### Class field theory and TGD: does TGD reduce to number theory?

The intriguing general result of class field theory) -something extremely abstract for physicist's brain - is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this).

Does this mean that it is not absolutely necessary to introduce p-adic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also p-adic integers and numbers are. What is intriguing that p-dic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2-adic units consisting of bit sequences which can have literally infinite non-vanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and p-adic norm approaching to zero expresses the reduction of the insignificance.

**1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions?**

Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.

- Could real physics and various p-adic physics on one hand, and number theoretic physics based on maximal Abelian extension of rational octonions and quaternions on one hand, define equivalent formulations of physics?

- Besides various p-adic physics all classical number fields (reals, complex numbers, quaternions, and octonions) are central in the number theoretical vision about TGD. The technical problem is that p-adic quaternions and octonions exist only as a ring unless one poses some additional conditions. Is it possible to pose such conditions so that one could define what might be called quaternionic and octonionic adeles and ideles?

It will be found that this is the case: p-adic quaternions/octonions would be products of rational quaternions/octonions with a p-adic unit. This definition applies also to algebraic extensions of rationals and makes it possible to define the notion of derivative for corresponding adeles. Furthermore, the rational quaternions define non-commutative automorphisms of quaternions and rational octonions at least formally define a non-associative analog of group of octonionic automorphisms (see this).

- I have already earlier considered the idea about Galois group as the ultimate symmetry group of physics. The representations of Galois group of maximal Abelian extension (or even that for algebraic numbers) would define the quantum states. The representation space could be group algebra of the Galois group and in Abelian case equivalently the group algebra of ideles or adeles. One would have wave functions in the space of ideles.

The Galois group of maximal Abelian extension would be the Cartan subgroup of the absolute Galois group of algebraic numbers associated with given extension of rationals and it would be natural to classify the quantum states by the corresponding quantum numbers (number theoretic observables).

If octonionic and quaternionic (associative) adeles make sense, the associativity condition would reduce the analogs of wave functions to those at 4-dimensional associative sub-manifolds of octonionic adeles identifable as space-time surfaces so that also space-time physics in various number fields would result as representations of Galois group in the maximal Abelian Galois group of rational octonions/quaternions. TGD would reduce to classical number theory!

- Absolute Galois group is the Galois group of the maximal algebraic extension and as such a poorly defined concept. One can however consider the hierarchy of all finite-dimensional algebraic extensions (including non-Abelian ones) and maximal Abelian extensions associated with these and obtain in this manner a hierarchy of physics defined as representations of these Galois groups homomorphic with the corresponding idele groups.

- In this approach the symmetries of the theory would have automatically adelic representations and one might hope about connection with Langlands program.

**2. Adelic variant of space-time dynamics and spinorial dynamics?**

As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the space-time dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).

- Could one think that adeles or ideles could extend reals in the formulation of the theory: note that reals are included as Cartesian factor to adeles. Could one speak about adelic or even idelic space-time surfaces endowed with adelic or idelic coordinates? Could one formulate variational principle in terms of adeles so that exponent of action would be product of actions exponents associated with various factors with Neper number replaced by p for Z
_{p}. The minimal interpretation would be that in adelic picture one collects under the same umbrella real physics and various p-adic physics.

- Number theoretic vision suggests that 4:th/8:th Cartesian powers of adeles have interpretation as adelic variants of quaternions/ octonions. If so, one can ask whether adelic quaternions and octonions could have some number theretical meaning. Note that adelic quaternions and octonions are not number fields without additional assumptions since the moduli squared for a p-adic analog of quaternion and octonion can vanish so that the inverse fails to exist.

If one can pose a condition guaranteing the existence of inverse, one could define the multiplicative group of ideles for quaternions. For octonions one would obtain non-associative analog of the multiplicative group. If this kind of structures exist then four-dimensional associative/co-associative sub-manifolds in the space of non-associative ideles define associative/co-associative ideles and one would end up with ideles formed by associative and

co-associative space-time surfaces.

- What about equations for space-time surfaces. Do field equations reduce to separate field equations for each factor? Can one pose as an additional condition the constraint that p-adic surfaces provide in some sense cognitive representations of real space-time surfaces: this idea is formulated more precisely in terms of p-adic manifold concept (see this). Or is this correspondence an outcome of evolution?

