An explicit formula for M8-H duality

M⁸-H duality is a generalization of momentum-position duality relating the number theoretic and geometric views of physics in TGD and, despite that it still involves poorly understood aspects, it has become a fundamental building block of TGD. One has 4-D surfaces Y⁴ ⊂ M⁸_c, where M⁸_c is complexified M⁸ having interpretation as an analog of complex momentum space and 4-D spacetime surfaces X⁴ ⊂ H = M⁴ × CP₂. M⁸_c, equivalently E⁸_c, can be regarded as complexified octonions. M⁸_c has a subspace M⁴_c containing M⁴.

Comment: One should be very cautious with the meaning of "complex". Complexified octonions involve a complex imaginary unit i commuting with the octonionic imaginary units I_k. i is assumed to also appear as an imaginary unit also in complex algebraic numbers defined by the roots of polynomials P defining holographic data in M⁸_c.

In the following M⁸-H duality and its twistor lift are discussed and an explicit formula for the dualities are deduced. Also possible variants of the duality are discussed.

1. Holography in H

X⁴ ⊂ H satisfies holography and is analogous to the Bohr orbit of a particle identified as a 3-surface. The proposal is that holography reduces to a 4-D generalization of holomorphy so that X⁴ is a simultaneous zero of two functions of complex CP₂ coordinates and of what I have called Hamilton-Jacobi coordinates of M⁴ with a generalized K hler structure.

The simplest choice of the Hamilton-Jacobi coordinates is defined by the decomposition M⁴ = M² × E², where M² is endowed with hypercomplex structure defined by light-like coordinates (u,v), which are analogous to z and \overline{z}. Any analytic map u → f(u) defines a new set of light-like coordinates and corresponds to a solution of the massless d'Alembert equation in M². E² has some complex coordinates with an imaginary unit defined by i.

The conjecture is that also more general Hamilton-Jacobi structures for which the tangent space decomposition is local are possible. Therefore, one would have M⁴ = M²(x) × E²(x). These would correspond to non-equivalent complex and K hler structures of M⁴ analogous to those possessed by 2-D Riemann surfaces and parametrized by the moduli space.

2. Number theoretic holography in M⁸_c

Y⁴ satisfies number theoretic holography defining dynamics, which should reduce to associativity in some sense. The Euclidean complexified normal space N⁴(y) at a given point y of Y⁴ is required to be associative, i.e. quaternionic. Besides this, N⁴(i) contains a preferred complex Euclidean 2-D subspace Y²(y). Also, the spaces Y²(x) define an integrable distribution. I have assumed that Y²(x) can depend on the point y of Y⁴.

These assumptions imply that the normal space N(y) of Y⁴ can be parameterized by a point of CP₂ = SU(3)/U(2). This distribution is always integrable, unlike quaternionic tangent space distributions. M⁸-H duality assigns to the normal space N(y) a point of CP₂. M⁴_c point y is mapped to a point x in M⁴ ⊂ M⁴ × CP₂ defined by the real part of its inversion (conformal transformation): this formula involves the effective Planck constant for dimensional reasons.

The 3-D holographic data, which partially fixes 4-surfaces Y⁴, is partially determined by a polynomial P with real integer coefficients smaller than the degree of P. The roots define mass squared values which are in general complex algebraic numbers and define complex analogs of mass shells in M⁴_c ⊂ M⁸_c, which are analogs of hyperbolic spaces H³. The 3-surfaces at these mass shells define 3-D holographic data continued to a surface Y⁴ by requiring that the normal space of Y⁴ is associative, i.e. quaternionic. These 3-surfaces are not completely fixed, but an interesting conjecture is that they correspond to fundamental domains of tessellations of H³.

