Monday, April 25, 2005

Quaternions, Octonions, and Hyperfinite Type II_1 Factors

Quaternions and octonions as well as their hyper counterparts obtained as subspace of complexified quanternions and octonions are central elements of the number theoretic vision about TGD. The latest progress in understanding of hyper-finite II_1 factors relates to the question of how could one understand quaternions and octonions and their possible quantum counterparts in this framework. The quantum counterparts of quaternions and octonions have been proposed around 1999 and would be directly relevant to the construction of vertices as quantum quaternionic (at least) multiplication and co-multiplication. The notion of quaternionic or octonionic Hilbert space is not attractive (problems with the notions of orthogonality, hermiticity, and tensor product). A more natural idea is that Hilbert space is hyper-Kaehler manifold in that operator algebra allows a representation of quaternion units. The fact that unit would have trace 1 would automatically imply that for inclusions N subset M quantum quaternion algebra would appear. The fact that for Jones index M:N=4 inclusion hyper-finite II_1 factor is nothing but infinite Connes tensor product of quaternionic algebra represented as 2x2 matrices indeed suggests that Hyper-Kahler property and its quantum version are inherent properties of II_1 factor. The choice of preferred Kahler structure is necessary and corresponds to the choice of the preferred octonionic imaginary unit at space-time level. Hyper Kähler structure in tangent space of configuration space makes it Hyper-Kähler manifold with vanishing Einstein tensor and Ricci scalar. If this were not the case, constant curvature property would imply infinite Ricci scalar and perturbative divergences in configuration space functional integral coming from perturbations of the metric determinant. This indeed happens in case of loop spaces, which is a very strong objection against string models. The octonionic structure is more intricate since non-associativity cannot be realized by linear operators. * operation is however anti-linear and the classical Cayley-Dickson construction uses it to build up a hierarchy of algebras by adding imaginary units one by one. It is possible to extend any * algebra (in particular von Neumann algebra) by adding * operation, call it J, as an imaginary unit to the algebra by posing 3 constraints. The restriction of the construction to quaternionic units of von Neumann algebra gives rise to quantum octonions. J acts time reversal T/CP operation and superpositions of states created by linear operators A and antilinear operators AJ are not allowed in general since they would break fermion number conservation. Only states created by operators of form A or JA are allowed and J would transform vacuum to a new one and give rise to negative energy states in TGD framework. One might perhaps say that A and JA create bra and ket type states. At space-time level the correlates for this are hyper-quaternionic and co-hyper-quaternionic space-time surfaces obeying different variant of Kahler calibration. For the first one the value of the Kahler action for a space-time region inside which action density is of fixed sign is as near as possible to zero and for the second one as far as possible from zero. The fact that CP and T are broken supports the view that these dual and non-equivalent dynamics correspond to opposite time orientations of space-time sheets. To sum up, hyper-finite factors of type II_1 seem to have very close relationship with quantum TGD and it remains to be seen how much they catch from axiomatics needed to fix TGD completely. Even M^4xCP_2 might emerge from extension of von Neumann algebra by * since M^8< --> M^4xCP_2 duality, which provides a number theoretic realization of compactification as wave particle duality in the cotangent bundle of the configuration space, is made possible by the choice of the preferred octonionic imaginary unit. Could the requirement that the hyper-finite II_1 factor has a classical space-time representation be enough to fix the theory? For more details see the chapter Was von Neumann Right After All? of TGD. Matti Pitkanen

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