During the last months dramatic progress in the understanding of TGD in terms of what I call HO-H duality, has occurred. HO denotes 8-D space of hyper-octonions obtained from octonions by multiplying the imaginary units with a commuting square root of -1. H denotes the imbedding space H=M^4xCP_2 obtained by replacing each point of Minkowski space with CP_2. The strongest form of HO-H duality is that TGD can be described either in H picture or in HO picture and one can speak about number theoretical compactification (no actual compactification of course occurs).
To get a glimpse about what is involved for the earlier postings and the chapter
Physics as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts.
HO-H duality seems to physically correspond to electroweak-color duality and I have tried to understand what is really involved. After reading emails and visiting at blog groups to find that nothing interesting had appeared (I do not find the usual battles between alpha males interesting) I suddenly got the feeling that I might be able to write something new about this duality. My gut feeling was correct.
The e(lectro)w(eak)-color duality associated with H-HO duality reflects the fact that in both pictures the dynamics of single space-time surface can provide only a partial description of quantum dynamics, and that configuration space level(configuration space is the world of classical worlds, space-time surfaces) is needed in order to code all quantum numbers and all interactions. The situation cries for a more precise formulation for the ew--color duality. The sought for formulation can be expressed as a single concise statement.
In H (HO) picture ew-spin (color) degrees freedom correspond to spin like quantum numbers and color (ew) degrees of freedom to classical conserved quantum numbers.
Consider now what this statement really means.
1. Spin-like quantum numbers and conserved charges in H-picture
In H picture ew quantum numbers and spin are manifestly present whereas color quantum numbers and interactions emerge as spin like quantum numbers only at configuration space level as does also four-momentum via Kac-Moody representations.
Classical(!) color and Poincare charges are well defined also in H picture. There is also a non-trivial interaction between color and ew degrees of freedom since color transformations are accompanied by ew rotations in accordance with the fact that U(2)_{ew} can be mapped to a subgroup of SU(3) via the coset construction.
2. Spin-like quantum numbers and conserved charges in HO-picture
Hyper-octonion HO spinors decompose to representations of color group whereas H spinors decompose to the representations of ew and Lorentz group. Hence for HO picture color is manifestly present as spin degrees of freedom but ew spin and spin are absent.
By ew-color duality at space-time level ew and spin charges should somehow emerge also in HO picture as
classical conserved quantities. The first observation is that the automorphism group G_2 corresponds to 2 conserved commuting charges. Translating H picture directly to HO level this would mean that the classical conserved charges associated with WZW + Dirac action have identification as ew charges. Also now a non-trivial relation between electro-weak and color quantum numbers is involved.
There are also symmetries not respecting hyper-octonion real-analyticity and analogous to those affecting the moduli characterizing complex structure. SO(7) leaves the spatial part of the hyper-octonionic norm invariant and this extends the number of conserved charges to 3 bringing in spin. The full isometry group of the hyper-octonionic norm is SO(7,1) so that also Lorentz boost would be included to the Cartan algebra.
Also translations are symmetries of HO picture since the shift of the origin gives rise to a new solution family which is however not hyper-octonion analytic in the original coordinate system. Four-momentum should emerge at quantum level via Kac-Moody type realization also now.
The correspondences M^4xSO(7,1)<--> PxSU(3) and SU(3)<--> U(2)_{ew} code for the ew--color duality. Interestingly, the four M^4 coordinates depending on X^4 define a local Kac-Moody algebra identifiable in terms of the Cartan algebra of SO(7,1) and extendable by k=1 vertex operator construction to a representation of SO(7,1) Kac-Moody algebra. On the other hand, the Euclidian stringy degrees of freedom in M^4 give rise to SU(3) Kac Moody algebra and to SU(3)/U(2) WZW model serving as a candidate for a model of ew interactions. A very tight web of correspondences between various symmetries is involved.
3. Ew-color duality and and parton-string duality
G_2/SU(3) coset WZW theory would correspond naturally to QCD and geometrically to the identification of a preferred hyper-octonion imaginary unit defining a foliation of HO by string like 2-surfaces. SU(3)/U(2) corresponds naturally to ew gauge theory. These WZW models are obviously mutually exclusive if defined at time-like stringy surfaces Y^2. One can however consider defining SU(3)/U(2) WZW type model at the partonic 2-surfaces X^2 in H picture. These two dual space-time pictures might allow a rather satisfactory quantum-classical correspondence.
That information from entire light-like causal determinants is needed, conforms with the quantum gravitational holography, and the objection that the theory is equivalent to a string model can be avoided. The duality corresponds two kinds of conformal symmetries: hyper-octonionic conformal invariance and the conformal invariance associated with the light-like causal determinants.
4. Ew-color duality and duality of long and short p-adic length scales
The
first formulation for ew-color duality was in terms of p-adic length scale hypothesis selecting the primes p\simeq 2^k, k positive integer, preferably prime or power of prime, as preferred p-adic length scales. L_p\propto \sqrt{p} corresponds to the p-adic length scale defining the size of the space-time sheet at which elementary particle represented as CP_2 type extremal is topologically condensed and is of order Compton length. L_k\propto \sqrt{k} represents the p-adic length scale of the wormhole contacts associated with the CP_2 type extremal and CP_2 size is the natural length unit now.
The proposal was that QCD type description based on quarks and gluons corresponds to a description in the ultra-short length scale L_k and the description in terms of hadrons possessing only electro-weak quantum numbers and spin corresponds to the hadronic length scale L_p. The order of magnitude for alpha_s is predicted correctly directly from the fact that it is proportional to alpha_K and as U(1) coupling increases towards short p-adic length scales in a manner predicted by heuristic arguments assuming that gravitational constant does not run appreciably as a function of p-adic length scale.
Indeed, HO picture describing color as a spin like quantum number is more appropriate near CP_2 length scale whereas H picture describing color classically (as in color flux tube models) is more appropriate in hadronic length scales. Perturbative--non-perturbative QCD duality would thus also correspond to HO-H duality. Strictly speaking, non-perturbative QCD would be a meaningless notion.
5. H-HO duality in the world of classical worlds
An interesting challenge is to translate H-HO duality to the level of configuration space geometry and spinor structure.
a) In H picture CH Hamiltonians correspond to Hamiltonians of \delta M^4_+xCP_2 in representations of SO(3)xSU(3) whereas spin and electro-weak spin correspond to spin degrees of freedom associated with complexified gamma matrices acting as super-generators.
b) In HO picture CH is replaced with what might be called CHO. The guess is that also CHO allows Kähler and symplectic structures. CHO Hamiltonians cannot correspond to Hamiltonians of E^7 (imaginary hyper-octonions) since E^7 has wrong dimension. 7-D light-cone is in turn metrically a 6-sphere. If S^6 does not allow complex structure as Chern's last theorem claims, it does not allow Kähler structure neither. Situation changes if onen considers \delta M^4_+xE^4 metrically equivalent to S^2xE^4, which certainly allows Kähler and symplectic structures. This choice is of course perfectly natural and consistent with the view that number theoretical compactification takes effectively E^4 to CP_2 by attaching to it a 2-sphere at infinity. SO(3)xSO(4) would assign to Hamiltonians spin and ew quantum numbers. Color quantum numbers would correspond to spin degrees of freedom associated with CHO gamma matrices acting also as super generators. H-HO duality could be also interpreted as a super-symmetry permuting bosonic and fermionic degrees of freedom at the level of configuration space. With a technical skills of a super-string dualist it would be probably easy to derive strong testable predictions from this duality.
For more details see
Physics as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts.
Matti Pitkanen
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