Very Special relativity and preferred role of M2 for generalized Feynman graphs

The preferred role of M² in the construction of generalized Feynman diagrams could be used as a criticism. Poincare invariance is lost. The first answer to the criticism is that one integrates of the choices of M² so that Poincare invariance is lost. One can however defend this assumption also from different view point. Actually Glashow and Cohen did this in their Very Special Relativity proposal! While scanning old files, I found an old text about Very Special Relativity of Glashow and Cohen, and realized that it relates very closely to the special role of M² in the construction of generalized Feynman diagrams. There is article Very Special Relativity and TGD at my homepage but for some reason the text has disappeared from the book that contained it. I add the article more or less as such here.

Configuration space ("world of classical worlds", WCW) decomposes into a union of sub-configuration spaces associated with future and past light-cones and these in turn decompose to sub-sub-configuration spaces characterized by selection of quantization axes of spin and color quantum numbers. At this level Poincare and even Lorentz group are reduced. The possibility that this kind of breaking might be directly relevant for physics is discussed below.

One might think that Poincare symmetry is something thoroughly understood but the Very Special Relativity proposed by nobelist Sheldon Glashow and Andrew Cohen suggests that this might belief might be wrong. Glashow and Cohen propose that instead of Poincare group, call it P, some subgroup of P might be physically more relevant than the whole P. To not lose four-momentum one must assume that this group is obtained as a semi-direct product of some subgroup of Lorentz group with translations. The smallest subgroup, call it L₂, is a 2-dimensional Abelian group generated by K_x+J_y and K_y-J_x. Here K refers to Lorentz boosts and J to rotations. This group leaves invariant light-like momentum in z direction. By adding J_z acting in L₂ like rotations in plane, one obtains L₃, the maximal subgroup leaving invariant light-like momentum in z direction. By adding also K_z one obtains the scalings of light-like momentum or equivalently, the isotropy group L₄ of a light-like ray.

The reasons why Glashow and Cohen regard these groups so interesting are following.

1. All kinematical tests of Lorentz invariance are consistent with the reduction of Lorentz invariance to these symmetries.

2. The representations of group L₃ are one-dimensional in both massive and massless case (the latter is familiar from massless representations of Poincare group where particle states are characterized by helicity). The mass is invariant only under the smaller group. This might allow to have left-handed massive neutrinos as well as massive fermions with spin dependent mass.

3. The requirement of CP invariance extends all these reduced symmetry groups to the full Poincare group. The observed very small breaking of CP symmetry might correlate with a small breaking of Lorentz symmetry. Matter antimatter asymmetry might relate to the reduced Lorentz invariance.

The idea is highly interesting from TGD point of view. The groups L₃ and L₄ indeed play a very prominent role in TGD.

1. The full Lorentz invariance is obtained in TGD only at the level of the entire configuration space ("world of classical worlds", WCW) which is union over sub-configuration spaces associated with what I call causal diamonds (see this). These sub-configuration spaces decompose further into a union of sub-sub-configuration spaces for which a choice of quantization axes of spin reflects itself at the level of generalized geometry of the imbedding space (quantum classical correspondence requires that the choice of quantization axes has imbedding space and space-time correlates) (see this). The construction of the geometry for these sub-worlds of classical worlds reduces to light-cone boundary so that the little group L₃ leaving a given point of light-cone boundary invariant is in a special role in TGD framework.

2. The selection of a preferred light-like momentum direction at light-cone boundary corresponds to the selection of quantization axis for angular momentum playing a key role in TGD view about hierarchy of Planck constants associated with a hierarchy of Jones inclusions implying a breaking of Lorentz invariance induced by the selection of quantization axis. The number theoretic vision about quantum TGD implies a selection of two preferred axes corresponding to time-like and space-like direction corresponding to real and preferred imaginary unit for hyper-octonions (see this). In both cases L₄ emerges naturally.

3. The TGD based identification of Kac-Moody symmetries as local isometries of the imbedding space acting on 3-D light-like orbits of partonic 2-surfaces involves a selection of a preferred light-like direction and thus the selection of L₄.

4. Also the so called massless extremals representing a precisely targeted propagation of patterns of classical gauge fields with light velocity along typically cylindrical tubes without a change in the shape involve L₄. A very general solution ansatz to classical field equations involves a local decomposition of M⁴ to longitudinal and transversal spaces and selection of a light-like direction (see this).

5. The parton model of hadrons assumes a preferred longitudinal direction of momentum and mass squared decomposes naturally to longitudinal and transversal mass squared. Also p-adic mass calculations rely heavily on this picture and thermodynamics mass squared might be regarded as a longitudinal mass squared (see this). In TGD framework right handed covariantly constant neutrino generates a super-symmetry in CP₂ degrees of freedom and it might be better to regard left-handed neutrino mass as a longitudinal mass.

This list justifies my own hunch that Glashow and Cohen might have discovered something very important.

Reader interested in background can consult to the article Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams and the new chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-Matrix".