The recent view involving strong form of holography would provide dramatically simplified view about how these representations are formed as continuations of representations of strings world sheets and partonic 2-surfaces in the intersection of real and p-adic variants of WCW ("World of Classical Worlds") in the sense that the parameters characterizing these representations are in the algebraic numbers in the algebraic extension of p-adic numbers involved.

For details see the article Intentions, Cognition, and Time

For a summary of earlier postings see Links to the latest progress in TGD.

## 4 comments:

Matti, I really think you are on to something (as if you weren't already) with the adelic approach to things. I was just looking at a paper in wrote on fractal strings and membranes a few years ago and realized that the adelic product defined by Lapidus and others, assigns to each element of the product a (square intehrable) Hilbert space. Also I think there must be something special about modular arithmetic , the special status of the integral numbers 12 and 24 hours there, and the approximately 24hr period of the standard day. Maybe life requires these almost ratios? If other planets had cycles not commensurate then perhaps life would not be favored? --crow

To Anonymous:

Thank you for a very stimulating comment.

Adelic approach is free of assumptions which require mathematics which need not exist: transformation of p-adic space-time surfaces to real ones was the questionable assumption. It always takes years to develop ability to see it from bigger perspective. 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. An excellent lesson!

I think you sent 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.

To be continued....

To Anonymous:

One can think that the real amplitude in the intersection of realities and p-adicities for all values of parameter is rational number. If given p-adic amplitude is just the p-adic norm of real amplitude, one would have the adelic identity. But this would require that p-adic variant of the amplitude is real number-valued: I have p-adic valued amplitudes. And Witten's identity holds for vacuum amplitude. I want it for entire S-matrix, M-matrix, and/or U-matrix and for all states of the basis in some sense.

a) Consider first vacuum amplitude. A weaker form of the identity would be that the *p-adic norm* of 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). This would make sense and give a non-trivial constraint. In particular, the p-adic norm of the real amplitude would be inverse of the the product of p-adic norms of p-adic amplitudes. Most of these amplitudes should have p-adic norm equal to one. This condition can make sense only if the p-adic norm of p-adic amplitude equals to 1 for most prime.

b) 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 the S-matrix.

*For S-matrix elements one should have SS^dagger=1. This states unitarity of S-matrix. Probability is conserved. Could it make sense to generalize this condition and demand that it holds true only adelically that is for the product of real and p-adic norms of SS^dagger in various number fields? For each state m of basis: the product of norm real X_mm =(SS^dagger)_{mm} and p-adic norms its p-adic counterparts would be qual to 1 in the intersection of reality and p-adicities. Strong condition would be that this holds for X_mm themselves. It could if the the p-adic norm of X_mm is X_mm itself that is power of p.

*For a given diagonal element of unit matrix defining 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 labelling the diagonal matrix element of SS^dagger a *finite number of preferred p-adic primes*!! The underlying deep reason for this assignment I have been looking for!!

*Unitary 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 thermo-dynamics which indeed means that SS^+ is density matrix defining the p-adic thermal state!

*The diagonal elements of SS^dagger_{mm} in given number field would define analogs of probabilities. Could these probabilities be interpreted as the probabilities, whose sum equals to 1? Probability conservation for a given number field. Adelic S-matrix would be the more sophisticated counterpart of M-matrix!

One can consider a variant of this. One could also consider the possibility that the p-adic norm of SS^dagger_mm is replaced with its image under canonical identification. The information loss wold not be so huge. This might be required by p-adic thermodynamics.

To Anonymous:

I forgot the summary. The outcome could be two breakthroughs. First principle understanding of how preferred p-adic primes are assigned to quantum states and first principle justification for p-adic thermodynamics.

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