^{8}-H duality (see this, this, and this) has remained open since one consider several variants for the duality in M

^{4}⊂ M

^{8}=M

^{4}× E

^{4}degrees of freedom mapped to M

^{4}⊂ M

^{4}× CP

_{2}degrees of freedom.

The model for gravitational hum involves diffraction in the tessellation of H^{3} formed by stars or rather, by their magnetic bodies (see this). Reciprocal lattice is closely related to diffraction and equals to the lattice only in the case of cubic lattice. This is expected to be true also for tessellations and suggests that M^{8} to space-time surfaces of H maps the tessellation of H^{3}⊂ M^{4}⊂ M^{8} to a reciprocal tessellation of H^{3}⊂ M^{4}⊂ H.

The first problem is that the momenta at the M^{8} 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.

- 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 M^{8}-H a duality only too the real momenta of physical states or also the virtual momenta. - For virtual momenta the M
^{8}-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 M^{8}-H duality to it. The square m_{R}^{2}for the real part Re(p) of momentum however varies for the points on the complex mass shell since only the real part Re(m^{2}) of mass squared is constant at the complex mass shell. If 3-momenta are real, one has (Re(p))^{2}= Re(p_{0})^{2}-p_{3}^{2}=m_{R}^{2}and is not constant and in general larger than Re(p^{2})=Re(p_{0})^{2}-Im(p_{0})^{2}-p_{3}^{2}=Re(m^{2}). Should one use m_{R}^{2}or Re(m^{2}) in M^{8}-H duality?Re(m

^{2}) is constant at M^{8}side in accordance with the definition of mass shell. The value of a^{2}= h_{eff}^{2}(m_{R}^{2})/Re(m^{2})^{2}at H side varies and has a width defined by the variation of Im(p_{0}) at the points of the mass shell. m_{R}^{2}is not constant at the M^{8}side. This might relate to the fact that particle masses have a width and would relate Im(p_{0}to a physical observable. The time shell is given by a^{2}= h_{eff}^{2}/m_{R}^{2}and is genuine H^{3}.

- 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 E
^{3}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 M^{8}-H duality. - 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 H
^{3}? 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 possible. - 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. M^{4}⊃ H^{3}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 M^{4}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 x_{3}^{i}∝ ε^{ijk}p_{3,j}× p_{3,k}.Physical intuition suggests that the components of spatial momentum are real for the basis vectors p

_{3,i}so that only the energy has imaginary part. For their discrete Lorentz books by Γ this cannot be the case in M^{8}_{c}. - Mass shell condition p
_{i}^{2}= M^{2}must be replaced with x^{2}_{i}= h_{eff}/M^{2}. The precise identification of M^{2}will be considered below. The image of the real part of the energy p_{0,i}is the time coordinate t^{0}_{i}= h_{eff}Re(p_{0,i})/M^{2}. For the naive option considered earlier, the time shell condition is satisfied if the dual position vectors x_{3}^{i}are of form x_{3}^{i}=h_{eff}[(p^{2}_{i})^{1/2}/M^{2}]e^{i}, where e^{i}is a unit vector in the direction of p_{i}. This option is correct if the momenta p_{i}are orthogonal since in this case the reciprocal unit vectors and vectors co-insider. In the general case, one must replace the unit vector e^{i}with the unit vector associated with the vector X^{i}= ε^{ijk}p_{3,j}p_{3,j}given by e^{i}=X^{i}/((X^{i})^{2})^{1/2}so that the formula for x^{3}_{i}remains otherwise the same. - The conditions have a similar form independently of whether one takes the mass squared parameter M
^{2}to be M^{2}=Re(m^{2}) or M^{2}=m_{R}^{2}. 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_{i}^{2}) =Re(m^{2}). Spatial coordinates in M^{4}must obey similar formula, which implies the length of the image vector is r= h_{eff}×p_{3,i}/M^{2}so that time shell condition t^{2}-r^{2}= h_{eff}^{2}/M^{2}conforms with the Uncertainty Principle. - Which option is correct: M
^{2}=Re(m^{2}) or M^{2}=m_{R}^{2}? For the first option the discretized real mass shell m_{R}^{2}is deformed and might be essential for having a non-trivial number theoretical holography implying by M^{8}-H duality a non-trivial holography at H side. One can however defend the second option by non-trivial holography at H side.

^{8}-H duality is indirect in that it is applied only to the (at least) 3 basic vectors and the action of Γ gives the tessellation in H

^{3}⊂ H. One can also apply the entire Lorentz group to these vectors to obtain the time shell.

The proposed construction assumed that the basis of 3 vectors is essential for the definition of tessellation and that it is possible to assign a set of reciprocal vectors to it in the proposed way involving cross product, which is essentially 3-D notion and relates to quaternions. Is this really the case for all tessellations of H^{3}?

- In Euclidian 3-space E
^{3}only cubic lattice defines a regular tessellation and for the reciprocal lattices is well-defined. In this case, the linear combinations of 3 basic vectors define the lattice. Note that Platonic solids have duals but this duality has nothing to do with the reciprocal lattice. - In hyperbolic 3-space H
^{3}one can have cubic tessellation, 2 icosahedral tessellations, and dodecahedral tessellation as regular tessellations. There is also icosa tetrahedral tessellation involving both tetrahedra, octahedra and icosahedra (see this). Linear combinations of the basic vectors do not exist now. If 3 basic vectors of the tessellation are known, it would be their orbit under the discrete group Γ which defines the tessellation. If Γ is transitive, a single point in principle defines the tessellation as the orbit of Γ. - One can assign to tessellations of H
^{3}what might be called a fundamental tetrahedron as a 4-simplex, which is not a regular tetrahedron in the general case. Could the loci of its vertices with respect to a selected vertex define the fundamental tetrahedron? For a cube in E^{3}this tetrahedron would correspond to a tetrahedron defined by the 3 nearest vertices of a selected vertex of the cube. At least in E^{3}one can select the vertex by requiring that the distances of the neighbouring vertices from the selected vertex are minimal. In this case, the remaining 3 vertices form an orbit under Z^{3}. - Could one reduce the situation from H
^{3}to E^{3}by considering the 3 basic vectors for the projection of the fundamental tetrahedron from H^{3}to t=constant hyperplane E^{3}?The 3 basic vectors would be from the selected vertex defining the origin to neighboring vertices: note that here the submanifold property H^{3}⊂ M^{4}is essential.Could one assign to this triplet a reciprocal in the same way as in the Euclidian case using the cross product induced by the quaternion structure? The notion of dual basis is also behind the bra-ket formalism of quantum mechanics and is based on the notion of vector space and its dual? The following considerations rely on this optimistic assumption.

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.

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