Monday, April 18, 2005

Feynman diagrams within Feynman diagrams and reflective levels of consciousness

Here is the little step forward of this day made in understanding of the role of Jones inclusions of hyper-finite factors of type II_1 as a key element in the construction quantum counterpart for the many-sheeted space-time. It is possible to assign to a given Jones inclusion N subset M an entire hierarchy of Jones inclusions M_0 subset M_1 subset M_2..., M_0=N, M_1=M. A natural interpretation for these inclusions would be as a sequence of topological condensations. This sequence also defines a hierarchy of Feynman diagrams inside Feynman diagrams. The factor M containing the Feynman diagram having as its lines the unitary orbits of N under Delta_{M} (, which defines a canonical automorphism in II_1 factor) becomes a parton in M_1 and its unitary orbits under Delta_{M_1} define lines of Feynman diagrams in M_1. The outcome is a hierarchy of Feynman diagrams within Feynman diagrams, a fractal structure for which many particle scattering events at a given level become particles at the next level. The particles at the next level represent dynamics at the lower level: they have the property of "being about" representing perhaps the most crucial element of conscious experience. Since net conserved quantum numbers can vanish for a system in TGD Universe, this kind of hierarchy indeed allows a realization as zero energy states. Crossing symmetry can be understood in terms of this picture and has been applied to construct a model for S-matrix at high energy limit. The quantum image for the orbit of parton has dimension log_2(M:N) +1<= 3. Two subsequent inclusions form a natural basic unit since the bipartite diagrams classifying Jones inclusions are duals of each other by black-white duality. In this double inclusion a two-parameter family of deformations of M counterpart of a partonic 2-surface is formed and has quantum dimension log_2(M:N) +2<=4. One might perhaps say that quantum space-time corresponds to a double inclusion and that further inclusions bring in N-parameter families of space-time surfaces. For more details see the new chapter Was von Neumann Right After All? Matti Pitkanen

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