About the precise form of M8-H duality

The precise form of M⁸-H duality (see this, this, and this) has remained open since one consider several variants for the duality in M⁴⊂ M⁸=M⁴× E⁴ degrees of freedom mapped to M⁴⊂ M⁴× CP₂ degrees of freedom.

The first problem is that the momenta at the M⁸ side are complex unlike the space-time points at the H side. The basic condition comes from the Uncertainty Principle in semiclassical form but does not complettessellationely fix the duality.

1. If the momenta are real, the simplest option is that the mass shell is mapped to a time shell a=h_eff/m, where a is light-cone proper time. For physical states the momenta are real by Galois confinement and have integer components when the momentum scale is defined by causal diamond (CD). For virtual fermions, the momenta are assumed to be algebraic integers and can be complex. The question is whether one should apply M⁸-H a duality only too the real momenta of physical states or also the virtual momenta.
2. For virtual momenta the M⁸-H duality must be consistent with that for real momenta and the simplest option is that one projects the real part of the virtual momentum and applies M⁸-H duality to it. The square m_R² for the real part Re(p) of momentum however varies for the points on the complex mass shell since only the real part Re(m²) of mass squared is constant at the complex mass shell. If 3-momenta are real, one has (Re(p))²= Re(p₀)² -p₃²=m_R² and is not constant and in general larger than Re(p²)=Re(p₀)²-Im(p₀)² -p₃²=Re(m²). Should one use m_R² or Re(m²) in M⁸-H duality?

Re(m²) is constant at M⁸ side in accordance with the definition of mass shell. The value of a²= h_eff²(m_R²)/Re(m²)² at H side varies and has a width defined by the variation of Im(p₀) at the points of the mass shell. m_R² is not constant at the M⁸ side. This might relate to the fact that particle masses have a width and would relate Im(p₀ to a physical observable. The time shell is given by a²= h_eff²/m_R² and is genuine H³. The model for the gravitational hum (see this) provides another problem, which can serve as an additional guideline.

1. The model was based on gravitational diffraction in the tessellation defined by a discrete subgroup of SL(2,C). This tessellation is a hyperbolic analog of a lattice in E³ with a discrete translation group replaced with a discrete subgroup Γ of the Lorentz group or its covering SL(2,C). The matrix elements of the matrices in Γ should belong to the extension of rationals defined by the polynomial P defining the space-time surface by M⁸-H duality.
2. For ordinary lattices, the reciprocal lattice assigns to a spatial lattice a momentum space lattice, which automatically satisfies the constraint from the Uncertainty Principle. Could the notion of the reciprocal lattice generalize to H³? What is needed are 3 basis vectors (at least) characterizing the position of a fundamental region and having components that must belong to the algebraic extension of rationals considered. The application of Γ would then produce the entire lattice. In this case a linear superposition of lattice vectors is not satisfied.
3. The (at least) 3 basic 3-vectors p_3,i need not be orthogonal or have the same length. They should have components, which are algebraic integers in the extension of rationals defined by P. M⁴⊃ H³ is a subspace of complexified quaternions with the space-like part of momentum vector, which is imaginary with respect to commuting imaginary unit i to transform the algebraic scalar product (no conjugation with respect to i). The ordinary cross product appearing in the definition of the reciprocal lattice appears in the quaternionic product. This suggests that the (at least) 3 reciprocal vectors p_3,i as M⁴ projections of four-momentum vectors are proportional to the cross products of the basis vectors apart from a normalization factor determined by the condition that the light-cone proper time is proportional to the inverse of mass. One would have x₃ⁱ∝ ε^ijkp_3,j× p_3,kk.
4. The conditions have a similar form independently of whether one takes a mass squared parameter, call it M² to be M²=Re(m²) or M²=m_R². The time components of the momentum vectors p_i associated with p_3,i, which are assumed to be real, are determined by the mass shell condition Re(p_0,i)²-Im(p_0,i)²- Re(p_i²) =Re(m²). One must use the real projection of the time component. The image of the energy is time coordinate t⁰_i= h_effp₀ⁱ/M². Spatial coordinates in M⁴ must obey similar formula, which implies the length of the image vector is r= h_eff×p_3,i/M² so that time shell condition reas t²-r²= h_eff²/M² and conforms with the Uncertainty Principle.
5. If one can satisfy these conditions for the 3 (at least) reciprocal vectors with suitable normalization factors k_i, the action by Γ on this triplet guarantees the on mass shell condition for the entire tessellation. The most general ansatz for the lattice vectors at H side is x₃ⁱ= h_eff k(i) ε^ijkp_3,j× p_3,k/M². The conditions give k(1)= p₁/p₂p₃, k(2)= p₂/p₃p₁, and k(3)= p₃/p₁p₂. Here one must be very careful with what one means with p_i. Number theoretic purely algebraic norm proportional to the commuting imaginary unit i so that x₃ⁱ is proportional to i so that the number theoretic scale product reduces at H side to the scalar product defined by Minkowski metric.
6. Which option is correct: M²=Re(m²) or M²=m_R²? For the first option the discretized real mass shell m_R² is deformed and might be essential for having a non-trivial number theoretical holography implying by M⁸-H duality a non-trivial holography at H side. One can however defend the second option by non-trivial holography at H side.

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.