Quantum arithmetics is a notion which emerged as a possible resolution of long-lived challenge of finding mathematical justification for the canonical identification mapping p-adics to reals playing key role in p-adic mass calculations. The model for Shnoll effect was the bridge leading to the discovery of quantum arithmetics.

- What quantum arithmetics suggests is a modification of p-adic numbers by replacing p-adic pinary expansions with their quantum counterparts allowing the coefficients of prime powers to be integers not divisible by p.
- A further constraint is that quantum integers respect the decomposition of integer to powers of prime. Quantum p-adic integers are to p-adic integers what the integers in the extension of number field are for the number field and one can indeed identify Galois group G
_{p}for each prime p and form adelic counterpart of this group as Cartesian product of all G_{p}:s. After various trials it turned out that quantum p-adics are indeed quantal in the sense that one can assign to given quantum p-adic integer n a wave function at the orbit of corresponding Galois group decomposing to Galois groups of its prime factors of n. The basic conditions are that ×_{q}and +_{q}satisfy the basic associativity and distributivity laws.One can interpret ×

_{q}and +_{q}and their co-algebra operations as 3-vertices 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). This allows to deduce very precise information about the symmetries of the vertices needed to satisfy the associativity and distributivity and actually fix them highly uniquely, and therefore determined corresponding zero energy states having collections of integers as counterparts of incoming positive energy (or negative energy) particles.This gives strong support for the old conjectures that generalized Feynman diagrams have number theoretic interpretation and allow moves transforming them to tree diagrams - also this generalization of old-fashioned string duality is old romantic idea of quantum TGD. The moves for generalized Feynman diagrams would code for associativity and distributivity of quantum arithmetics. Also braidings with strands labelled by the primes dividing the integer emerge naturally so that the connection with quantum TGD proper becomes very strong.

- Canonical identification finds a fundamental role in the definition of the norm for both quantum p-adics and quantum adeles.
- There are arguments suggesting that quantum p-adics form a field so that also differential calculus and even integral calculus would make sense since quantum p-adics inherit well-ordering from reals via canonical identification.

The ring of adeles is essentially Cartesian product of different p-adic number fields and reals.

- The proposal is that adeles can be replaced with quantum adeles. G
_{p}has natural action on quantum adeles allowing to construct representations of G_{p}. This norm for quantum adeles is the ordinary Hilbert space norm obtained by first mapping quantum p-adic numbers in each factor of quantum adele by canonical identification to reals. - Also quantum adeles could form form a field rather than only ring so that also differential calculus and even integral calculus could make sense. This would allow to replace reals by quantum adeles and in this manner to achieve number theoretical universality. The natural applications would be to quantum TGD, in particular to construction of generalized Feynman graphs as amplitudes which have values in quantum adele valued function spaces associated with quantum adelic objects. Quantum p-adics and quantum adeles suggest also solutions to a number of nasty little inconsistencies, which have plagued to p-adicization program.
- One must of course admit that quantum arithmetics is far from a polished mathematical notion. It would require a lot of work to see whether the dream about associative and distributive function field like structure allowing to construct differential and integral calculus is realized in terms of quantum p-adics and even in terms of quantum adeles. This would provide a realization of number theoretical universality.

Ordinary adeles play a fundamental technical tool in Langlands correspondence. 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 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.

The crazy question is whether quantum adeles could be isomorphic with algebraic numbers and whether the Galois group of quantum adeles could be isomorphic with AGG or with its commutator group. If so, AGG would naturally act is symmetries of quantum TGD. The connection with infinite primes leads to a proposal what quantum p-adics and quantum adeles associated with algebraic extensions of rationals could be and provides support for the conjecture. The Galois group of quantum p-adic prime p would be isomorphic with the ordinary Galois group permuting the factors in the representation of this prime as product of primes of algebraic extension in which the prime splits.

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 2-surfaces, and could be important in the intersection of real and p-adic 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. Quantum Adeles would make these representations very concrete by bringing in cognition represented in terms of quantum p-adics.

Quantum Adeles could allow to realize number theoretical universality in TGD framework and would be essential in the construction of generalized Feynman diagrams as amplitudes in the tensor product of state spaces assignable to real and p-adic number fields. Canonical identification would allow to map the amplitudes to reals and complex numbers. Quantum Adeles also provide a fresh view to conjectured M^{8}-M^{4}×CP_{2} duality, and the two suggested realizations for the decomposition of space-time surfaces to associative/quaternionic and co-associative/co-quaternionic regions.

For detais see the new chapter Quantum Adeles of "Physics as Generalized Number Theory".

## 5 comments:

Matti,

I mentioned you on my blog:

"A personal or social consequence in that I have had some direction on the frontiers with fellow bloggers is that in the case of Matti Pitkanen who's lifework I imagine is as hard to read like mine and understand, where he relates consciousness as dark matter phenomena, there is a similarity of these ideas possible in that the idea while to me vague or vaguely communicated or some lack on my part consequently has some weight after all on the frontier of our age of new physics."

Now, your current topic here does show some general sense of advanced ideas in the depths of things but it was quite a wormhole for me to make the connection as something rather more concrete. Such effects, such a way to see numbers- and well, for me something a little further than a quantum foundation as such.

What do you think? I hope this informal post Universal Mind, can help... I hope we come to understand consciousness a little more than we are now aware of but cannot express in words or formuli.

But while on the science blogs any mention of effects like this for your topic was immediately dismissed as not scientific. We know better :-)

The Pe Sla

The mathematics behind the current topic is really hard for any-one. Adeles appear in mathematics in the framework of Langlands program and reading even the introductory text by Frenkel is really hard work. No-one would bother to read them unless he were an authority in the field. Usually these things are of course learned in face-to-face communications but this is not possible for those thrown outside the system.

