Massless quarks as modes of the Dirac equation for the induced spinors at the space-time surface

I wrote some time ago an article describing the results from solving the Dirac equation in H= M⁴×CP₂ assuming that M⁴ has an analog of Kähler structure. Apart from covariantly constant right handed neutrino (covariantly constant in CP₂) the color partial waves have huge masses with mass scale given by CP₂ mass scale. It is however possible to obtain massless color singlet many-quark states with the difference of quark and antiquark numbers equal to a multiple of 3. These would develop p-adic thermal mass squared. The conclusion was that only hadrons are possible in this framework.

There is however an objection against this view. The successes of QCD suggests that also the description in terms of massless quarks should make sense and should correspond to a phase different from the hadronic phase.

1. The induction of the spinor structure to the space-time surface is a fundamental piece of TGD. This gives a induced/modified Dirac equation at the space-time surface X⁴ and the generalized holomorphic solutions of this equation are massless in the sense that the square of the modified Dirac operator annihilates them (see this and this). The conjugates of the holomorphic gamma matrices annihilate these modes and implies that the spin term involving the induced Kähler form vanishes and does not give rise to mass squared term. Somehow I failed to realize that the modes of this Dirac equation represent massless quarks. One can speak of a phase dual to the hadronic phase.
2. The induction of spinor structure (see this) by restricting the H modes to the space-time surface however requires a generalized holomorphic solution basis for H, which makes sense only in finite regions of M⁴ and CP₂ inside which the holomorphic modes remain finite. It is not clear whether this basis is locally orthogonal to the solution basis of ordinary Dirac equation in H. These modes must remain finite in X⁴. Since space-time surfaces are enclosed inside CDs with a finite size scale and CP₂ type Euclidean regions connecting two Minkowskian space-time sheets (see this and this) have holes as 3-D light-like partonic orbits, these modes can remain finite.
3. If the notion of the induced/modified spinor structure really makes sense, one can speak of two phases: free quarks and gluons and hadrons. The hadronic phase corresponds to the modes of the Dirac operator of H massless in 8-D sense but extremely massive in M⁴ sense. The holomorphic modes of the X⁴ Dirac operator correspond to massless quarks and gluons and also leptons. These descriptions should be dual to each other. The challenge is to understand this phase transition, which means breaking of conformal invariance and can be seen as a generalization of the phase transition from Higgs=0 phase to a phase with non-vanishing Higgs expectation.

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.