Could adelic approach allow to understand the origin of preferred p-adic primes?

The comment of Crow to the posting Intentions, cognitions, and time stimulated rather interesting ideas about the adelization of quantum TGD.

First two questions.

1. What is Adelic quantum TGD? The basic vision is that scattering amplitudes are obtained by algebraic continuation
   to various number fields from the intersection of realities and p-adicities (briefly intersection in what follows) represented at the space-time level by string world sheets and partonic 2-surfaces for which defining parameters (WCW coordinates) are in rational or in in some algebraic extension of p-adic numbers. This principle is a combination of strong form of holography and algebraic continuation as a manner to achieve number theoretic universality.
   
   
2. Why Adelic quantum TGD? Adelic approach is free of the earlier assumptions, which require mathematics which need not exist: transformation of p-adic space-time surfaces to real ones as a realization of intentional actions was the questionable assumption, which is un-necessary if cognition and matter are two different aspects of existence as already the success of p-adic mass calculations strongly suggests. It always takes years to develop ability to see things from bigger perspective and distill discoveries from clever inventions. Now adelicity is totally obvious. Being a conservative radical - not radical radical or radical conservative - is the correct strategy which I have been gradually learning. This particular lesson was excellent!
   

For some years ago Crow sent to me the book of Lapidus about adelic strings. Witten wrote for long time ago an article in which the demonstrated that that the product of real stringy vacuum amplitude and its p-adic variants equals to 1. This is a generalisation of the adelic identity for a rational number stating that the product of the norm of rational number with its p-adic norms equals to one.

The real amplitude in the intersection of realities and p-adicities for all values of parameter is rational number or in an appropriate algebraic extension of rationals. If given p-adic amplitude is just the p-adic norm of real amplitude, one would have the adelic identity. This would however require that p-adic variant of the amplitude is real number-valued: I want p-adic valued amplitudes. A further restriction is that Witten's adelic identity holds for vacuum amplitude. I live in Zero Energy Ontology (ZEO) and want it for entire S-matrix, M-matrix, and/or U-matrix and for all states of the basis in some sense.

Consider first the vacuum amplitude. A weaker form of the identity would be that the p-adic norm of a given p-adic valued amplitude is same as that p-adic norm for the rational-valued real amplitude (this generalizes to algebraic extensions, I dare to guess) in the intersection. This would make sense and give a non-trivial constraint: algebraic continuation would guarantee this constraint. In particular, the p-adic norm of the real amplitude would be inverse of the product of p-adic norms of p-adic amplitudes. Most of these amplitudes should have p-adic norm equal to one in other words, real amplitude is product of finite number of powers of prime. This because the p-adic norms must approach rapidly to unity as p-adic prime increases and for large p-adic primes this means that the norm is exactly unity. Hence the p-adic norm of p-adic amplitude equals to 1 for most primes.

In ZEO one must consider S-, M-, or U-matrix elements. U and S are unitary. M is product of hermitian square root of density matrix times unitary S-matrix. Consider next S-matrix.

1. For S-matrix elements one should have p_m=(SS^†)_mm=1. This states the unitarity of S-matrix. Probability is conserved. Could it make sense to generalize this condition and demand that it holds true only adelically that only for the product of real and p-adic norms of p_m equals to one: N_R(p_m)(R)∏_p N_p(p_m(p))=1. This could be actually true identically in the intersection if algebraic continuation principle holds true. Despite the triviality of the adelicity condition, one need not have anymore unitarity separately for reals and p-adic number fields. Notice that the numbers p_m would be arbitrary
   rationals in the most general cased.
   
   
2. Could one even replace N_p with canonical identification or some form of it with cutoffs reflecting the length scale cutoffs? Canonical identification behaves for powers of p like p-adic norm and means only
   more precise map of p-adics to reals.
   
   
3. For a given diagonal element of unit matrix characterizing particular state m one would have a product of real norm and p-adic norms. The number of the norms, which differ from unity would be finite. This condition would give finite number of exceptional p-adic primes, that is assign to a given quantum state m a finite number of preferred p-adic primes! I have been searching for a long time the underlying deep reason for this assignment forced by the p-adic mass calculations and here it might be.
   
   
4. Unitarity might thus fail in real sector and in a finite number of p-adic sectors (otherwise the product of p-adic norms would be infinite or zero). In some sense the failures would compensate each other in the adelic picture. The failure of course brings in mind p-adic thermodynamics, which indeed means that adelic SS^†, or should it be called MM^†, is not unitary but defines the density matrix defining the p-adic thermal state! Recall that M-matrix is defined as hermitian square root of density matrix and unitary S-matrix.
   
   
5. The weakness of these arguments is that states are assumed to be labelled by discrete indices. Finite measurement resolution implies discretization and could justify this.
   

The p-adic norms of p_m or the images of p_m under canonical identification in a given number field would define analogs of probabilities. Could one indeed have ∑_m p_m=1 so that SS^† would define a density matrix?

1. For the ordinary S-matrix this cannot be the case since the sum of the probabilities p_m equals to the dimension N of the state space: ∑ p_m=N. In this case one could accept p_m>1 both in real and p-adic sectors. For this option adelic unitarity would make sense and would be highly non-trivial condition allowing perhaps to understand how preferred p-adic primes emerge at the fundamental level.
   
   
2. If S-matrix is multiplied by a hermitian square root of density matrix to get M-matrix, the situation changes and one indeed obtains ∑ p_m=1. MM†=1 does not make sense anymore and must be replaced with MM†=ρ, in special case a projector to a N-dimensional subspace proportional to 1/N. In this case the numbers p(m) would have p-adic norm larger than one for the divisors of N and would define preferred p-adic primes. For these primes the sum N_p(p(m)) would not be equal to 1 but to NN_p(1/N.
   
   
3. Situation is different for hyper-finite factors of type II₁ for which the trace of unit matrix equals to one by definition and MM^†=1 and ∑ p_m=1 with sum defined appropriately could make sense. If MM† could be also a projector to an infinite-D subspace. Could the M-matrix using the ordinary definition of dimension of Hilbert space be equivalent with S-matrix for the state space using the definition of dimension assignable to HFFs? Could these notions be dual of each other? Could the adelic S-matrix define the counterpart of M-matrix for HFFs?
   
   

This looks like a nice idea but usually good looking ideas do not live long in the crossfire of counter arguments. The following is my own. The reader is encouraged to invent his or her own objections.

1. The most obvious objection against the very attractive direct algebraic continuation} from real to p-adic sector is that if the real norm or real amplitude is small then the p-adic norm of its p-adic counterpart is large so that p-adic variants of p_m(p) can become larger than 1 so that probability interpretation fails. As noticed there is no actually no need to pose probability interpretation. The only way to overcome the "problem" is to assume that unitarity holds separately in each sector so that one would have p(m)=1 in all number fields but this would lead to the loss of preferred primes.
   
   
2. Should p-adic variants of the real amplitude be defined by canonical identification or its variant with cutoffs? This is mildly suggested by p-adic thermodynamics. In this case it might be possible to satisfy the condition p_m(R)∏_p N_p(p_m(p))=1. One can however argue that the adelic condition is an ad hoc condition in this
   case.
   
   

To sum up, if the above idea survives all the objections, it could give rise to a considerable progress. A first principle understanding of how preferred p-adic primes are assigned to quantum states and thus a first principle justification for p-adic thermodynamics. For the ordinary definition of S-matrix this picture makes sense and also for M-matrix. One would still need the justification of canonical identification map playing a key role in p-adic thermodynamics allowing to map p-adic mass squared to its real counterpart.