Wednesday, August 23, 2023

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 ⊂ M8c, where M8c is complexified M8 having interpretation as an analog of complex momentum space and 4-D spacetime surfaces X4 ⊂ H = M4 × CP2. M8c, equivalently E8c, can be regarded as complexified octonions. M8c has a subspace M4c 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 Ik. 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 M8c.

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 M8c

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. M4c 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 M4c ⊂ M8c, 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 Eeff=(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 Eeff. 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(m2/21/2 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 ⊂ M8c 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 M4c → SU(3)c ⊂ G2,c, where G2,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 AR=g-1dg to H is determined by g using M4 coordinates for Y4 . The M4 coordinates pk(M8) having interpretation as momenta are mapped to the coordinates mk of H by the inversion

    I: mk= ℏeff Re(pk/p2) , p2== pkpk ,

    where pk is complex momentum. Re(A)H is obtained by the action of the Jacobian

    dIkl= ∂ pk/∂ ml


    AH,k = dIkl Re(AM8,l) .

    dIkl can be calculated as the inverse of the Jacobian ∂ mk/∂Re(p)l. Note that Im(pk) is expressible in terms of Re(pk). This gives the formula

    For Im(pk)=0, the Jacobian for I reduces to that for mk= ℏeff mk/m2 and one has

    ∂ mk/∂ pl= (ℏeff/m2)(δkl- mkml/m2) .

    This becomes singular for m2=0. The nonvanishing of Im(pk) 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= ℏeff/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 M4c ⊂ M8c 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 M8c and M4 × CP2?

    1. The selection of SU(3) ⊂ G2 for ordinary M8-H duality means that the G2,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 ⊂ M8c 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 G2,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.
    4. 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.

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