The QCD description of hadronic reactions is statistical and is in terms of quark and gluon distribution functions characterizing hadrons and fragmentation functions to hadrons for quarks and gluons. What could the TGD counterparts of these functions be and could an analogous description at quantum level be possible? What happens in the transitions hadron phase and free quark phase and how to describe this in TGD?
1. Hadron phase ↔ quark phase transition as a transition between phases characterized by 8-D and 4-D masslessness
In the transition to the X4 phase with free massless quarks, the colored H spinor modes are replaced with holomorphic X4 spinor modes. The opposite transition takes place in hadronization. X4→ H transition is analogous with the Higgs mechanism in which transition occurs from a massless phase to a massive phase (in M4 sense). Transition is also between deconfined and confined phases. This description applies also to leptons which also move in H spinor partial waves.
A more general view is based on conformal symmetry breaking. In hadronization 4-D light-likeness replaced with 8-D light-likeness in H. Propagation takes place along the space-time surface and propagator is determined by the induced/modified Dirac operator. What is of crucial importance is that fermionic oscillator operators for inducd spinors fields are expressed in terms of those for the H spinor field.
What about the description of color in the X4 phase? Does one obtain color triplets in the holomorphic basis? Could the color partial waves {ξ1,ξ2,1} proportional form a counterpart of color triplet? Does the color triplet correspond to the 3 coordinate patches for the complex structure of CP2 as a complex projective space? Why color triplets are special for quarks and color singlets for leptons. Does this relate to conformal invariance making higher partial waves gauge degrees of freedom? What about Kac-Moody type gauge invariance? Could only the lowest modes matter. Fixed H spinor modes as ground states for Kac-Moody representations.
2. Quantum measurement theory in ZEO as a guideline
Quantum measurement can be seen as a Hilbert space projection.Could this projection be induced by a geometric projection from H to the space-time surface for the spinor modes. The modes of the X4 Dirac operator have a fixed M4 chirality and this is the signature of masslessness. Apart from the covariantly constant right-handed neutrino, H modes have only a fixed H chirality and are therefore massive. Therefore also M4 chirality would be measured in the transition to the quark phase. Note that also projections to lower dimensional surfaces, such as partonic orbits, string world sheets and fermion lines make sense if this interpretation is correct.
In this picture, the overlap between H modes and X4 modes would characterize the transition from hadrons to quarks and vice versa. The ZEO based description of any particle reaction involves a pair of BSFRs. In the case of hadronic reactions this would involve the transition of hadrons to quarks in BSFR, time evolution with opposite arrow of time, and second BSFR leading from quark phase to hadron phase.
- In ZEO, the deconfinement phase transition H→ X4 from hadron to quark phase would involve a localization from H to X4. This also means a localization in the "world of classical worlds" (WCW). In the deconfined state localized to single X4, one would have an analog of QFT in a fixed background space-time. Note however that every 3-surface defines its own space-time surface as its Bohr orbit, which is however not quite unique, which in fact forces ZEO. Therefore one has a superposition of scattering amplitudes over the space-time surfaces satisfying holography= holomorphy principle.
- Hadronization as a transition X4→ H would in turn mean a delocalization in WCW and could be interpreted as a localization in the analog of momentum space for WCW. The observables measured would be quantum numbers of WCW spinor modes. This includes measurement of H quantum numbers but the light states are color singlet many fermion states. Color partial waves have the CP2 mass scale.
3. What does the interaction of particles as space-time surfaces mean?
What does the interaction of particles as space-time surfaces obeying holography= holomorphy principle mean? When do the particle interactions lead to the transition to the phase corresponding to a localization in WCW? In strong interactions this kind of interaction requires a high collision energy implying that the interactions occur in a scale smaller than the geometric size scale of the colliding particles so that the internal geometric structure of the particle become visible. In the TGD framework, these details naturally correspond to lower dimensional structure consisting of the light-like parton orbits and string world sheets having their boundaries at the parton orbits. Note that this picture might apply to to all interactions. Topological considerations allow to make this picture rather concrete (see this).
