The symmetry between gravitational and gauge interactions

The beauty of the proposal is that implies a complete symmetry between gravitational and gauge interactions.

1. Weak interactions and gravitation couple to weak isospin and spin respectively. Color interactions couple to the isometry charges of CP₂ and gravitational interactions coupling to the isometry charges of M⁴. The extreme weakness of the gravitation can be understood as the presence of the CP₂ contribution to the induced metric in the gravitational vertices.

Does color confinement have any counterpart at the level of M⁴? The idea that physical states have vanishing four-momenta does not look attractive.

1. In ZEO, the finite-D space of causal diamonds (CDs) forms (see this) the backbone of WCW and Poincare invariance and Poincare quantum numbers can be assigned with wave functions in this space. For CD, the infinite-D unitary representations of SO(1,3) satisfying appropriate boundary conditions are a highly attractive identification for the counterparts of finite-D unitary representations associated with gauge multiplets. The basic objection against gravitation as SO(1,3) gauge theory would fail.

   One could replace the spinor fields of H with spinor fields restricted to CD with spinor fields for which M⁴ parts sinor nodes as plane waves are replaced with spinor modes in CD labelled by spin and its hyperbolic counterpart assignable to Lorentz boosts with respect to either tip of CD. One could also express these modes as superpositions of the plane wave modes defined in the entire H.

   The analog of color confinement would hold true for particles as unitary representations of SO(1,3) in CD. One could say that SO(1,3) appears as an internal isometry group of an observer’s perceptive field represented by CD and Poincare group as an external symmetry group treating the observer as a physical object.

2. By separation of variables the spinor harmonics in CD factorize phases depending on the mass of the particle determined by CP₂ and spinor harmonic of hyperbolic 3-space H³=SO(1,3)/SO(3). SO(1,3) allows an extremely rich set of representations in the hyperbolic space H³ analogous to spherical harmonics. A given infinite discrete subgroup Γ⊂ SO(1,3) defines a fundamental domain of Γ as a double coset space ΓSO(1,3)/SO(3). This fundamental domain is analogous to a lattice cell of condensed matter lattice defined by periodic boundary conditions. The graphics of Escher give an idea about these structures in the case of H². The products of wave functions defined in Γ⊂ SO(1,3) and of wave functions in Γ define a wave function basis analogous to the space states in condensed matter lattice.
3. TGD allows gravitational quantum coherence in arbitrarily long scales and I have proposed that the tessellations of H³ define the analogs of condensed matter lattices at the level of cosmology and astrophysics (see this). The unitary representations of SO(1,3) would be central for quantum gravitation at the level of gravitationally dark matter. They would closely relate to the unitary representations of the supersymplectic group of δ M⁴₊× CP₂ in M⁴ degrees of freedom and define their continuations to the entire CD.
4. There exists a completely unique tessellation known as icosa tetrahedral tessellation consisting of icosahedrons, tetrahedrons, and octahedrons glued along boundaries together. I have proposed that it gives rise to a universal realization of the genetic code of which biochemical realizations is only a particular example (see this and this). Also this supports a deep connection between biology and quantum gravitation emerging also in classical TGD (see this and this). Also electromagnetic long range classical fields are predicted to be involved with long length scale quantum coherence (see this).

The challenge is to understand the implications of this picture for M⁸-H duality (see this). The discretization of M⁸ identified as octonions O with the Minkowskian norm defined by Re(Im(o²)) is linear M⁸ coordinates natural for octonions. The discretization obtained by the requirement that the coordinates of the points of M⁸ (momenta) are algebraic integers in an algebraic extension of rationals would make sense also in p-adic number fields.

In the Robertson-Walker coordinates for the future light-cone M⁴₊ sliced by H³:s the coordinates define by mass (light-cone proper time in H), hyperbolic angle and spherical angles, the discretizations defined by the spaces ΓSO(1,3)/SO(3) would define a discretization and one can define an infinite hierarchy of discretizations defined by the discrete subgroups of SO(1,3) with matrix elements belonging to an extension of rationals. This number theoretically universal discretization defines a natural alternative for the linear discretization. Maybe the linear resp. non-linear discretization could be assigned to the moduli space of CDs resp. CD.

See the article What gravitons are and could one detect them in the TGD Universe?) and the chapter the chapter About the Relationships Between Electroweak and Strong Interactions and Quantum Gravity in the TGD Universe.