Unitarity constraint and the construction of S-matrix in the TGD framework

The recent TGD based view of particle reactions (see this) replaces QCD type approach with its stringy version and allows the construction of S-matrix for arbitrary initial and final states.

1. The construction of S-matrix in elementary particle degrees of freedom reduces to that for fundamental fermions. There are two levels involved. External particles are constructed as bound states of fundamental fermions giving rise to hadrons, leptons, gauge bosons, and gravitons. Number theoretic vision, in particular Galois confinement (see this and this) plays a key role in the construction of the bound states.

   The fundamental fermions correspond to the modes of the Dirac equation in H, being massless in the 8-D sense. If M⁴ has hypercomplex Kähler structure the Dirac equation in H allows massless light color singlet states as many-fermion states (see this).

   The analog of the quark phase corresponds to modes of the X⁴ Dirac operator for fundamental fermions, which are massless in 4-D sense: color triplets can be understood in terms of CP₂ geometry. The oscillator operators for X⁴ modes are expressible in terms of those for H modes (see this).

   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.

2. The S-matrix is determined by the overlap of these two fermionic state bases and the unitary matrix describing the scattering in the quark phase. Fermion pair creation in the induced classical fields is the basic vertex and reduces to a defect of the standard smooth structure: these defects give rise to an exotic smooth structure (see this, this and this). In the vertex, fermion current fails to be conserved for the standard smooth structure but is proposed to be conserved for the exotic smooth structure (see this, this, this and this). The non-vanishing divergence at the defect determines various vertices.
3. Besides the fermionic degrees of freedom, also the geometric degrees of freedom of WCW are included. Holography = holomorphy vision (H-H) (see this, this, this and this) implies that the path integral disappears and there is only a functional integral over 3-surfaces X³ and the sum over the Bohr orbits for each X³. Does the role of the functional integral become trivial with respect to unitarity? Locality in WCW suggests that this is the case. Let us assume in the following that this is indeed the case.

Unitarity is a strong constraint in the construction of S-matrix and will be considered in the sequel.

Two T-matrices corresponding to hadronic phase in H and quark phase in X⁴

How could the T-matrix for hadronic phase relate to the T-matrix for the quark phase, call it briefly t?

1. t would be related to scattering in the string phase, where the quarks would be free or rather at the boundary lines of string world sheets at light-like partonic orbits. The phase would consist only of conformally massless quarks and leptons at the fermionic lines. H-H would determine the space-time surfaces X⁴ and fermionic modes.
2. We can start from unitarity. In the hadron phase, the scattering amplitude satisfies the conditionT-T^† = TT^†. Unitarity would also hold for t in the quark phase. In the forward direction, a cut for T in the forward direction essentially gives the total cross section.
3. The scattering would correspond to two “big” state function reductions (BSFRs) changing the arrow of time (see this and this). T would be between the hadron phases and T^† between their time reversals. The same applies to t. This suggests a concrete interpretation of unitarity. T and T^† would correspond to opposite time directions. Analogously, t and t^† would be associated with a sequence of SSFRs in opposite time directions, increasing the size of the CD as a correlate for the geometric time. This would give a concrete geometric meaning for the unitary conditions.
4. T would decompose into a product of three operators. The first one would be the operator O, which would project from the hadron phase to the quark phase. It could, and actually should, be a 1-1 map. The second operator would be t or t^†, which would describe the scattering operator in the quark phase. The third would be the inverse operation of O. It should be possible to identify it uniquely, but if O is not 1-1, then there might be problems.
5. H-H gives strong conditions. t would correspond to a sequence of SSFRs and classical non-determinism would determine t. The creation of quark pairs is the basic process created by t, and here exotic smooth structures would come into play (see this)/masterformula,insectform}.

Could the unitarity for T reduce to unitarity for t?

1. O projects the hadronic state into a state consisting of quarks and the latter evolves according to t. After that, the quark state would to a hadronic state and the inverse of O would be included. The reduction T→ t from the hadronic level to the quark level takes place if an inverse of O exists.
2. If quark states can be mapped in 1-1 way to hadronic states, then the classical non-determinism, which can be interpreted as a cognitive non-determinism, would completely determine t. Everything would be discrete and extremely simple at the quark level. Note however that quark pair production occurs and the defect of the standard smooth structure as a classical correlate.

The transition involves the usual quantum physical non-determinism, which naturally to O and its inverse. O would be completely determined by the overlap of the spinor modes of H and X⁴ determined by H-H.

Can the matrix O be invertible?

Can O define an isometry between two different state spaces? The analog of a projection from the hadron phase to the quark phase is in question, and it need not be an isometry. The analog of projection, or rather, the map, of O from the hadron phase to the quark phase is well-defined. Can O have a unique inverse? Light-likeness in H and light-likeness in X⁴ are very different notions physically: is a 1-1 correspondence between hadronic and quark states possible?

1. Could the additional degrees of freedom in the quark phase come from the fact that X⁴ is not closed like CP₂ and CD is finite? Conformal modes would diverge in H but not in X⁴ and increase the number of the fermion modes. The argument does not seem convincing to me.
2. Classical non-determinism (see this)/memorytgd} brings in additional degrees of freedom identified as cognitive degrees of freedom. Could this make isometry possible?
3. Could additional degrees of freedom in the quark phase emerge from an improved measurement resolution needed to “see” the quarks. This would correspond to a larger extension to rationals and that in turn to cognitive non-determinism so that this option is equivalent with the third option.

About the role of hyperfinite factors (HFFs)?

1. HFF (see this, this and this) is a fractal and contains hierarchies of subalgebras isomorphic with HFF itself. The number-theoretic vision assigns such hierarchies as hierarchies of algebraic extensions of rationals. Also measurement accuracy can be defined in terms of algebraic complexity.
2. The concept of inclusion is central. A subfactor corresponds to a subalgebra of the factor. Inclusion is not a 1-1 correspondence nor isometry. For a factor, the trace of the unitary operator is Tr(Id)=1 and for a sub-factor, the trace of the projector to it is Tr(P)= q≤ 1. q is quantized. There is a close connection with quantum groups and related concepts. The concept of HFF is particularly natural for fermions, so that it nicely fits into TGD.
3. Does the quark phase correspond to a subfactor of the hadron phase? Could classical non-determinism increase the value of q to unity and make the correspondence an isometric embedding of the quark operator algebra to hadronic operator algebra?

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.