Why the number of visible particle families is three?

The assumption that family replication phenomenon for elementary particles has a topological origin reduces the construction of TOE to that of finding a theory which explains the symmetries of the standard model. A simple ten line argument leads to the hypothesis that if space-times are representable as 4-surfaces in H=M⁴×CP₂ this is achieved. The genus of the 2-surface identified as a time=constant cross section of 3-D light-like causal determinant is the topological quantum number in question. Particles are in this picture analogous to shockwaves. This theory extends the conformal symmetries of super string models and predicts the space-time dimension to be four.

During these 27 years I have gone through the entire physics to develop contact of TGD with reality and also extended quantum measurement theory to a quantum theory of consciousness and of living systems. For instance, elementary particle mass spectrum is predicted correctly and the theory predicts a lot of new physics: actually entire hierarchy of scaled down copies of standard model governing the physics of dark matter hierarhcy. Regrettably, the extremely arrogant culture of present day theoretical physics makes for most of my colleagues it impossible to even consider getting familiar with the seven books about TGD and 8 books about TGD inspired theory of consciousness at my homepage providing a thorough documentation about TGD. This despite that an increasing number of theoretical physicists admits that theoretical physics is in a deep crisis and agrees with me about the underlying reasons.

Returning to the topic: the problem is to understand why only three lowest genera g=0,1,2 are light. For long time ago I realized that the hyper-ellipticity is the explanation but it took more than decade to develop the argument to what seems to be the its final and amazingly simple form. The latest simplifications came from the realization that only hyper-ellipticity is needed since g> 2 genera behave in any case like dark matter and can therefore be light. I attach below the abstract of the appropriate chapter.

Genus-generation correspondence is one of the basic ideas of TGD approach. In order to answer various questions concerning the plausibility of the idea, one should know something about the dependence of the elementary particle vacuum functionals on the vibrational degrees of freedom for the boundary component. The construction of the elementary particle vacuum functionals based on Diff invariance, 2-dimensional conformal symmetry, modular invariance plus natural stability requirements indeed leads to an essentially unique form of the vacuum functionals and one can understand why g >2 bosonic families are experimentally absent and why lepton numbers are conserved separately.

An argument suggesting that the number of the light fermion families is three, is developed. The argument goes as follows. Elementary particle vacuum functionals represent bound states of g handles and vanish identically for hyper-elliptic surfaces having g > 2. Since all g≤ 2 surfaces are hyper-elliptic, g ≤ 2 and g > 2 elementary particles cannot appear in same non-vanishing vertex and therefore decouple. The g>2 vacuum functionals not vanishing for hyper-elliptic surfaces represent many particle states of g≤ 2 elementary particle states being thus unstable against the decay to g≤ 2 states. The failure of Z₂ conformal symmetry for g>2 elementary particle vacuum functionals could in turn explain why they are heavy: this however not absolutely necessary since these particles would behave like dark matter in any case.

For details see the chapter Elementary Particle Vacuum functionals of "p-Adic Length Scale and Dark Matter Hierarchy" at my re-organized homepage.

Matti Pitkänen