Thursday, September 27, 2012

Fresh view about hyper-finite factors in TGD framework


For about seven years ago there was a burst of ideas related to the mathematics and physics of TGD. The notions of zero energy ontology, hierarchy of (effective as it turned out) Planck constants, and hyperfinite factors and their inclusions as a realization of finite measurement resolution emerged at that time. I have just looked for the vision about hyper-finite finite factors in the light of wisdom gained during these years. I do not bother to type more but just give the abstract of a little article Fresh view about hyper-finite factors in TGD framework representing some new results: in particular, a more precise view about how the quantum spaces identified as factors spaces of including and included HFF can be defined.

In this article I will discuss the basic ideas about the role of hyper-finite factors in TGD with the background given by a work of more than half decade. First I summarize the input ideas which I combine with the TGD inspired intuitive wisdom about HFFs of type II1 and their inclusions allowing to represent finite measurement resolution and leading to notion of quantum spaces with algebraic number valued dimension defined by the index of the inclusion.

Also an argument suggesting that the inclusions define "skewed" inclusions of lattices to larger lattices giving rise to quasicrystals is proposed. The core of the argument is that the included HFF of type II1 algebra is a projection of the including algebra to a subspace with dimension D≤ 1. The projection operator defines the analog of a projection of a bigger lattice to the included lattice. Also the fact that the dimension of the tensor product is product of dimensions of factors just like the number of elements in finite group is product of numbers of elements of coset space and subgroup, supports this interpretation.

One also ends up with a detailed identification of the hyper-finite factors in "orbital degrees of freedom" in terms of symplectic group associated with δ M4+/-× CP2 and the group algebras of their discrete subgroups define what could be called "orbital degrees of freedom" for WCW spinor fields. By very general argument this group algebra is HFF of type II, maybe even II1.

For more details see the chapter Was von Neumann Right after All? of "Towards M-Matrix" or the already mentioned article.

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