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Tuesday, November 8, 2016 2:49

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The concrete model for elementary particles has developed gradually during years and is by no means final. In the recent model elementary particle corresponds to a pair of wormhole contacts and monopole flux runs between the throats of of the two contacts at the two space-time sheets and through the contacts between space-time sheets.

The first criticism relates to twistor lift of TGD. In the case of Kähler action the wormhole contacts correspond to deformations for pieces of CP_{2} type vacuum extremals for which the 1-D M^{4} projection is light-like random curve. Twistor lift adds to Kähler action a volume term proportional to cosmological constant and forces the vacuum extremal to be a minimal surface carrying non-vanishing light-like momentum (this is of course very natural): one could call this surface CP_{2} extremal. This implies that M^{4} projection is light-like geodesic: this is physically rather natural.

Twistor lift leads to a loss of the proposed space-time correlate of massivation used also to justify p-adic thermodynamics: the average velocity for a light-like random curve is smaller than maximal signal velocity – this would be a clear classical signal for massivation. One could however conjecture that the M^{4} projection for the light-like boundaries of string world sheets becomes light-like geodesic of M^{4}× CP_{2} instead light-like geodesic of M^{4} and that this serves as the correlate for the massivation in 4-D sense.

Second criticism is that I have not considered in detail what the monopole flux hypothesis really means at the level of detail. Since the monopole flux is due to the CP_{2} topology, there must be a closed 2-surface which carries this flux. This implies that the flux tube cannot have boundaries at larger space-time surface but one has just the flux tube which closed cross section obtained as a deformation of a cosmic string like object X^{2}× Y^{2}, where X^{2} is minimal surface in M^{4} and Y^{2} a complex surface of CP_{2} characterized by genus. Deformation would have 4-D M^{4} projection instead of 2-D string world sheet.

** Note:** One can also consider objects for which the flux is not monopole flux: in this case one would have deformations of surfaces of type X^{2}× S^{2}, S^{2} homologically trivial geodesic sphere: these are non-vacuum extremals for the twistor lift of Kähler action (volume term). The net magnetic flux would vanish – as a matter fact, the induced Kähler form would vanish identically for the simplest situation. These objects might serve as correlates for gravitons since the induced metric is the only field degree of freedom. One could also have non-vanishing fluxes for flux tubes with disk-like cross section.

If this is the case, the elementary particles would be much simpler than I have though hitherto.

- Elementary particles would be simply closed flux tubes which look like very long flattened squares. Short sides with length of order CP
_{2}radius would be identifiable as pieces of deformed CP_{2}type extremals having Euclidian signature of the induced metric. Long sides would be deformed cosmic strings with Minkowskian signature with apparent ends, which are light-like 3-surfaces at which the induced 4-metric is degenerate. Both Minkowskian and Euclidian regions of closed flux tubes would be accompanied by fermionic strings. These objects would topologically condense at larger space-time sheets with wormhole contacts that do not carry monopole flux: touching the larger space-time surface but not sticking to it. - One could understand why the genus for all wormhole throats must be the same for the simplest states as the TGD explanation of family replication phenomenon demands. Of course, the change of the topology along string like object cannot be excluded but very probably corresponds to an unstable higher mass excitation.
- The basic particle reactions would include re-connections of closed string like objects and their reversals. The replication of 3-surfaces would remain a new element brought by TGD. The basic processes at fermionic level would be reconnections of closed fermionic strings. The new element would be the presence of Euclidian regions allowing to talk about effective boundaries of strings as boundaries between the Minkowskian or Euclidian regions. This would simplify enormously the description of particle reactions by bringing in description topologically highly analogous to that provided by closed strings.

See the article How the hierarchy of Planck constants might relate to the almost vacuum degeneracy for twistor lift of TGD?.

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