## Sunday, September 25, 2022

### About the TGD based notions of mass, of twistors and hyperbolic counterpart of Fermi torus

The notion of mass in the TGD framework is discussed from the perspective of M8-H duality (see this, this, and this).
1. In TGD, space-time regions are characterized by polynomials P with rational coefficients (see this). Galois confinement defines a universal mechanism for the formation of bound states. Momenta for virtual fermions have components, which are algebraic integers in an extension of rationals defined by a polynomial P characterizing a space-time region. For the physical many fermion states, the total momentum as the sum of fermion momenta has components, which are integers using the unit defined by the size of the causal diamond (CD) (see this, this, and this).
2. This defines a universal number theoretical mechanism for the formation of bound states as Galois singlets. The condition is very strong but for rational coefficients it can be satisfied since the sum of all roots is always a rational number as the coefficient of the first order term.
3. Galois confinement implies that the sum of the mass squared values, which are in general complex algebraic numbers in E, is also an integer. Since the mass squared values correspond to conformal weights as also in string models, one has conformal confinement: states are conformal singlets. This condition replaces the masslessness condition of gauge theories (see this).
Also the TGD based notion of twistor space is considered at concrete geometric level.
1. Twistor lift of TGD means that space-time surfaces X4 is H=M4× CP2 are replaced with 6-surfaces in the twistor space with induced twistor structure of T(H)= T(M4)× T(CP2) identified as twistor space T(X4). This proposal requires that T(H) has Kähler structure and this selects M4× CP2 as a unique candidate (see this) so that TGD is unique.
2. One ends up to a more precise understanding of the fiber of the twistor space of CP2 as a space of "light-like" geodesics emanating from a given point. Also a more precise view of the induced twistor spaces for preferred extremals with varying dimensions of M4 and CP2 projections emerges. Also the identification of the twistor space of the space-time surface as the space of light-like geodesics itself is considered.
3. Twistor lift leads to a concrete proposal for the construction of scattering amplitudes. Scattering can be seen as a mere re-organization of the physical many-fermion states as Galois singlets to new Galois singlets. There are no primary gauge fields and both fermions and bosons are bound states of fundamental fermions. 4-fermion vertices are not needed so that there are no divergences.
4. There is however a technical problem: fermion and antifermion numbers are separately conserved in the simplest picture, in which momenta in M4⊂ M8 are mapped to geodesics of M4⊂ H.The led to a proposal for the modification of M8-H duality (see this and this). The modification would map the 4-momenta to geodesics of X4. Since X4 allows both Minkowskian and Euclidean regions, one can have geodesics, whose M4 projection turns backwards in time. The emission of a boson as a fermion-antifermion pair would correspond to a fermion turning backwards in time. A more precise formulation of the modification shows that it indeed works
The third topic of this article is the hyperbolic generalization of the Fermi torus to hyperbolic 3-manifold H3/Γ. Here H3=SO(1,3)/SO(3) identifiable the mass shell M4\subset M8 or its M8-H dual in H=M4× CP2. Γ denotes an infinite subgroup of SO(1,3) acting completely discontinuously in H3. For virtual fermions also complexified mass shells are required and the question is whether the generalization of H3/Γ, defining besides hyperbolic 3-manifold also tessellation of H3 analogous to a cubic lattice of E3.

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