I have not even tried to really understand intuitively ideas of the geometric Langlands. I just understand a little bit about Galois group and Adeles. The only reason why I make attempts to build something like Quantum Adeles with my minor skills is and without the help of helpful discussions with colleagues (because they are not possible) is that it is something which quantum TGD quite obviously needs.

I have got accustomed to the stupidity of the average colleague and it bothers me just enough to guarantee the optimal adrenaline level;-). Average colleague has to earn his living and it requires some ethical compromises to get a formal status of expert.

Academic world is great theatre. Most of those appearing on the stage as professionals are nothing but good actors. And as actors they are able to make the most impressive gestures;-). When you hear some authority to ridicule cold fusion, or claim all theoreticians not believing in M-theory as idiots, you have met an actor.

.Dear Matti,

Yesterday, I struggle with Quantum Arithmetics, Absolute Galois Group and Quantum adeles, at end I confused a lot and didn't understand them:( but it was interesting. Perhaps it is enough for me a concise overall view about them, at this level. And I should not hurry :-)

-TGD leads to existence of U-matrix in two main ways (?):

1-In a non-straight way: Non-determinism of Kahler action in real WCW.

2-In a straight way: Generalization of WCW to a fusion of p-adic and real WCW.

Then What is U-Matrix in the view of Quantum adeles, briefly?

-How inertial-gravitational dichotomy is a direct correlate for the geometric-subjective

dichotomy for time?

Dear Hamed,

thank you for questions. I do my best to answer.

1. Quantum Arithmetics and related things are the newest layer in the development and I would not take too seriously anything about it before few years have passed and I (or maybe you;-)) have found and corrected all the mistakes or killed the idea;-).

Quantum Adeles are basically just a Cartesian product of quantum variants of p-adic number fields so that everything reduces to quantum p-adics.

Do these quantum p-adic fields really make sense mathematically as analogs of local number fields? Are they internally consistent structures: really number fields with all desired properties and does differential and integral

calculus make sense for them? Canonical identification would be the key element: it would define norm instead of p-adic norm and it would induced well-ordering making definite integral possible. Could canonical identification map solutions of quantum p-adic field equations to those of real field equations? This would be fantastic! Physical vision requires and strongly suggests the existence of quantum p-adics and I have not found any killer argument yet. Requires a lot of argumentation forth and back.

2. U-matrix follows from ZEO. M-matrices by definition entangle the positive and negative energy parts of zero energy states. U-matrix is a collection of mutually orthogonal M-matrices assignable to an orthogonalize WCW spinor field basis. Each M-matrix appearing as a row of U-matrix define one particular mode of WCW spinor field.

One could say that U-matrix and M-matrices characterize the modes of free spinor fields in WCW. All interactions reduce to WCW geometry and are coded by these free spinor fields.

Note that spinor field here is something different from what one might think on basis of finite-dimensional intuition. It is superposition of pairs of positive and negative energy states made out of fermions at opposite boundaryies of CD as functional of 3-surface defined by the ends of space-time surface.

3. I would not pose the questions about how TGD leads to U-matrix in the manner you did. Rather:

a) It is ZEO which is motivated by the non-determinism of Kaehler action rather than U-matrix.

b) Number theoretical universality of U-matrix is motivated by the generalization of WCW by fusion of real and p-adic WCWs.

Dear Hamed,

here an attempt to answer you remaining questions.

1. Quantum Adelic U-matrix would be quantum adelic valued spinor field in quantum adelic WCW.

If the notion of quantum adele makes sense, it provides a convenient manner to talk formally about entire collection of space-time sheets as a single object defined as quantum adele. The psychedelic quantum dream would be something like follows.

a) The basic rule would be simple: replaces real everywhere by quantum adeles.

b) Adelic WCW spinor fields would correspond to a tensor product of WCW spinor fields associated with various quantum p-adic sectors and would be quantum p-adic valued (also real sector would be included). One would have tensor product of Hilbert spaces belong to different (quantum) number fields. These quantum adelic quantum states would not have physical meaning as such. One would have entanglement between different number fields for instance.

c) To get real probability amplitudes one would map quantum adelic amplitudes to complex valued amplitudes by canonical identification. After that one could calculate probabilities using ordinary calculus.

2. Is inertial-gravitational dichotomy is a direct correlate for geometric-subjective dichotomy of time? I might have made this question somewhere some time. Certainly I would leave the question unanswered now;-). Inertial-gravitational dichotomy would correspond to the dichotomy with the geometric times assignable with the imbedding space and space-time surface. One could see the geometric time assigned to a space-time sheet as something "subjective" since it is associated with a particular physical system. The geometric time coordinate assignable to imbedding space could be seen as something "objective" not assignable to any particular physical subsystem. Maybe I am playing with the meanings of "subjective" and "objective" now;-). Be critical!

In any case, inertial-gravitational dichotomy is essentially the content of Equivalence Principle and whether EP is realized in TGD or not has been one of the most longstanding open problems. At this moment I believe that EP is realized in a generalized form and its Einsteinian form emerges only in long length scales. Just as in string models.

Coset representations for conformal algebras of symplective algebra of delta M^4_+xCP_2 (--inertial) and Kac-Moody algebras associated with light-like 3-surfaces (---gravitational) imply that inertial and gravitational quantum numbers are identical. Gravitational = inertial applies not only to four-momentum and mass but to all quantum numbers.

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