Monday, May 29, 2023

The newest piece to the TGD inspired model of family replication

The TGD vision about family replication phenomenon of fermions is as follows.
  1. Fermion families correspond to the genera for partonic 2-surfaces. This predicts generation-genus correspondence. Electron and its neutrino correspond to a sphere with genus g=0; muon and its neutrino to a torus with g=1; τ and its neutrino to to with g=2. Similar picture applies to quarks. CKM mixing corresponds to topological mixings of genera, which are different for different charged states and CKM mixing is the difference of these mixings.

    The problem is that TGD suggests an infinite number of genera. Only 3 fermion families are observed. Why?

  2. The first piece of the answer is Z2 conformal symmetry. It is present for the genera g=0,1,2 but only for hyperelliptic Riemann surfaces for g>2.
  3. The second piece of the answer is that one regards the genera g>q 2 as many-handle states. For g> 2 many-handle states would have a continuous mass spectrum and would not be elementary particles. For g=2 a bound state of two handles would be possible by Z2 symmetry.
Consider now the new building brick for the explanation.
  1. Quantum classical correspondence is the basic principle of TGD and requires that quantum states have classical counterparts.
  2. Assume that in a suitable region of moduli space it makes sense to talk of a handle as a particle moving in the geometry defined by g-1 handles. One can imagine that the handle is glued by a small wormhole contact to the background defined by g-1 handles and behaves like a free point-like particle moving along a geodesic line of the background.

    This relationship must be symmetric so that the background must move along the geodesic line of the handle. This means that particles and background are glued together along the geodesic lines of both.

  3. Consider now various cases.
    1. The case g=0 is trivial since one has a handle vacuum.
    2. For g=1, one has the motion of a handle in spherical geometry along a great circle, which corresponds to a geodesic line of the sphere. The torus can rotate like a rigid body and this corresponds to a geodesic line of torus characterized by two winding numbers (m,n). Alternatively, one can say that the sphere rotates along a geodesic of the torus. There is an infinite but discrete number of orbits. The simplest solution is the stationary solution (m,n)=(0,0).
    3. For g=2, one has a geodesic motion of a handle in the toric geometry defined by the second handle. Now one can speak of bound states of two handles.

      One would have a gluing of two tori along geodesic lines (m,n) and (r,s). The ratios of these integers are rational so that one obtains a closed orbit. The simplest solution is (m,n)= (r,s)=0.

      Stationary solutions are stable for constant curvature case since curvature of torus vanishes. Locally the stationary solution is like a particle at rest in Euclidian plane.

    4. For g=3 one has a geodesic motion of the handle in g=2 geometry or vice versa. g=2 geometry has negative total scalar curvature and as a special case a constant negative curvature. This implies that all points are saddle points and therefore unstable geodesics so that two geodesics going through a given point in general diverge. This strongly suggests that only unstable geodesics are possible for g=2 whether it is regarded as background or as a particle. This suggests a butterfly effect and a chaotic behavior. Even if g=2 particle represents a classical bound state the third handle must move along a chaotic geodesics of g=2 geometry.This could explain the absence of bound states at quantum level.
    See the article About the TGD based views of family replication phenomenon and color confinement or the chapter Elementary Particle Vacuum Functionals.

    For a summary of earlier postings see Latest progress in TGD.

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