Critizing the view about elementary particles

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 CP₂ type vacuum extremals for which the 1-D M⁴ 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 CP₂ extremal. This implies that M⁴ 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 M⁴ projection for the light-like boundaries of string world sheets becomes light-like geodesic of M⁴× CP₂ instead light-like geodesic of M⁴ 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 CP₂ 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 X²× Y², where X² is minimal surface in M⁴ and Y² a complex surface of CP₂ characterized by genus. Deformation would have 4-D M⁴ 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 X²× S², S² 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.

1. Elementary particles would be simply closed flux tubes which look like very long flattened squares. Short sides with length of order CP₂ radius would be identifiable as pieces of deformed CP₂ 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.
2. 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.
3. 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.
4. The original picture need not of course be wrong: it is only slightly more complex than the above proposal. One would have two space-time sheets connected by a pair of wormhole contacts between, which most of the magnetic flux would flow like in flux tube. The flux from the throat could emerges more or less spherically but eventually end up to the second wormhole throat. The sheets would be connected along their boundaries so that 3-space would be connected. The absence of boundary terms in the action implies this. The monopole fluxes would sum up to a vanishing flux at the boundary, where gluing of the sheets of the covering takes place.

If only Minkowskian portions are present, particles could be seen as pairs of open fermionic strings and the counterparts of open string vertices would be possible besides reconnection of closed strings. For this option one can also consider single fermionic open strings connecting wormhole contacts: now possible flux tube would not carry monopole flux.

For background see Particle Massivation in TGD Universe.