Updated view about Quantum Adeles
I have been working last weeks with quantum adeles. This has involved several wrong tracks and about five days ago a catastrophe splitting the chapter "Quantum Adeles" to two pieces entitled "Quantum Adeles" and "About Absolute Galois Group" took place, and simplified dramatically the view about what adeles are and led to the notion of quantum mathematics. At least now the situation seems to be settled down and I see no signs about possible new catastrophes. I glue the abstract of the reincarnated "Quantum Adeles" below.
Quantum arithmetics provides a possible resolution of a longlasting challenge of finding a mathematical justification for the canonical identification mapping padics to reals playing a key role in TGD  in particular in padic mass calculations. pAdic numbers have padic pinary expansions ∑ a_{n}p^{n} satisfying a_{n}<p. of powers p^{n} to be products of primes p_{1}<p satisfying a_{n}<p for ordinary padic numbers. One could map this expansion to its quantum counterpart by replacing a_{n} with their counterpart and by canonical identification map p→ 1/p the expansion to real number. This definition might be criticized as being essentially equivalent with ordinary padic numbers since one can argue that the map of coefficients a_{n} to their quantum counterparts takes place only in the canonical identification map to reals.
One could however modify this recipe. Represent integer n as a product of primes l and allow for l all expansions for which the coefficients a_{n} consist of primes p_{1}<p but give up the condition a_{n}<p. This would give 1tomany correspondence between ordinary padic numbers and their quantum counterparts.
It took time to realize that l<p condition might be necessary in which case the quantization in this sense  if present at all  could be associated with the canonical identification map to reals. It would correspond only to the process taking into account finite measurement resolution rather than replacement of padic number field with something new, hopefully a field. At this step one might perhaps allow l>p so that one would obtain several real images under canonical identification.
This did not however mean giving up the notion of the idea of generalizing number concept. One can replace integer n with ndimensional Hilbert space and sum + and product × with direct sum ⊕ and tensor product ⊗ and introduce their cooperations, the definition of which is highly nontrivial.
This procedure yields also Hilbert space variants of rationals, algebraic numbers, padic number fields, and even complex, quaternionic and octonionic algebraics. Also adeles can be replaced with their Hilbert space counterparts. Even more, one can replace the points of Hilbert spaces with Hilbert spaces and repeat this process, which is very similar to the construction of infinite primes having interpretation in terms of repeated second quantization. This process could be the counterpart for construction of n^{th} order logics and one might speak of Hilbert or quantum mathematics. The construction would also generalize the notion of algebraic holography and provide selfreferential cognitive representation of mathematics.
This vision emerged from the connections with generalized Feynman diagrams, braids, and with the hierarchy of Planck constants realized in terms of coverings of the imbedding space. Hilbert space generalization of number concept seems to be extremely well suited for the purposes of TGD. For instance, generalized Feynman diagrams could be identifiable as arithmetic Feynman diagrams describing sequences of arithmetic operations and their cooperations. One could interpret ×_{q} and +_{q} and their coalgebra operations as 3vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD (stringy vertices and vertices of Feynman diagrams). The definition of cooperations would characterize quantum dynamics. Physical states would correspond to the Hilbert space states assignable to numbers. One prediction is that all loops can be eliminated from generalized Feynman diagrams and diagrams are in projective sense invariant under permutations of incoming (outgoing legs).
I glue also the abstract for the second chapter "About Absolute Galois" group which came out from the catastrophe. The reason for the splitting out was that the question whether Absolute Galois group might be isomorphic with the analog of Galois group assigned to quantum padics ceased to make sense.
Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals  Absolute Galois Group (AGG)  through its representations. Invertible adeles ideles  define Gl_{1} which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry.
I have asked already earlier whether AGG could act is symmetries of quantum TGD. The basis idea was that AGG could be identified as a permutation group for a braid having infinite number of strands. The notion of quantum adele leads to the interpretation of the analog of Galois group for quantum adeles in terms of permutation groups assignable to finite l braids. One can also assign to infinite primes braid structures and Galois groups have lift to braid groups (see this).
Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2surfaces, and could be important in the intersection of real and padic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Adeles would make these representations very concrete by bringing in cognition represented in terms of padics and there is also a generalization to Hilbert adeles.
For details see the new chapters Quantum Adeles and About Absolute Galois Group of "Physics as Generalized Number Theory".
6 Comments:
Matti:
In this book
http://tgdtheory.com/public_html/pdfpool/genememec.pdf
you mention humanplant communications through memes and genes. I read it and began to think from my mystical standpoint.



The entheogenic molecules would represent communications between plant/molecule magnetic body and human magnetic body. The plant molecule would be considered just as much of an explorer as the human is!
Matti:
http://noosphere.princeton.edu/
Global Consciousness Project and the emerging Noosphere.
Noosphere in TGD context as magnetic body of Mother Gaia? A superstructure of magnetic bodies?
Regards.
Fractality:
you might be right about entheogens. We think that we are at the top because of our ability to change and control our environment. If spiritual intelligence is as criterion our position in the hierarchy might change.
Magnetic body could be taken as a particular model for noosphere.
http://www.menrvagames.com/forum/http/Occult%20%5B7977%20books%5D/Stephen%20Schneider%20%20Scientists%20Debate%20Gaia%20The%20Next%20Century.pdf
Scientists debate Gaia.
Stephen Schneider et al.
http://www.sciencedaily.com/releases/2012/03/120321142903.htm
The basic units of today's data processing are the bit states "0" and "1," which differ in their electrical voltage. To encode these states, only the charge of the electrons is crucial. "Electrons also have other properties though" says Wieck, and these are exactly what you need for quantum bits. "The extension from bits to quantum bits can dramatically increase the computational power of computers"
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