Monday, November 14, 2022

Could the exotic smooth structures have a physical interpretation in the TGD framework?

The exotic smooth structures appearing for 4-D manifolds, even R4, challenges both the classical GRT and quantum GRT based on path integral approach. They could also cause problems in the TGD framework. I have already considered the possibility that the exotics could be eliminated by what could be called holography of smoothness meaning that the smooth structure for the boundary dictates it for the entire 4-surface. One can however argue that the atlas of charts for 3-D boundary cannot define uniquely the atlas of charges for 4-D surface.

In the TGD framework, exotic smooth structures could also have a physical interpretation. As noticed, the failure of the standard smooth structure can be thought to occur at a point set of dimension zero and correspond to a set of point defects in condensed matter physics. This could have a deep physical meaning.

  1. The space-time surfaces in H=M4× CP2 are images of 4-D surfaces of M8 by M8-H-duality. The proposal is that they reduce to minimal surfaces analogous to soap films spanned by frames. Regions of both Minkowskian and Euclidean signature are predicted and the latter correspond to wormhole contacts represented by CP2 type extremals. The boundary between the Minkowskian and Euclidean region is a light-like 3-surface representing the orbit of partonic 2-surface identified as wormhole throat carrying fermionic lines as boundaries of string world sheets connecting orbits of partonic 2-surfaces.
  2. These fermionic lines are counterparts of the lines of ordinary Feynman graphs, and have ends at the partonic 2-surfaces located at the light-like boundaries of CD and in the interior of the space-time surface. The partonic surfaces, actually a pair of them as opposite throats of wormhole contact, in the interior define topological vertices, at which light-like partonic orbits meet along their ends.
  3. These points should be somehow special. Number theoretically they should correspond points with coordinates in an extension of rationals for a polynomial P defining 4-surface in H and space-time surface in H by M8-H duality. What comes first in mind is that the throats touch each other at these points so that the distance between Minkowskian space-time sheets vanishes. This is analogous to singularities of Fermi surface encountered in topological condensed matter physics: the energy bands touch each other. In TGD, the partonic 2-surfaces at the mass shells of M4 defined by the roots of P are indeed analogs of Fermi surfaces at the level of M4⊂ M8, having interpretation as analog of momentum space.

    Could these points correspond to the defects of the standard smooth structure in X4? Note that the branching at the partonic 2-surface defining a topological vertex implies the local failure of the manifold property. Note that the vertices of an ordinary Feynman diagram imply that it is not a smooth 1-manifold.

  4. Could the interpretation be that the 4-manifold obtained by removing the partonic 2-surface has exotic smooth structure with the defect of ordinary smooth structure assignable to the partonic 2-surface at its end. The situation would be rather similar to that for the representation of exotic R4 as a surface in CP2 with the sphere at infinity removed (see this).
  5. The failure of the cosmic censorship would make possible a pair creation. As explained, the fermionic lines can indeed turn backwards in time by going through the wormhole throat and turn backwards in time. The above picture suggests that this turning occurs only at the singularities at which the partonic throats touch each other. The QFT analog would be as a local vertex for pair creation.
  6. If all fermions at a given boundary of CD have the same sign of energy, fermions which have returned back to the boundary of CD, should correspond to antifermions without a change in the sign of energy. This would make pair creation without fermionic 4-vertices possible.

    If only the total energy has a fixed sign at a given boundary of CD, the returned fermion could have a negative energy and correspond to an annihilation operator. This view is nearer to the QFT picture and the idea that physical states are Galois confined states of virtual fundamental fermions with momentum components, which are algebraic integers. One can also ask whether the reversal of the arrow of time for the fermionic lines could give rise to gravitational quantum computation as proposed here.

See the article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II.

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

Articles related to TGD.

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