How to realize Lorentz invariance for on-mass-shell states in M^8-H duality?

M⁸-H duality corresponds physically to a generalization of momentum-position duality due to the replacement of point-like particles with 3-surfaces whose orbits are slightly non-determinic space-time surfaces analogous to Bohr orbits satisfying holography = holomorphy principle at the level of H=M⁴× CP₂. Mathematical M⁸-H duality can be seen as a generalization of the geometric Langlands duality from dimension D=2 to dimension D=4.

Recently, considerable progress (see this) has occurred in the understanding of M⁸-H duality. This has meant in some sense a return to the roots and generalization of the earlier too narrow approach and the emergence of a sound physical interpretation. This has led to an exact solution of the duality in terms of local G₂ invariance.

1. A given Y⁴⊂ M⁸, having interpretation as 8-D momentum space, is determined as roots of an octonion analytic function f(o). The roots of Re(o) or I(o) define 3-D holographic data for Y⁴ which can have quaternionic tangent space or normal space and correspondingly Minkowskian or Euclidean signature with respect to the number theoretic metric defined by Re(o₁o₂).
2. The general solution of the octonionic holography is in terms of the action of local G₂ transformations acting on the simplest solutions, which are pieces of M⁴ identified as quaternions or of its orthogonal complement E⁴. having quaternionic tangent resp. normal space. This picture leads to Feynman diagram type structures in which Minkowskian and Euclidean pieces are glued together at vertices which the conditions Re(f)=0 and Im(f)=0 hold true simultaneously.
3. The 4-D surfaces Y⁴ in M⁸ having interpretation as octonions have interpretation for a representation of dispersion relation, which is however not Lorentz invariant but is invariant under 3-D rotation group. Is it possible to obtain Lorentz invariant dispersion relation in a natural way at some 3-surfaces defining on-mass-shell states and having interpretation in terms of hyperbolic 3-space H³? Can one assign a mass squared spectrum to a given Y⁴? How does this spectrum relate to the mass squared spectrum of the Dirac operator of H=M⁴× CP₂? Does 8-D light-likeness fix this spectrum?

If the tangent spaces of Y⁴ are quaternionic, the condition Re(f) =0 or Im(f) =0 has as a solution the union of ∪_o0S⁶(o₀) of 6-spheres with radius r₇(o₀). The intersection Y³= E³(o₀)∩ ∪_o0S⁶(o₀)= ∪_o0S²(o₀) defines the holographic data. For the 2-spheres S²(o₀), the 3-momentum squared is constant but depends on the energy o₀ via a dispersion relation that is in general not Lorentz invariant.

M⁸-H duality suggests how to obtain Lorentz invariant mass shell conditions E²-p²= m².

1. The modes of the Dirac equation in H (see this and this)are massless in the 8-D sense. This is a natural additional condition also in M⁸ and could define on mass shell states consistent with Lorentz invariance and distinguish them from the other points of Y⁴ having an interpretation as off-mass-shell momenta allowed by Y⁴ as a representation of a dispersion relation.
2. 8-D masslessness corresponds in M⁸ to the condition o₀²-r₇²=0, where r₇² is the counterpart of the CP₂ mass squared as the eigenvalue of the CP₂ spinor Laplacian. The additional condition o₀²-r₇²=0 picks up a discrete set of values (o₀(r₇),r₇). The 4-D mass squared would be m₄²= r₇² and a discrete mass spectrum is predicted for a given f(o) and a given selection a Re(f)=0 or Im(f)=0.
3. An interesting question is whether the eigenvalue spectrum of CP₂ spinor Laplacian is realized at the level of M⁸ as on-mass-shell states.
4. A natural guess would be that the eigenvalue spectrum of CP₂ spinor Laplacian is realized at the level of M⁸ as on-mass-shell states.

The TGD based proposal cite{btart/dirprop,HJdireq} for color confinement producing light states involves tachyonic states. These states would naturally correspond to 4-surfaces Y⁴ with Euclidean signature and bound states would be formed by gluing together the tachyonic and non-tachyonics states to Feynman graph-like structures. Note that the on-mass-shell 2-spheres are in general different from those satisfying the conditions (Re(f), Im(f))=(0,0) proposed to define vertices for the generalized Feynman graphs.

Note that the on mass shell 2-spheres are in general different from those satisfying the conditions (Re(f),Im(f))=(0,0) proposed to define vertices for the generalized Feynman graphs. See the article Does M⁸-H duality reduce to local G₂ symmetry? 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.