About Dirac equation in H=M4×CP2 assuming Kähler structure for M4

The holography= holomorphy vision has led also to a dramatic progress in the understanding of Dirac equation in the TGD framework, in particular its variant involving M⁴ Kähler form, and this article summarizes the latest discoveries.

The first topic of this article is Hamilton-Jacobi (H-J) structure, in particular the its hyper-complex aspects. The physical implications of M⁴ Kähler structure are discussed and the classification of Hamilton-Jacobi structures at the level of H=M⁴× CP₂ and of space-time surfaces are discussed.

The main topic of the article are the solutions with non-trivial M⁴ Kähler structure.

1. The mass squared spectrum of the M⁴ Dirac operator turns out to be integer valued harmonic oscillator spectrum, being therefore “stringy”. Mass squared has also negative tachyonic values.

By 8-D masslessness, M⁴ mass squared spectrum must be identical with CP₂ mass squared spectrum given by eigenvalues of SU(3) Casimir operator with a spin term added. For the color partial waves, the Casimir operator is always integer valued whereas the spin term, which is proportional to the Kähler coupling q_K need not be integer valued. For CP₂, the ratio of Kähler couplings q_K of leptons and quarks is 3.

The possibility of M⁴ tachyons leads to a proposal that, apart from the p-adic thermodynamic contribution to the mass squared, physical states are massless and that the addition of tachyonic fermion pairs to the states guarantees this. In the model proposed earlier, they would be pairs of left-handed and right-handed neutrinos, inducing a weak screening and guaranteeing that leptons/quarks correspond to color singlets/triplets. The different values of the spin terms for quarks and leptons however imply that this mechanism requires that the total number of quarks and antiquarks is a multiple of 3! In particular, massless bosons can be either lepton pairs or superposition of lepton pairs and baryon pairs. This condition also implies that the sum of spin terms for quarks is integer valued and the total M⁴ and CP₂ contributions to the mass squared can be identical.

To sum up, the earlier work with Dirac equation and propagator (see this) without M⁴ Kähler structure, when combined with the work of this article, allows to understand color confinement, predicts that quarks can appear only as baryon-like bound states of three quarks, predicts that bosons are superpositions of lepton pairs and baryon pairs (rather than quark pairs). Also the mechanism transforming ultraheavy colored quarks and leptons with CP₂ mass scale to massless color singlet leptons and baryons by addition of pairs of left- and right-handed tachyonic neutrinos is understood at the level of the Dirac equation.

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