What does the complexity of the mass shells mean? The simplest interpretation is that the space-like M⁴ coordinates (3-momentum components) are real, whereas the time-like coordinate (energy) is complex and determined by the mass shell condition. One would have Re²(E) - Im(E)² - p² = Re(m²) and 2Re(E)Im(E) = Im(m²). The condition for the real parts gives H³ when E_eff=(Re²(E) - Im(E)²)^1/2 is taken as energy coordinate. The second condition allows solving Im(E) in terms of Re(E), so that the first condition reduces to a modifed equation of the mass shell when (Re(E)² - Im(E)²)^1/2, expressed in terms of Re(E), is used in E_eff. Is this deformation of H³ in the imaginary time direction equivalent to a region of the hyperbolic 3-space H³?

One can look at the formula in more detail. Mass shell condition gives Re²(E)-Im(E)²- p²= Re(m²) and 2Re(E)Im(E)= Im(m²). The condition for the real parts gives H^3, when [Re²(E)-Im(E)²]^1/2 is taken as an effective energy. The second condition allows to solve Im(E) in terms of Re(E) so that the first condition reduces to a dispersion relation for Re(E)².

Re(E)² = (1/2)×(Re(m²)-Im(m²) +p²)(1 +/- [1+2Im(m²)²/(Re(m²)-Im(m²)+p²)²]^1/2]

Only the positive root gives a non-tachyonic result for Re(m²)-Im(m²)>0. For real roots with Im(m²)=0 and at the high momentum limit the formula coincides with the standard formula. For Re(m²)= Im(m²) one obtains Re(E)²→ Im(m²/2^1/2 at the low momentum limit p²→ 0. Energy does not depend on momentum at all: the situation resembles that for plasma waves.

3. Can one find an explicit formula for M⁸-H duality?

The dream is an explicit formula for the M⁸-H duality mapping Y⁴ ⊂ M⁸_c to X⁴ ⊂ H. This formula should be consistent with the assumption that the generalized holomorphy holds true for X⁴.

The following proposal is a more detailed variant of the earlier proposal for which Y⁴ is determined by a map g of M⁴_c → SU(3)_c ⊂ G_2,c, where G_2,c is the complexified automorphism group of octonions and SU(3)_c is interpreted as a complexified color group.

1. This map defines a trivial SU(3)_c gauge field. The real part of g however defines a non-trivial real color gauge field due to the non-linearity of the non-abelian gauge field with respect to the gauge potential. The quadratic terms involving the imaginary part of the gauge potential give an additional condition to the real part in the complex situation and cancel it. If only the real part of g contributes, this contribution would be absent, and the gauge field is non-vanishing.

2. A physically motivated proposal is that the real parts of SU(3)_c gauge potential and color gauge field can be lifted to H and the lifts are equal to the classical gauge potentials and color gauge field proposed in H. Color gauge potentials in H are proportional to the isometry generators of the color gauge field and the components of the color gauge field are proportional to the products of color Hamiltonians with the induced K\"ahler form.
3. The color gauge field Re(G) obeys the formula Re(G)= dRe(A) +[Re(A),Re(A)]=[Re(A),Re(A)] and does not vanish since the contribution of [Im(A),Im(A)] cancelling the real part is absent. The lift of A_R=g^-1dg to H is determined by g using M⁴ coordinates for Y⁴ . The M⁴ coordinates p^k(M⁸) having interpretation as momenta are mapped to the coordinates m^k of H by the inversion

   I: m^k= ℏ_eff Re(p^k/p²) , p²== p^kp_k ,

   where p^k is complex momentum. Re(A)_H is obtained by the action of the Jacobian

   dI^k_l= ∂ p^k/∂ m^l

   as

   A_H,k = dI_k^l Re(A_M⁸,l) .

   dI^k_l can be calculated as the inverse of the Jacobian ∂ m^k/∂Re(p)^l. Note that Im(p^k) is expressible in terms of Re(p^k). This gives the formula

   For Im(p^k)=0, the Jacobian for I reduces to that for m^k= ℏ_eff m^k/m² and one has

   ∂ m^k/∂ p^l= (ℏ_eff/m²)(δ^k_l- m^km_l/m²) .

