Configuration space gamma matrices as hyperoctonionic
conformal fields having values in HFF?
The fantastic properties of HFFs of type II_{1} inspire the idea that a localized version of Clifford algebra of configuration space might allow to see spacetime, embedding space, and configuration space as emergent structures.
Configuration space gamma matrices act only in vibrational degrees of freedom of 3surface. One must also include center of mass degrees of freedom which appear as zero modes. The natural idea is that the resulting local gamma matrices define a local version of HFF of type II_{1} as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyperoctonions identified as a subspace of complexified octonions. As a matter fact, one can generalize octonions to quantum octonions for which quantum commutativity means restriction to a hyperoctonionic subspace of quantum octonions . Nonassociativity is essential for obtaining something nontrivial: otherwise this algebra reduces to HFF of type II_{1} since matrix algebra as a tensor factor would give an algebra isomorphic with the original one. The octonionic variant of conformal invariance fixes the dependence of local gamma matrix field on the coordinate of HO. The coefficients of Laurent expansion of this field must commute with octonions.
The world of classical worlds has been identified as a union of configuration spaces associated with M^{4}_{�} labeled by points of H or equivalently HO. The choice of quantization axes certainly fixes a point of H (HO) as a point remaining fixed under SO(1,3)×U(2) (SO(1,3)×SO(4)). The condition that hyperquaternionic inverses of M^{4} � HO points exist suggest a restriction of arguments of the npoint function to the interior of M^{4}_{�}.
Associativity condition for the npoint functions forces to restrict the arguments to a hyperquaternionic plane HQ=M^{4} of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative subspace HC of HO. Fixing preferred real and imaginary units means a choice of M^{2}=HC interpreted as a partial choice of quantization axes. This has quite strong implications.
 The hyperquaternionic planes with a fixed choice of M^{2} are labeled by points of CP_{2}. If the condition M^{2} � T^{4} characterizes the tangent planes of all points of X^{4} � HO it is possible to map X^{4} � HO to X^{4} � H so that HOH duality ("number theoretic compactification") emerges. X^{4} � H should correspond to a preferred extremal of Kähler action. The physical interpretation would be as a global fixing of the plane of nonphysical polarizations in M^{8}: it is not quite clear whether this choice of polarization need not have direct counterpart for X^{4} � H. Standard model symmetries emerge naturally. The resulting surface in X^{4} � H would be analogous to a warped plane in E^{3}. This new result suggests rather direct connection with super string models. In super string models one can choose the polarization plane freely and one expects also now that the generalized choice M^{2} � M^{4} � M^{8} of polarization plane can be made freely without losing Poincare invariance with reasonable assumption about zero energy states.
 One would like to fix local tangent planes T^{4} of X^{4} at 3D lightlike surfaces X^{3}_{l} fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T^{3} of X^{3}_{l} to give T^{4}. This might be achieved if t belongs to M^{2} and perhaps corresponds to a lightlike vector in M^{2}.
 Assume that partonic 2surfaces X belong to dM^{4}_{�} � HO defining ends of the causal diamond. This is obviously an additional boundary condition. Hence the points of partonic 2surfaces are associative and can appear as arguments of npoint functions. One thus finds an explanation for the special role of partonic 2surfaces and a reason why for the role of lightcone boundary. Note that only the ends of lightlike 3surfaces need intersect M^{4}_{�} � HO. A stronger condition is that the preimages of lightlike 3surfaces in H belong to M^{4}_{�} � HO.
 Commutativity condition is satisfied if the arguments of the npoint function belong to an intersection X^{2}�M^{2} � HQ and this gives a discrete set of points as intersection of lightlike radial geodesic and X^{2} perhaps identifiable in terms of points in the intersection of number theoretic braids with dH_{�}. One should show that this set of points consists of rational or at most algebraic points. Here the possibility to choose X^{2} to some degree could be essential. As a matter fact, any radial light ray from the tip of lightcone allows commutativity and one can consider the possibility of integrating over npoint functions with arguments at light ray to obtain maximal information. For the preimages of lightlike 3surfaces commutativity would allow onedimensional curves having interpretation as braid strands. These curves would be contained in plane M^{2} and it is not clear whether a unique interpretation as braid strands is possible (how to tell whether the strand crossing another one is infinitesimally above or below it?). The alternative assumption consistent with virtual parton interpretation is that lightlike geodesics of X^{3} are in question.
To sum up, this picture implies HOH duality with a choice of a preferred imaginary unit fixing the plane of nonphysical polarizations globally, standard model symmetries, and number theoretic braids. The introduction of hyperoctonions could be however criticized: could octonions and quaternions be enough after all? Could HOH duality be replaced with OH duality and be interpreted as the analog of Wick rotation? This would mean that quaternionic 4surfaces in E^{8} containing global polarization plane E^{2} in their tangent spaces would be mapped by essentially by the same map to their counterparts in M^{4}×CP_{2},and the time coordinate in E^{8} would be identified as the real coordinate. Also lightcones in E^{8} would make sense as the inverse images of M^{4}_{�}.
For background see the chapter Was von Neumann right after all? . See also the article "Topological Geometrodynamics: an Overall View".
