Wednesday, May 07, 2008

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 X3l fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T3 of X3l 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".

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