How could strong interactions emerge at the level of scattering amplitudes?

The above considerations are dangerous in that the intuitive QFT based thinking based is applied in TGD context where all interactions reduced to the dynamics of 3-surfaces and fields are geometrized by reducing them to the induced geometry at the level of space-time surface. Quantum field theory limit is obtained as an approximation and the applications of its notions at the fundamental level might be dangerous. In any case, it seems that only electroweak gauge potentials appear in the fermionic vertices and this might be a problem.

1. By holography perturbation series is not needed in TGD. Scattering amplitudes are sums of amplitudes associated with Bohr orbits, which are not completely deterministic: there is no path integral. Whether path integral could be an approximate approximation for this sum under some conditions is an interesting question.
2. It is best to start from a concrete problem. Is pair creation possible in TGD? The problem is that fermion and antifermion numbers are separately conserved for the most obvious proposals for scattering amplitudes. This essentially due to the fact that gauge bosons correspond to fermion-antifermion pairs. Intuitively, fermion pair creation means that fermion turns backwards in time. If one considers fermions in classical background fields this turning back corresponds to a 2-particle vertex. Could pair creation in classical fields be a fundamental process rather than a mere approximation in the TGD framework. This would conform with the vision that classical physics is an exact part of quantum physics.

   The turning back in time means a sharp corner of the fermion line, which is light-like elsewhere. M⁴ time coordinate has a discontinuous derivative with respect to the internal time coordinate of the line. I have propoeed (see this and this) that this kind of singularities are associated with vertices involving pair creation and that they correspond to local defects making the differentiable structure of X⁴ exotic. The basic problem of GRT would become a victory in the TGD framework and also mean that pair creation is possible only in 4-D space-time.

One can imagine two kinds of turning backs in time.

1. The turning back in time could occur for a 3-D surface such as monopole flux tube and induce the same process the string world sheets associated with the flux tubes and for the ends of the string world sheets as fermion lines ending at the 3-D light-like orbits of partonic 2-surfaces.
2. In the fusion of two 2-sheeted monopole flux tubes along their "ends" identifiable as partonic 2-surfaces wormhole contacts, the ends would fuse instantaneously (this process is analogous to "join along boundaries". The time reversal of this process would correspond to the splitting of the monopole flux tube inducing a turning back in time for a partonic 2-surface and for fermionic lines as boundaries of string world sheets at the partonic 2-surface.

   This would be analogous to a creation of a fermion pair in a classical induced gauge field, which is electroweak. The same would occur for the partonic 2-surfaces as opposite wormhole throats and for the string world sheets having light-like boundaries at the orbits of partonic 2-suraces.

3. The light-like orbit of a partonic 2-surface contains fermionic lines as light-like boundaries of string world sheets. A good guess is that the singularity is a cusp catastrophe so that the surface turns back in time in exactly the opposite direction. One would have an infinitely sharp knife edge.

What one can say about the scattering amplitudes on the basis of this picture? Can one obtain the analog for the 2-vertex describing a creation of a fermion pair in a classical external field?

1. The action for a geometric object of a given dimension defines modified gamma matrices in terms of canonical momentum currents as Γ^α= T^αkΓ_k, T^α_k= ∂ L/∂(∂_α h^k). By hermiticity, the covariant divergence D_αΓ^α of the vector defined by modified gamma matrices must vanish. This is true if the field equations are satisfied. This implies supersymmetry between fermionic and bosonic degrees of freedom.

   For space-time surfaces, the action is Kähler action plus volume term. For the 3-D light-partonic orbits one has Chern-Simons-Kähler action. For string world sheets one has area action plus the analog of Kähler magnetic flux. For the light-boundaries of string world sheets defining fermion lines one has the integral ∫ A_μdx^μ. The induced spinors are restrictions of the second quantized spinors fields of H=M⁴× CP₂ and the argument is that the modified Dirac equation holds true everywhere, except possibly at the turning points.

