Friday, October 09, 2009

Octonionic approach to the modified Dirac equation

The recent progress in the understanding of the modified Dirac equation defining quantum TGD at fundamental level (see this and this) stimulated a further progress. I managed to find a general ansatz for the modified Dirac equation solving it with very general assumptions about the preferred extremal of Kähler action.

A key role in the ansatz is played by the assumption that modified Dirac equation can be formulated using an octonionic representation of imbedding space gamma matrices. Associativity requires that the space-time surface is associative in the sense that the modified gamma matrices expressible in terms of octonionic gamma matrices of H span quaternionic sub-algebra at each point of space-time surface. Also octonionic spinors at given point of space-time surface must be associative: that is they span same quaternionic subspace of octonions as gamma matrices do. Besides this the 4-D modified Dirac operator defined by Kähler action and the 3-D Dirac operator defined by Chern-Simons action and corresponding measurement interaction term must commute: this condition must hold true in any case. The point is that associativity conditions fix the solution ansatz highly uniquely since the action of various operators in Dirac equation is not allowed to lead out from the quaternionic sub-space and the resulting ansatz makes sense also for ordinary gamma matrices.

It must be emphasized that octonionization is far from a trivial process. The mapping of sigma matrices of imbedding space to their octonionic counterparts means projection of the vielbein group SO(7,1) to G2 acting as automorphism group of octonions and only the right handed parts of electroweak gauge potentials survive so that only neutral Abelian part of classical electroweak gauge field defined in terms of CP2 remains. More over, electroweak holonomy group is mapped to rotation group so that electroweak interactions transform to gravitational interactions in the octonionic context! If octonionic and ordinary representations of gamma matrices are physically equivalent this represents kind of number theoretical variant for the possiblity to represent gauge interactions as gravitational interactions. This effective reduction to electrodynamics is absolutely essential for the associativity and simplifies the situation enormously. The conjecture is that the resulting solutions as such define also solutions of the modified Dirac equation for ordinary gamma matrices.

The additional outcome is a nice formulation for the notion of octo-twistor using the fact that octonion units define a natural analog of Pauli spin matrices having interpretation as quaternions. Associativity condition reduces the octo-twistors locally to quaternionic twistors which are more or less equivalent with the ordinary twistors and their construction recipe might work almost as such. It must be however emphasized that this notion of twistor is local unlike the standard notion of twistor since projections of momentum and color charge vector to space-time surface are considered. The two spinors defining the octo-twistor correspond to quark and lepton like spinors having different chirality as 8-D spinors.

The basic motivation for octo-twistors is that they might allow to overcome the problems caused by the massivation in the case of ordinary twistors. One might think that 4-D massive particles correspond to 8-D massless particles. A more refined idea emerges from modified Dirac equation. The space-time vector field obtained by contracting the space-time projections of four-momentum and the vector defined by Cartan color charges might be light-like with respect to the effective metric defined by the anticommutators of the modified gamma matrices. Whether this additional condition is consistent with field equations for the preferred extremals of Kähler action remains to be seen. Note that the geometry of the space-time sheet depends on momentum and color quantum numbers in accordance with quantum classical correspondence: this is what makes possible entanglement of classical and quantum degrees of freedom essential for quantum measurement theory.

Since it not much point in typing the detailed equations I give a link to a ten page pdf file Octo-twistors and modified Dirac equation representing the calculations. For details and background see the chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of the book "Towards M-matrix".

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