In yesterday's posting "Precise form of ew-color duality" an important step in the understanding of HO-H duality was summarized. The short and long p-adic length scales L_k and L_p, p=about 2^k correspond to short and long distance limits in HO-H duality. In particular, the duality was formulated in terms of symmetries of the world of the classical worlds as a kind of super-symmetry permuting configuration space degrees of freedom with corresponding spin degrees of freedom.
The description of duality at the configuration space level can be applied to gain a view about color confinement and its dual for electro-weak interactions at short distance limit as a process in which configuration space degrees of freedom begin to dominate, field theory picture becomes obsolete, and it is better to go the dual description. The correct prediction is that SO(4) should appear as dynamical symmetry group of low energy hadron physics.
There are two basic types of vacuum extremals: CP_2 type extremals representing elementary particles, and vacuum extremals having CP_2 projection which is at most 2-dimensional Lagrange manifold representing for instance hadrons. It is not surprising that HO-H duality can be interpreted in terms of these vacuum extremals and they provide a more precise view about what happens at the limits when either CH or CHO degrees of freedom begin to dominate over space-time degrees of freedom describable ordinary quantum field theory.

### 1. Short distance limit

Consider first the short distance limit at which electro-weak confinement is expected and HO picture becomes more appropriate.
a) Ew-color duality would suggests that at the limit of short distances something analogous to color confinement occurs for electro-weak interactions. Also the large value of U(1) coupling supports this expectation. The vacuum property of CP_2 type extremals means that induced spinor fields become vacuum spinor fields with an identically vanishing Dirac action. Therefore these spinor fields effectively disappear at space-time level for the maxima of Kähler function, and contribute only via quantum fluctuations, which correspond to configuration space dynamics. Color partial waves are left as a genuine configuration space degree of freedom and the expectation is that only the lowest color partial waves corresponding to singlet and triplet remain and become spin like degrees of freedom analogous to QCD color in HO picture.
b) Duality suggests SO(4) confinement in E^4 degrees of freedom at this limit. The nearly vacuum property should allow very large fluctuations of the ordinary fermion and anti-fermion numbers at the limit when the fermions become pure vacuons for which creation and annihilation operators reduce to anti-commuting Grassmann numbers. In HO picture this would mean that high SO(4) partial waves in E^4 are possible for composites although net ew quantum numbers vanish. Hence electro-weak spins become analogous to classical angular momentum at this limit.

### 2. Long distance limit

Consider next color confinement at the long length scale limit as a dual of this picture.
a) In the case of color interactions very high color partial waves for quarks and gluons appear at the confinement limit. For instance, vacuum extremals representable as maps M^4--> CP_2 identifiable as hadronic space-time sheets correspond to color confinement limit. Strong fluctuations due to high color partial waves in CH appear, and corresponds in CHO$picture to the presence of high colored hyper-octonionic fermion and anti-fermion numbers. Since configuration space degrees of freedom begin to dominate, color confinement limit transcends the descriptive power of QCD just as high energy limit transcends the descriptive power of standard model of electro-weak interactions.
b) The success of SO(4) sigma model in the description of low lying hadrons could directly relate to the fact that this group labels also the E^4 Hamiltonians in HO picture. SO(4) quantum numbers can be identified as right and left handed strong isospin for which no natural interpretation exists in H picture. Family replication phenomenon is described in the same manner in both cases so that quantum numbers like strangeness and charm are not fundamental. Indeed, p-adic mass calculations allowing fractally scaled up versions of various quarks allow to replace Gell-Mann mass formula with highly successful predictions for

hadron masses.
c) Ordinary fermion numbers do not fluctuate at the color confinement limit. That this does not occur must relate to the facts that modified Dirac action relates by super-symmetry to Kähler action and the variations of Kähler action vanish up to third order around canonically embedded M^4 whereas for the CP_2 type extremals the situation is completely different. The absence of fluctuations corresponds to the possibility to describe low energy hadrons using simple valence quark model without the inclusion of the quark and gluon sea. At the asymptotic freedom limit sea becomes increasingly important.
d) Baryons should be analogous to color partial waves of quarks, and just as CP_2 spinors allow at CH level color triplet partial waves also hyper-octonionic fermions should allow at CHO level SO(4) partial waves transforming as doublets under SU(2)_L or SU(2)_R.
e) Ordinary fermion numbers do not fluctuate at the color confinement limit. That this does not occur must relate to the facts that modified Dirac action relates by super-symmetry to Kähler action and the variations of Kähler action vanish up to third order around canonically embedded M^4 whereas for the CP_2 type extremals the situation is completely different. The absence of fluctuations in ew spin degrees of freedom suggests the possibility of describing low energy hadrons using simple valence quark model without quark color with quark and gluon sea modelling the presence hyper-octonionic quark pairs. The problems due to statistics might be resolved by anyonic statistic possible for 2-D partonic surfaces. At the asymptotic freedom limit hyper-octonionic sea becomes less and less important.

### 3. Proton spin crisis as a signature of
hyper-octonionic quarks

Hyper-octonionic quarks carry neither ordinary nor electro-weak spin since these quantum numbers correspond to orbital quantum numbers in HO. Hence in the ideal colored quark description the contribution of quarks to both spin and electro-weak spin of proton should vanish whereas in H quark description quarks should give proton spin. Obviously, these descriptions correspond to colored current quark description and to a static color singlet quark descriptions possible for anyonic statistics. This prediction would sound crazy unless the essence of proton spin crisis were just the finding that the contribution of quarks to proton spin is small.
Spin-statistics paradox is avoided if configuration space degrees of freedom are taken into account. Quantum-classical correspondence, if taken at extreme, would suggest that configuration space degrees of freedom might have some kind of space-time correlate. The 2-dimensionality of stringy and partonic surfaces suggests that anyons might provide this correlate. In HO picture spin-statistics paradox at space-time level would be avoided by the 2-dimensionality of partonic surfaces allowing to have braid representations of the rotation group and colored quarks can have half-odd integer valued anyonic spin and electro-weak spin.
A possible physical mechanism transforming H quarks without color spin but with ew- and ordinary spin to HO quarks having only color spin is following. Anyonic and ordinary contributions to ew- and ordinary spin of H quark cancel each other and color spin is generated anyonically. In TGD framework anyons are associated with punctures assignable to the thin flux threads connecting partonic 2-surfaces and these punctures appear always as pairs with the ends of thread carrying opposite anyonic quantum numbers. OH fermions would correspond to fermion plus the second end of the anyon thread.
In H picture the approach to confinement means large fluctuations also in SO(3) degrees of freedom and the emergence of Regge trajectories. In HO picture the angular momentum of hadron would be due the angular momentum of a large number of colored quark pairs.
For more details see the chapter

TGD as a Generalized Number Theory: Quaternions, Octonions, and their Hyper Counterparts.
Matti Pitkänen

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