Infinite primes as an answer to the question of Manin

Kea quotes Manin in her blog: If numbers are similar to polynomials in one variable over a finite field, what is the analogue of polynomials in several variables? Or, in more geometric terms, does there exist a category in which one can define absolute Descartes powers SpecZ×⋯×SpecZ?

Well, I answered this question for about 15 years ago. I of course did not know that this question had been posed by some-one (and still do not know when it has been posed for the first time). The answer came as I introduced the notion of infinite prime with motivations which were purely physical. I started from the following observation.

If evolution means a gradual increase of the p-adic prime P assumed to characterize the entire Universe and if there has been no first quantum jump, P should be literally infinite! Puzzling! Could infinite primes exist and what they could be? The answer came in few minutes from the observation that taking the product of all finite primes -call it simply X to avoid too long a posting;-)- and adding +/- 1 to it you get the simplest possible infinite primes P₊=X+1 and P_-=X-1. As a matter fact, infinite primes always come in pairs differing by 2 units.

This led to a series of amazing observations.

1. The simplest lowest level infinite primes correspond to quantum states of a second quantized super-symmetric arithmetic QFT with bosons and fermions labelled by finite primes. Their products define infinite integers having interpretation as free many-particle Fock states.

2. One can map infinite primes to polynomials and this leads to a surprising revelations. One obtains also the analogs of bound states! This is quite an emotional kick for anyone knowing how hard it is to understand non-perturbative effects - in particular bound states- in quantum field theories. Single particle states correspond first order polynomials of single variable and many particle states to irreducible higher order polynomials with algebraic roots!

3. Second quantization can be repeated again and again by taking many particle states of the previous level single particle states of the next level in the hierarchy. Every step brings in an additional variable to the polynomials (essentially as the product of the infinite primes of the previous level) and one obtains infinite algebraics at the higher levels as roots of the polynomials besides algebraics. In TGD framework this hierarchy corresponds to the hierarchy of space-time sheets in many-sheeted space-time.

4. This construction generalizes to other classical number fields and I have proposed a concrete identification of pairs of what I call hyper-octonion infinite primes in terms of standard model quantum numbers. The motivation comes from the observation that one can understand standard model symmetries in terms of octonions and quaternions and M⁴×CP₂ has a number theoretic interpretation and the preferred extremals of Kähler action could be hyper-quaternionic space-time surfaces with the additional property that at each point the tangent space contains preferred hyper-octonionic imaginary unit.

This procedure gives a generalization of the notion of real number different from non-standard numbers of Robinson.

1. Very roughly, infinitesimal and infinite numbers of Robinson are replaced with their exponentials which define infinite number of real units so that real number is replaced with infinite number of its copies with arbitrary complex number theoretic anatomy. There are neither infinitesimals nor infinities but there are infinite number of copies of real number equivalent as far as the magnitude of the number is considered.

2. This number theoretcal anatomy is so incredibly complex that can quite seriously consider the possibility that one can ask whether the quantum states of the Universe and the world of classical worlds (WCW) could allow a concrete representation in terms of this anatomy. These infinite-dimensional spaces would be absolutely real- not fictions of quantum theorist! One can think that the evolution at the level of WCW is realized concretely as evolution at the level of space-time and imbedding space making the number theoretical anatomy of space-time points more and more complex.

3. An intriguing possibility is that the sums of the units normalized so that unity comes out could be interpretation as representation of quantum superposition of zero energy states so that quantum superposition would not be needed separately. Physical existence would reduce to generalized octonions and subjective existence to quantum jumps!

4. In zero energy ontology zero energy quantum states assignable to given causal diamond analogous to Penrose diagram. The infinite integers appearing as the numerator and denominator of the rational reducing to real unity would represent positive and negative energy parts of zero energy state and real unity property would code for the conservation of number theoretic momentum expressible as the sum ∑n_ilog(p_i): each n_i is separately conserved since the numbers log(p_i) are algebraically independent. Number theoretic Brahman= Atman identity or algebraic holography would perhaps be the proper term here.

I have compared the notion of real number inspired by the notion of infinite prime to non-standard numbers in previous posting and in the new chapter Non-Standard Numbers and TGD of "Physics as Generalized Number Theory".