   This becomes singular for m²=0. The nonvanishing of Im(p^k) however saves from the singularity.

4. The M⁸-H duality obeys a different formula at the light-cone boundaries associated with the causal diamond: now one has p⁰= ℏ_eff/m⁰. This formula should be applied for m²=0 if this case is encountered. Note that number theoretic evolution for masses and classical color gauge fields is directly coded by the mass squared values and holography.

   How could the automorphism g(x) ⊂ SU(3) ⊂ G₂ give rise to M⁸-H duality?

   1. The interpretation is that g(y) at a given point y of Y⁴ relates the normal space at y to a fixed quaternionic/associative normal space at point y₀, which corresponds to being fixed by some subgroup U(2)₀ ⊂ SU(3). The automorphism property of g guarantees that the normal space is quaternionic/associative at y. This simplifies the construction dramatically.
   2. The quaternionic normal sub-space (which has Euclidean signature) contains a complex sub-space corresponding to a point of the sphere S² = SO(3)/O(2), where SO(3) is the quaternionic automorphism group. The interpretation could be in terms of a selection of spin quantization axes. The local choice of the preferred complex plane would not be unique and is analogous to the possibility of having non-trivial Hamilton Jacobi structures in M⁴ characterized by the choice of M²(x) and equivalently its normal subspace E²(x).
   3. The real part Re(g(y)) defines a point of SU(3), and the bundle projection SU(3) → CP₂ in turn defines a point of CP₂ = SU(3)/U(2). Hence one can assign to g a point of CP₂ as M⁸-H duality requires and deduce an explicit formula for the point. This means a realization of the dream.
   4. The construction requires a fixing of a quaternionic normal space N₀ at y₀ containing a preferred complex subspace at a single point of Y⁴ plus a selection of the function g. If M⁴ coordinates are possible for Y⁴, the first guess is that g as a function of complexified M⁴ coordinates obeys generalized holomorphy with respect to complexified M⁴ coordinates in the same sense and in the case of X⁴. This might guarantee that the M⁸-H image of Y⁴ satisfies the generalized holomorphy.
   5. Also, space-time surfaces X⁴ with M⁴ projection having a dimension smaller than 4 are allowed. I have proposed that they might correspond to singular cases for the above formula: a kind of blow-up would be involved. One can also consider a more general definition of Y⁴ allowing it to have a M⁴ projection with dimension smaller than 4 (say cosmic strings). Could one have implicit equations for the surface Y⁴ in terms of the complex coordinates of SU(3)_c and M⁴? Could this give, for instance, cosmic strings with a 2-D M⁴ projection and CP₂ type extremals with 4-D CP₂ projection and 1-D light-like M⁴ projection?

   4. What could the number theoretic holography mean physically?

   What could be the physical meaning of the number theoretic holography? The condition that has been assumed is that the CP₂ coordinates at the mass shells of M⁴_c ⊂ M⁸_c mapped to mass shells H³ of M⁴ ⊂ M⁴ × CP₂ are constant at the H³. This is true if the g(y) defines the same CP₂ point for a given component X³_i of the 3-surface at a given mass shell. g is therefore fixed apart from a local U(2) transformation leaving the CP₂ point invariant. A stronger condition would be that the CP₂ point is the same for each component of X³_i and even at each mass shell, but this condition seems to be unnecessarily strong.

   Comment: One can criticize this condition as too strong, and one can consider giving up this condition. The motivation for this condition is that the number of algebraic points at the 3-surfaces associated with H³ explodes since the coordinates associated with normal directions vanish. Kind of cognitive explosion would be in question.

   SU(3) corresponds to a subgroup of G₂ and one can wonder what the fixing of this subgroup could mean physically. G₂ is 14-D, and the coset space G₂/SU(3) is 6-D, and a good guess is that it is just the 6-D twistor space SU(3)/U(1) × U(1) of CP₂: at least the isometries are the same. The fixing of the SU(3) subgroup means fixing of a CP₂ twistor. Physically, this means the fixing of the quantization axis of color isospin and hypercharge.

