How could the transitions between hadronic and quark phases occur in the TGD framework?

The solutions of the ordinary Dirac equation in H with M⁴ Kähler structure have masses of order CP₂ and light states are color singlets whereas the solutions of the induced/modified Dirac equation for quarks in X⁴ are massless. In the case of quarks this suggests an interpretation in terms of hadrons and massless quarks. This picture also applies to leptons.

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.

1. 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.
2. 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. 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.

Inner product between the spinor modes of X⁴ and H involves an integration over the space-time surface. Overlap integrals between the c-number valued modes of X⁴ spinor field and the second quantized H spinor field give X⁴ oscillator operators in terms of H oscillator operators. 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.

4. Description of hadron reactions in ZEO

Zero energy ontology suggests a universal description of all particle reactions. The particle reaction involves a temporary time reversal involving two BSFRs.

1. 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.
2. 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.

See the article About Dirac equation in H= M⁴ × CP₂ assuming Kähler structure for M⁴ 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.

Source: https://matpitka.blogspot.com/2025/07/how-could-transitions-between-hadronic.html