- For topological reasons, the intersection of the generic space-time surfaces is a discrete set of points. The systems should fuse somehow and form a quantum coherent interaction region. If the H-J structures of the space-time surfaces are identical meaning that in the interaction region both space-time surfaces have the same coordinates (u,v, w, ξ1, ξ2), the intersection is 2-D string world sheet, containing point-like fermions as fermion lines at its boundaries assignable to light-like 3-D parton orbit (see this). This makes possible a string model type description for the interactions of the fundamental fermions. By the hypercomplex holomorphy, the description would be rather simple since the second light-like coordinate of the string world sheet is non-dynamical.
- Only fermions and their bound states appear as fundamental quantum objects in the TGD framework. If they emerge in the formation of states delocalized in WCW they would correspond to hadrons, leptons and electroweak bosons. In particular, bosons as incoming and outgoing are identified as bound states of fermions and antifermions. The stringy view of the interactions implies that bosons need not appear at all in the deconfined phase.
If so, there would be no gluons and the fundamental vertex would correspond to a creation or annihilation of a fermion pair from "vacuum" and the classical induced gauge fields would define the vertices. This would take place when string world sheets fuse or split and a pair of fermion lines at separate string world sheets is created or disappears.
- The notion of exotic smooth structure (see this, this, and this) possible only in 4-D space-time and reducing to the standard smooth structure apart from defects identifiable as this kind of singularities allows these kinds of edges (see this, this, and this). This allows also to consider scattering events in which the fermion line has an edge serving as vertex giving rise to momentum exchange. These edges would correspond to failure of holomorphy at a single point.
4. The relationship between the oscillator operators of spinor modes in H and X4
It is possible to express X4 oscillator operators in terms of H oscillator operators (see this). Induction means the restriction of the mode expansion of the second quantized H spinor field to the space-time surface X4. Similar expansion for X4 spinor field in terms of conformal modes makes sense. The two representations must be identical. This implies that the oscillator operators at X4 are expressible as inner products of conformal modes and H spinor field. H oscillator operators are fundamental and no separate second quantization at X4 is needed.
The inner products between the spinor modes of X4 and H involve an integration over the space-time surface or a lower-dimensional space-time region. By the 8-D chiral symmetry the matrix element must involve gamma matrices and reduce to an integral of an inner product of the conformal mode of the induced Dirac operator with the fermionic super current over the parton orbit. The 3-D intersection of the space-time surface with the light-like boundary of CD cannot be excluded. The integral over the parton orbit is natural since the transversal hypercomplex coordinate for the associated string world sheet is not dynamical. These integrals characterize the transition between the two phases and its reversal and would replace the parton distribution functions and fragmentation functions in TGD. The conservation of color quantum numbers and corresponding M4 quantum numbers in holomorphic basis in which X4 complex coordinates correspond to those of H.
What about propagators in the quark phase? The propagation would be restricted to X4 rather than occurring in H. X4 spinor field would be defined as the sum over its conformal modes and the Dirac propagator would be defined as a two point function, which can be calculated because oscillator operators are expressible in terms of H oscillator operators.
5. Description of hadron reactions in ZEO
As found, zero energy ontology and holomorphy= holography vision suggest a universal description of all particle reactions. The particle reaction involves a temporary time reversal involving two BSFRs.
- In the first BSFR a projection from the space of hadron states in H to free many-quark states in X4 would occur. This localization in WCW would also involve a measurement of M4 chirality by an external observer. The resulting state would consist of free massless quarks in X4 and evolve by SSFRs backwards in geometric time. The interactions would be mediated by string world sheets having fermion lines at their boundaries and the notion of exotic smooth structure would be essential making possible fermion scattering and pair creation in the absence of fundamental bosonic quantum fields.
- After that a second BSFR would occur inducing a delocalization in WCW and a hadronic state would emerge and evolve by SSFRs. One can say that the states delocalized in WCW correspond to hadrons (and quite generally color singlet states). WCW observables, which include the observables associated with H, would be measured. Concerning the calculation of the scattering amplitudes, this means that the quark oscillator oscillator operators would be expressed in terms of H oscillator operators and a Hilbert space projection to a state of hadrons would take place.
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.