## The recent vision about preferred extremals and solutions of the modified Dirac equationDuring last months a considerable progress in understanding of both the preferred extremals of Kähler action and solutions of modified Dirac equation has taken place and there are good reasons to believe that various approaches are converging. Instead of trying to describe the results in detail here I just give the abstract of the article The recent vision about preferred extremals and solutions of the modified Dirac equation. The text appears also in the chapter Does the modified Dirac action define the fundamental variational principle? of the online book "TGD: Physics as Infinite-dimensional Geometry". Here is the short abstract.
- For preferred extremals generalization of conformal invariance to 4-D situation is very attractive approach and leads to concrete conditions formally similar to those encountered in string model (see this). In particular, Einstein's equations with cosmological constant follow as consistency conditions and field equations reduce to a purely algebraic statements analogous to those appearing in equations for minimal surfaces if one assumes that space-time surface has Hermitian structure or its Minkowskian variant Hamilton-Jacobi structure. The older approach based on basic heuristics for massless equations, on effective 3-dimensionality, and weak form of electric magnetic duality, and Beltrami flows is also promising. An alternative approach is inspired by number theoretical considerations and identifies space-time surfaces as associative or co-associative sub-manifolds of octonionic imbedding space (see this).
- There are also several approaches for solving the modified Dirac equation. The most promising approach is assumes that the solutions are restricted on 2-D stringy world sheets and/or partonic 2-surfaces. This strange looking view is a rather natural consequence of both strong form of holography and of number theoretic vision, and also follows from the notion of finite measurement resolution having discretization at partonic 2-surfaces as a geometric correlate. The conditions stating that electric charge is conserved for preferred extremals is an alternative very promising approach. One expects that stringy approach based on 4-D generalization of conformal invariance or its 2-D variant at 2-D preferred surfaces should also allow to understand the modified Dirac equation.
In the following the question whether these various approaches are mutually consistent is discussed. It indeed turns out that the approach based on the conservation of electric charge leads under rather general assumptions to the proposal that solutions of the modified Dirac equation are localized on 2-dimensional string world sheets and/or partonic 2-surfaces. One can also apply the approach at imbedding space level to the solutions of ordinary Dirac equation and this might be actually relevant for the representations of symplectic group of δ M
^{4}_{+/-}× CP_{2}. The implication seems to be that only color singlet representation is allowed for leptons and color triplet/antitriplet for quarks/anti-quarks. The result is physically very nice but would almost totally eliminate spinorial CP_{2} partial waves.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article already mentioned. |