   5. Twistor lift of the holography

   What is interesting is that by replacing SU(3) with G₂, one obtains an explicit formula from the generalization of M⁸-H duality to that of the twistorial lift of TGD!

   One can also consider a twistorial generalization of the above proposal for the number theoretic holography by allowing local G₂ automorphisms interpreted as local choices of the color quantization axis. G₂ elements would be fixed apart from a local SU(3) transformation at the components of 3-surfaces at mass shells. The choice of the color quantization axes for a connected 3-surface at a given mass shell would be the same everywhere. This choice is indeed very natural physically since a 3-surface corresponds to a particle.

   Is this proposal consistent with the boundary condition of the number theoretical holography in the case of 4-surfaces in M⁸_c and M⁴ × CP₂?

   1. The selection of SU(3) ⊂ G₂ for ordinary M⁸-H duality means that the G_2,c gauge field vanishes everywhere, and the choice of color quantization axis is the same at all points of the 4-surface. The fixing of the CP₂ point to be constant at H³ implies that the color gauge field at H³ ⊂ M⁸_c and its image H³ ⊂ H vanish. One would have color confinement at the mass shells H³_i, where the observations are made. Is this condition too strong?
   2. The constancy of the G₂ element at mass shells makes sense physically and means a fixed color quantization axis. The selection of a fixed SU(3) ⊂ G₂ for the entire space-time surface is in conflict with the non-constancy of the G₂ element unless the G₂ element differs at different points of the 4-surface only by a multiplication of a local SU(3)₀ element, which is a local SU(3) transformation. This kind of variation of the G₂ element would mean a fixed color group but a varying choice of the color quantization axis.
   3. Could one consider the possibility that the local G_2,c element is free and defines the twistor lift of M⁸-H duality as something more fundamental than the ordinary M⁸-H duality based on SU(3)_c? This duality would make sense only at the mass shells, so that only the spaces H³ × CP₂ assignable to mass shells would make sense physically? In the interior, CP₂ would be replaced with the twistor space SU(3)/U(1) × U(1). Color gauge fields would be non-vanishing at the mass shells, but outside the mass shells, one would have G₂ gauge fields. This does not look like an attractive option physically.

   There is also  a physical objection against the G₂  option. The 14-D Lie algebra representation of G₂ acts on the imaginary octonions which decompose with respect to the color group to 1⊕ 3⊕ 3^*. The automorphism property requires that 1 can be transformed to 3  or 3^* to themselves: this requires that the decomposition contains  3⊕ 3^*. Furthermore, it must be possible to transform 3 and 3^*, which requires the presence of 8. This leaves only the decomposition 8⊕ 3⊕ 3^*. G₂ gluons would both color octet and triplets. In the TDG framework the only conceivable interpretation would be in terms of ordinary gluons and leptoquark-like gluons. This does not fit with the basic vision   of TGD.

   The choice of a twistor as a selection of quantization axes should make sense also in the M⁴ degrees of freedom. M⁴ twistor corresponds to a choice of a light-like direction at a given point of M⁴. The spatial component of the light-like vector fixes the spin quantization axis. Its choice together with the light-likeness fixes the time direction and therefore the rest system and energy quantization axis. The light-like vector also fixes the choice of M² and of E² as its orthogonal complement. Therefore, the fixing of M⁴ twistor as a point of SU(4)/SU(3) × U(1) corresponds to a choice of the spin quantization axis and the time-like axis defining the rest system in which the energy is measured. This choice would naturally correspond to the Hamilton-Jacobi structure fixing the decompositions M²(x) × E²(x). At a given mass shell, the choice of the quantization axis would be constant for a given X³_i.

   See the article New findings related to the number theoretical view of TGD 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.