^{8}(see this). In the simplest picture, scattering would be just a re-organization of Galois singlets to new Galois singlets. Fundamental fermions would move as free particles.

The components of the 4-momentum of virtual fundamental fermion with mass m would be algebraic integers and therefore complex. The real projection of 4-momentum would be mapped by M^{8}-H duality to a geodesic of M^{4} starting from either vertex of the causal diamond (CD) . Uncertainty Principle at classical level requires inversion so that one has a= ℏ_{eff}/m, where ab denotes light-cone proper time assignable to either half-cone of CD and m is the mass assignable to the point of the mass shell H^{3}⊂ M^{4}⊂ M^{8}.

The geodesic would intersect the a=ℏ_{eff}/m 3-surface and also other mass shells and the opposite light-cone boundaries of CDs involved. The mass shells and CDs containing them would have a common center but Uncertainty Principle at quantum level requires that for each CD and its contents there is an analog of plane wave in CD cm degrees of freedom.

One can however criticize this framework. Does it really allow us to understand pair creation at the level of the space-time surfaces X^{4}⊂ H?

- All elementary particles consist of fundamental fermions in the proposed picture. Conservation of fermion number allows pair creation occurring for instance in the emission of a boson as fermion-antifermion pair in f→ f+b vertex.
- The problem is that if only non-space-like geodesics of H are allowed, both fermion and antifermion numbers are conserved separately so that pair creation does not look possible. Pair creation is both the central idea and source of divergence problems in QFTs.
- One can identify also a second problem: what are the anticommutation relations for the fermionic oscillator operators labelled by tachyonic and complex valued momenta? Is it possible to analytically continue the anticommutators to complexified M
^{4}⊂ H and M^{4}⊂ M^{8}? Only the first problem will be considered in the following.

One can first try to identify the key ingredients of the possible solution of the problem.

- Crossing symmetry is fundamental in QFTs and also in TGD. For non-trivial scattering amplitudes, crossing moves particles between initial and final states. How should one define the crossing at the space-time level in the TGD framework? What could the transfer of the end of a geodesic line at the boundary of CDs to the opposite boundary mean geometrically?
- At the level of H, particles have CP
_{2}type extremals - wormhole contacts - as building bricks. They have an Euclidean signature (of the induced metric) and connect two space-time sheets with a Minkowskian signature.The opposite throats of the wormhole contacts correspond to the boundaries between Euclidean and Minkowskian regions and their orbits are light-like. Their light-like boundaries, orbits of partonic 2-surfaces, are assumed to carry fundamental fermions. Partonic orbits allow light-like geodesics as possible representation of massless fundamental fermions.

Elementary particles consist of at least two wormhole contacts. This is necessary because the wormhole contacts behave like Kähler magnetic charges and one must have closed magnetic field lines. At both space-time sheets, the particle could look like a monopole pair.

- The generalization of the particle concept allows a geometric realization of vertices. At a given space-time sheet a diagram involving a topological 3-vertex would correspond to 3 light-like partonic orbits meeting at the partonic 2-surface located in the interior of X
^{4}. Could the topological 3-vertex be enough to avoid the introduction of the 4-fermion vertex?

^{4}so that the classical interactions at the space-time surface would make it possible to describe pair creation without introducing a 4-fermion vertex? Could the creation of a fermion pair mean that a virtual fundamental fermion moving along a space-like geodesics of a wormhole throat turns backwards in time at the partonic 3-vertex. If this is the case, it would correspond to a tachyon. Indeed, in M

^{8}picture tachyons are building bricks of physical particles identified as Galois singlets.

- If fundamental fermion lines are geodesics at the light-like partonic orbits, they can be light-like but are space-like if there is motion in transversal degrees of freedom.
- Consider a geodesic carrying a fundamental fermion, starting from a partonic 2-surface at either light-like boundary of CD. As a free fermion, it would propagate to the opposite boundary of CD along the wormhole throat.
What happens if the fermion goes through a topological 3-vertex? Could it turn backwards in time at the vertex by transforming first to a space-like geodesic inside the wormhole contact leading to the opposite throat and return back to the original boundary of CD? It could return along the opposite throat or the throat of a second wormhole contact emerging from the 3-vertex. Could this kind of process be regarded as a bifurcation so that it would correspond to a classical non-determinism as a correlate of quantum non-determinism?

- It is not clear whether one can assign a conserved space-like M
^{4}momentum to the geodesics at the partonic orbits. It is not possible to assign to the partonic 2-orbit a 3-momentum, which would be well-defined in the Noether sense but the component of momentum in the light-like direction would be well-defined and non-vanishing.By Lorentz invariance, the definition of conserved mass squared as an eigenvalue of d'Alembertian could be possible. For light-like 3-surfaces the d'Alembertian reduces to the d'Alembertian for the Euclidean partonic 2-surface having only non-positive eigenvalues. If this process is possible and conserves M

^{4}mass squared, the geodesic must be space-like and therefore tachyonic.The non-conservation of M

^{4}momentum at single particle level (but not classically at n-particle level) would be due to the interaction with the classical fields. - In the M
^{8}picture, tachyons are unavoidable since there is no reason why the roots of the polynomials with integer coefficients could not correspond to negative and even complex mass squared values. Could the tachyonic real parts of mass squared values at M^{8}level, correspond to tachyonic geodesics along wormhole throats possibly returning backwards along the another wormhole throat?

- The description of massivation in terms of p-adic thermodynamics suggests that at the fundamental level massive particles involve non-observable tachyonic contribution to the mass squared assignable to the wormhole contact, which cancels the non-tachyonic contribution.
All articles, and for the most general option all quantum states could be massless in this sense, and the massivation would be due the restriction of the consideration to the non-tachyonic part of the mass squared assignable to the Minkowskian regions of X

^{4}. - p-Adic thermodynamics would describe the tachyonic part of the state as "environment" in terms of the density matrix dictated to a high degree by conformal invariance, which this description would break. A generalization of the blackhole entropy applying to any system emerges and the interpretation for the fact that blackhole entropy is proportional to mass squared. Also gauge bosons and Higgs as fermion-antifermion pairs would involve the tachyonic contribution and would be massless in the fundamental description.
- This could solve a possible and old problem related to massless spin 1 bosons. If they consist of a collinear fermion and antifermion, which are massless, they have a vanishing helicity and would be scalars, because the fermion and antifermion with parallel momenta have opposite helicities. If the fermion and antifermion are antiparallel, the boson has correct helicity but is massive.
Massivation could solve the problem and p-adic thermodynamics indeed predicts that even photons have a very small thermal mass. Massless gauge bosons (and particles in general) would be possible in the sense that the positive mass squared is compensated by equally small tachyonic contribution.

- It should be noted however that the roots of the polynomials in M
^{8}can also correspond to energies of massless states. This phase would be analogous to the Higgs=0 phase. In this phase, Galois symmetries would not be broken: for the massive phase Galois group permutes different mass shells (and different a=constant hyperboloids) and one must restrict Galois symmetries to the isotropy group of a given root. In the massless phase, Galois symmetries permute different massless momenta and no symmetry breaking takes place.

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

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