s$ for a real 2-form $\Lambda$ and an $su(2)$-valued function $s$.
In the lepton block, the off-diagonal $SU(2)$ gauge potentials are associated with a new type of potential, called nil potential, which couples to the right-handed component.
These results give a strong indication that the principle of the fermionic projector is of physical significance.
