Gravitational hum and 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 model for gravitational hum involves diffraction in the tessellation of H³ 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 M8-H duality mapping momentum 4-surfaces of M⁸ to space-time surfaces of H maps the tessellation of H³⊂ M⁴⊂ M⁸ to a reciprocal tessellation of H³⊂ M⁴⊂ H.

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 possible.
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,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 M⁸_c.

4. Mass shell condition p_i²= M² must be replaced with x²_i= h_eff/M². The precise identification of M² will be considered below. The image of the real part of the energy p_0,i is the time coordinate t⁰_i= h_effRe(p_0,i)/M². For the naive option considered earlier, the time shell condition is satisfied if the dual position vectors x₃ⁱ are of form x₃ⁱ =h_eff [(p²_i)^1/2/M²]eⁱ, where eⁱ 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 eⁱ with the unit vector associated with the vector Xⁱ= ε^ijkp_3,j p_3,j given by eⁱ=Xⁱ/((Xⁱ)²)^1/2 so that the formula for x³_i remains otherwise the same.
5. The conditions have a similar form independently of whether one takes the mass squared parameter 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). 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 t²-r²= h_eff²/M² conforms with the Uncertainty Principle.
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.

Note that the proposed definition of M⁸-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 H³⊂ 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 H³?

1. In Euclidian 3-space E³ 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.
2. In hyperbolic 3-space H³ 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.
3. One can assign to tessellations of H³ 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 E³ this tetrahedron would correspond to a tetrahedron defined by the 3 nearest vertices of a selected vertex of the cube. At least in E³ 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 Z³ .
4. Could one reduce the situation from H³ to E³ by considering the 3 basic vectors for the projection of the fundamental tetrahedron from H³ to t=constant hyperplane E³?The 3 basic vectors would be from the selected vertex defining the origin to neighboring vertices: note that here the submanifold property H³⊂ M⁴ 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.

See the article The TGD view of the recently discovered gravitational hum as gravitational diffraction, the articles Magnetic Bubbles in TGD Universe: Part I and Magnetic Bubbles in TGD Universe: Part II or the chapter Quantum Astrophysics .

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.