Palmer's Invariant Set Theory and TGD
Tony Smith told me about the Invariant Set Theory of T. N. Palmer involving p-adic numbers and asked how it relates to TGD. As a rule this kind of questions are very useful and also now the questions forced to refresh my understanding about the notion of p-adic imbedding space and I realized a possible connection between finite measurement resolution, p-adicization, and hierarchy of inclusions of hyper-finite factors. The work of Palmer involves rather original ideas although our views about physics are radically different.
- What makes Palmer's work interesting from TGD point of view is that it involves p-adic number fields. p-Adic topology is assumed to provide a natural description for a space U of 3-D Universes. U could be seen as analog of Wheeler's superspace formed by 3-geometries or the "World of Classical Worlds" of TGD. The space IU is identified as an invariant set (IS) for an iterative dynamics in U assigned to a quasi-cyclic cosmology. IS would be expressible in terms of Cantor sets in the space U. Space-time would correspond to an orbit MU of 3-space in U.
- Palmer assumes that the physics is basically classical and deterministic albeit non-computable and that this picture about dynamics could reproduce the predictions of quantum theory. To show that this is the case is a formidable challenge: since one should deduce not only the description in terms of quantum states but also quantum measurement theory and non-deterministic state function reduction with its strange rules.
In TGD Universe classical physics is exact part of quantum physics: the very definition of WCW geometry assigns to 3-surface a 4-surface as analog of Bohr orbit associated with it - the interpretation is in terms of holography. This implies the replacement of path integral with functional integral. There is no attempt to reduce quantum to classical.
- In Palmer's approach p-adic distance function for the points of invariant set (IS) is introduced and single large p-adic prime is suggested to characterize the topology. This brings strongly in mind models for spin glass energy landscape, which has ultrametric topology (also p-adic topologies are ultrametric). The 3-spaces have metric with Euclidian signature. The challenge is to deduce Einsteinian space-time picture for the orbits MU in U. The Minkowskian signature of space-time metric is the challenge. Also Einstein's equations should follow from this framework.
In TGD framework p-adic physics is identified as a physical correlate of cognition and p-adicization of physics is carried at all levels: imbedding space level, space-time level, and WCW level. Single state space characterizes quantum states and can be interpreted as real or p-adic since the coefficient field is assumed to be extension of rationals. All number fields are fused to single structure and one obtains what might be called adelic physics (for the latest progress in the construction of p-adic geometries as "smooth" discretizations with discrete algebraic points of real geometric objects replaced with p-adic monads see the article). This makes it possible to satisfy field equation also at the p-adic level.
For a summary of earlier postings see Latest progress in TGD.