Configuration space gamma matrices as hyper-octonionic conformal fields with values in HFF

The fantastic properties of HFFs of type II₁ inspire the idea that a localized version of Clifford algebra of configuration space might allow to see space-time, embedding space, and configuration space as emergent structures.

Configuration space gamma matrices act only in vibrational degrees of freedom of 3-surface. 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 II₁ as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyper-octonions 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 hyper-octonionic subspace of quantum octonions . Non-associativity is essential for obtaining something non-trivial: otherwise this algebra reduces to HFF of type II₁ 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 M⁴_± 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 hyper-quaternionic inverses of M⁴ Ì HO points exist suggest a restriction of arguments of the n-point function to the interior of M⁴_±.

Associativity condition for the n-point functions forces to restrict the arguments to a hyper-quaternionic plane HQ=M⁴ of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative sub-space HC of HO. Fixing preferred real and imaginary units means a choice of M²=HC interpreted as a partial choice of quantization axes. This has quite strong implications.

1. The hyper-quaternionic planes with a fixed choice of M² are labeled by points of CP₂. If the condition M² Ì T⁴ characterizes the tangent planes of all points of X⁴ Ì HO it is possible to map X⁴ Ì HO to X⁴ Ì H so that HO-H duality ("number theoretic compactification") emerges. X⁴ Ì H should correspond to a preferred extremal of Kähler action. The physical interpretation would be as a global fixing of the plane of non-physical polarizations in M⁸: it is not quite clear whether this choice of polarization need not have direct counterpart for X⁴ Ì H. Standard model symmetries emerge naturally. The resulting surface in X⁴ Ì H would be analogous to a warped plane in E³. 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 M² Ì M⁴ Ì M⁸ of polarization plane can be made freely without losing Poincare invariance with reasonable assumption about zero energy states.
2. One would like to fix local tangent planes T⁴ of X⁴ at 3-D light-like surfaces X³_l fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T³ of X³_l to give T⁴. This might be achieved if t belongs to M² and perhaps corresponds to a light-like vector in M².
3. Assume that partonic 2-surfaces X belong to dM⁴_± Ì HO defining ends of the causal diamond. This is obviously an additional boundary condition. Hence the points of partonic 2-surfaces are associative and can appear as arguments of n-point functions. One thus finds an explanation for the special role of partonic 2-surfaces and a reason why for the role of light-cone boundary. Note that only the ends of lightlike 3-surfaces need intersect M⁴_± Ì HO. A stronger condition is that the pre-images of light-like 3-surfaces in H belong to M⁴_± Ì subset HO.
4. Commutativity condition is satisfied if the arguments of the n-point function belong to an intersection X²ÇM² Ì HQ and this gives a discrete set of points as intersection of light-like radial geodesic and X² 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 X² to some degree could be essential. For the pre-images of light-like 3-surfaces commutativity would allow one-dimensional curves having interpretation as braid strands. These curves would be contained in plane M² 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 light-like geodesics of X³ are in question.

To sum up, this picture implies HO-H duality with a choice of a preferred imaginary unit fixing the plane of non-physical polarizations globally, standard model symmetries, and number theoretic braids. The introduction of hyper-octonions could be however criticized: could octonions and quaternions be enough after all? Could HO-H duality be replaced with O-H duality and be interpreted as the analog of Wick rotation? This would mean that quaternionic 4-surfaces in E⁸ containing global polarization plane E² in their tangent spaces would be mapped by essentially by the same map to their counterparts in M⁴×CP₂,and the time coordinate in E⁸ would be identified as the real coordinate. Also light-cones in E⁸ would make sense as the inverse images of M⁴_±.

For background see the chapter Was von Neumann right after all? of "Towards S-matrix". See also the article "Topological Geometrodynamics: an Overall View".