What Could Be the Role of Complexity Theory in TGD?

Abstract

Chaotic systems seem to be the diametrical opposite of completely integrable systems about which TGD is a possible example. There is however also something common: in completely integrable classical systems all orbits are cyclic and in chaotic systems they form a dense set in the space of orbits. Furthermore, in chaotic systems the approach to chaos occurs via steps as a control parameter is changed. Same would take place in adelic TGD fusing the descriptions of matter and cognition. In TGD Universe the hierarchy of extensions of rationals inducing finite-dimensional extension of p-adic number fields defines a hierarchy of adelic physics and provides a natural correlate for evolution. Galois groups and ramified primes appear as characterizers of the extensions. The sequences of Galois groups could characterize an evolution by phase transitions increasing the dimension of the extension associated with the coordinates of WCW in turn inducing the extension used at space-time and Hilbert space level. WCW decomposes to sectors characterized by Galois groups G₃ of extensions associated with the 3-surfaces at the ends of space-time surface at boundaries of causal diamond (CD) and G₄ characterizing the space-time surface itself. G₃ (G₄) acts on the discretization and induces a covering structure of the 3-surface (space-time surface). If the state function reduction to the opposite boundary of CD involves localization into a sector with fixed G₃, evolution is indeed mapped to a sequence of G₃s. Also the cognitive representation defined by the intersection of real and p-adic surfaces with coordinates of points in an extension of rationals evolve.