Connes' Idea about Inherent Time Evolution of Certain Algebraic Structures from TGD Viewpoint

Abstract

Alan Connes has proposed that certain mathematical structures known as hyperfinite factors (HFFs) contain in their structure inherent time evolution.This time evolution is determined only modulo unitary automorphism analogous to a time evolution determined by Hamiltonian  so that this time evolution  seems to be too  general for the purposes of a physicist. Zero energy ontology of TGD combined with adelic physics leads to a vision that the sequences of state function reductions implies a mathematical evolution in the sense that the extensions of rationals characterizing  the space-time region increases gradually. This induces the increase of algebraic complexity   implying  time  evolution as the analog of biological evolution. The dimension of extension corresponds to  an effective Planck constant assumed to label dark matter as phases of ordinary matter. Therefore quantum coherence lengths increase in this evolution. This generalization of the idea of Connes is discussed in the framework provided by the recent view about TGD. In particular, the  inclusion hierarchies of  hyper-finite factors, the extension hierarchies of rationals, and fractal   inclusion hierarchies of subalgebras of supersymplectic algebra isomorphic with the entire algebra are proposed to be more or less one and the same thing in TGD framework. The time evolution operator of Connes could corresponds to super-symplectic algebra (SSA) to the time evolution generated by exp(iL₀τ) so that the operator Δ of Connes would be identified as  Δ = exp(L₀). This identification allows number theoretical universality if τ is quantized. Furthermore, one ends up with a model for the subjective time evolution by small state function reductions (SSFRs) for SSA with SSA_n gauge conditions: the unitary time evolution for given SSFR would be generated by a linear combination of Virasoro generators not annihilating the states.  This model would generalize the model for harmonic oscillator in external force allowing exact S-matrix.