### 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*(*iL _{0}τ*) so that the operator Δ of Connes would be identified as Δ =

*exp*(

*L*). 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

_{0}*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.*

_{n}