2. Consider now the interaction part of the action defining the fermionic vertices. The basic problem is that the entire modified Dirac action density is present and vanishes if the modified Dirac equation holds true everywhere. In perturbative QFT, one separates the interaction term from the action and obtains essentially ΨbarΓ^α D_αΨ. This is not possible now.

   The key observation is that the modified Dirac equation could fail at the turning points! QFT vertices would have purely geometric interpretation. The gamma matrices appearing in the modified Dirac action would be continuous but at the singularity the derivative ∂_μΨ= ∂_μm^k ∂_kΨ of the induced free second quantized spinor field of H would become discontinuous. For a Fourier mode with momentum p^k, one obtains

   ∂_μΨ_p= p_k∂_μ m^kΨ_p == p_μΨ_p.

   This derivative changes sign in the blade singularity. At the singularity one can define this derivative as an average and this leaves from the action Ψbar Γ^α D_αΨ only the term ΨbarΓ^α A_αΨ. This is just the interaction part of the action!

3. This argument can be applied to singularities of various dimensions. For D=3, the action contains the modified gamma matrices for the Kähler action plus volume term. For D=2, Chern-Simons-Kähler action defines the modified gamma matrices. For string world sheets the action could be induced from area action plus Kähler magnetic flux. For fermion lines from the 1-D action for fermion in induced gauge potential so that standard QFT result would be obtained in this case.

How does this picture relate to perturbative QFT?

1. The first thing to notice is that in the TGD framework gauge couplings do not appear at all in the interaction vertices. The induced gauge potentials do not correspond to A but to gA. The couplings emerge only at the level of scattering amplitudes when one goes to the QFT limit. Only the Kähler coupling strength and cosmological constant appear in the action.
2. The basic implication is that only the electroweak gauge potentials appear in the vertices. This conforms with the dangerous looking intuition that also strong interactions can be described in terms of electroweak vertices but this is of course a potential killer prediction. One should be able to show that the presence of WCW degrees of freedom taken into account minimally in terms of fermionic color partial waves in CP₂ predicts strong interactions and predicts the value of α_s. Note that the restriction of spinor harmonics of CP₂ to a homologically non-trivial geodesic sphere gives U(2) partial waves with the same quantum numbers as SU(3) color partial waves have.
3. TGD approach differs dramatically from the perturbative QFT. Since 1/α_s appears in the vertex, the increase of h_eff in the vertex increases it: just the opposite occurs in the perturbative QFT! This seems to be in conflict with QFT intuition suggesting a perturbation series in α_s ∝ 1/ℏ_eff. The explanation is that 1/α_K appears as a coupling parameter instead of α_s.

   This reminds of the electric-magnetic duality between perturbative and non-perturbative phases of gauge theories, where magnetic coupling strength is proportional to the inverse of the electric coupling strength. The description in terms of monopole flux tubes is therefore analogous to the description in terms of magnetic monopoles in the QFT framework. In TGD, it is the only natural description at the fundamental level. The decrease of α_K by increase of h_eff would indeed correspond to the QFT type description reduction of α_s.

   Could the description based on Maxwellian non-monopole flux tubes correspond to the usual perturbative phase without magnetic monopoles? In the Maxwellian phase there is huge vacuum degeneracy due to the presence of vacuum extremals with a vanishing Kähler form at the limit of vanishing volume action. Could this degeneracy allow path integral as a practical approximation at QFT limit.

4. h_eff/h₀ = n is proposed to correspond to the dimension of algebraic extension of rationals associated with the space-time surface and serve as a measure for algebraic complexity. The increase of algebraic complexity of the space-time region defining the strong interaction volume would also make interactions strong. In TGD, the fundamental coupling strength would be α_K and the increase of α_K for ordinary value of h would force the increase of h. This should happen below the electroweak scale or at least the confinement scales and make perturbation theory describing strong interactions possible. This description would involve monopole flux tubes and their reconnections.

   See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.