Infinite primes, integers, and rationals have a precise number theoretic anatomy. For instance, the simplest infinite primes corresponds to the numbers P+/-= X+/- 1, where X=∏kpk is the product of all finite primes. Indeed, P+/-mod p=1 holds true for all finite primes.
The construction of infinite primes at the first level of the hierarchy is structurally analogous to the quantization of super-symmetric arithmetic quantum field theory with finite primes playing the role of momenta associated with fermions and bosons. Also the counterparts of bound states emerge. This process can be iterated: at the second level the product of infinite primes constructed at the first level replaces X and so on.
The structural similarity with repeatedly second quantized quantum field theory suggests that physics might in some sense reduce to a number theory for infinite rationals M/N and that second quantization could be followed by further quantizations. As a matter fact, the hierarchy of space-time sheets could realize this endless second quantization geometrically and have also a direct connection with the hierarchy of logics labelled by their order. This could have rather breathtaking implications.
- Could this hierarchy correspond to a hierarchy of realities for which level below corresponds in a literal sense infinitesimals and the level next above to infinity?
- There is an infinite number of infinite rationals behaving like real units (M/N=1 in real sense) so that space-time points could have infinitely rich number theoretical anatomy not detectable at the level of real physics. Infinite integers would correspond to positive energy many particle states and their inverses (infinitesimals with number theoretic structure) to negative energy many particle states and M/N= 1 would be a counterpart for zero energy ontology to which oneness and emptiness are assigned in mysticism.
- Single space-time point, which is usually regarded as the most primitive and completely irreducible structure of mathematics, would take the role of Platonia of mathematical ideas being able to represent in its number theoretical structure even the quantum state of entire Universe. Algebraic Brahman=Atman identity and algebraic holography would be realized in a rather literal sense .
Therefore quantum jumps would correspond to changes in the anatomy of the space-time points. Imbedding space would be experiencing genuine number theoretical evolution. Physics would reduce to the anatomy of numbers. All mathematical notions which are more than mere human inventions would be imbeddable to the Platonia realized as the number theoretical anatomies of single imbedding space point.
This picture give also a justification for the decomposition of WCW to a union of WCW:s associated with imbedding spaces with preferred point (tip of the lightcone and point of CP2 fixing U(2) subgroup as isotropy group). Given point of space-time would provide representation for the spinors fields in WCW associated with the future and/or past light-cone at this point. The "big bang" singularity would code all the information about the quantum state of this particular sub-universe in its number theoretical anatomy.
Interestingly, this picture can be deduced by taking into extreme quantum-classical correspondence and by requiring that both configuration space and configuration space spinor fields have not only space-time correlates but representation at the level of space-time: the only reasonable identification is in terms of algebraic structure of space-time point.
For a brief summary of quantum TGD inspired theory of consciousness see the article TGD Inspired Theory of Consciousness.