Physical intuition would suggest that in most p-adic factors space-time surface corresponds to a point, or at least to a vacuum extremal. One can consider also the possibility that same algebraic equation describes the surface in various factors of the adele. Could this hold true in the intersection of real and p-adic worlds for which rationals appear in the polynomials defining the preferred extremals.

- To define field equations one must have the notion of derivative. Derivative is an operation involving division and can be tricky since adeles are not number field. If one can guarantee that the p-adic variants of octonions and quaternions are number fields, there are good hopes about well-defined derivative. Derivative as limiting value df/dx= lim ( f(x+dx)-f(x))/dx for a function decomposing to Cartesian product of real function f(x) and p-adic valued functions f
_{p}(x_{p}) would require that f_{p}(x) is non-constant only for a finite number of primes: this is in accordance with the physical picture that only finite number of p-adic primes are active and define "cognitive representations" of real space-time surface. The second condition is that dx is proportional to product dx × ∏ dx_{p}of differentials dx and dx_{p}, which are rational numbers. dx goes to xero as a real number but not p-adically for any of the primes involved. dx_{p}in turn goes to zero p-adically only for Q_{p}.

- The idea about rationals as points commont to all number fields is central in number theoretical vision. This vision is realized for adeles in the minimal sense that the action of rationals is well-defined in all Cartesian factors of the adeles. Number theoretical vision allows also to talk about common rational points of real and various p-adic space-time surfaces in preferred coordinate choices made possible by symmetries of the imbedding space, and one ends up to the vision about life as something residing in the intersection of real and p-adic number fields. It is not clear whether and how adeles could allow to formulate this idea.

- For adelic variants of imbedding space spinors Cartesian product of real and p-adc variants of imbedding spaces is mapped to their tensor product. This gives justification for the physical vision that various p-adic physics appear as tensor factors. Does this mean that the generalized induced spinors are infinite tensor products of real and various p-adic spinors and Clifford algebra generated by induced gamma matrices is obtained by tensor product construction? Does the generalization of massless Dirac equation reduce to a sum of d'Alembertians for the factors? Does each of them annihilate the appropriate spinor? If only finite number of Cartesian factors corresponds to a space-time surface which is not vacuum extremal vanishing induced Kähler form, Kähler Dirac equation is non-trivial only in finite number of adelic factors.

**3. Objections**

The basic idea is that apporopriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.

- The first objection is that p-adic quaternions and octonions do not make sense since p-adic variants of quaternions and octonions do not exist in general. The reason is that the p-adic norm squared ∑ x
_{i}^{2}for p-adic variant of quaternion, octonion, or even complex number can vanish so that its inverse does not exist.

- Second objection is that automorphisms of the ring of quaternions (octonions) in the maximal Abelian extension are products of transformations of the subgroup of SO(3) (G
_{2}) represented by matrices with elements in the extension and in the Galois group of the extension itself. Ideles separate out as 1-dimensional Cartesian factor from this group so that one does not obtain 4-field (8-fold) Cartesian power of this Galois group.

- This condition indeed allows to construct the inverse of p-adic quaternion/octonion as a product of inverses for rational quaternion/octonion and p-adic number! The reason is that the solutions to ∑ x
_{i}^{2}=0 involve always p-adic numbers with an infinite number of pinary digits - at least one and the identification excludes this possibility.

- This restriction would give a rather precise content for the idea of rational physics since all p-adic space-time surfaces would have a rational backbone in well-defined sense.

- One can interpret also the quaternionicity/octonionicity in terms of Galois group. The 7-dimensional non-associative counterparts for octonionic automorphisms act as transformations x→ gxg
^{-1}. Therefore octonions represent this group like structure and the p-adic octonions would have interpretation as combination of octonionic automorphisms with those of rationals.

Adelic variants of of octonions would represent a generalization of these transformations so that they would act in all number fields. Quaternionic 4-surfaces would define associative local sub-groups of this group-like structure. Thus a generalization of symmetry concept reducing for solutions of field equations to the standard one would allow to realize the vision about the reduction of physics to number theory.

For background see the chapter About Absolute Galois group of "TGD as Generalized Number Theory".

## 17 Comments:

Perleman and Poincare conjencture , Ricci flow. http://arxiv.org/pdf/0803.0150.pdf

Can you explain the difference between Kähler geometry and Kähler-Einstein geometry in a simple way?

Hi,

I assume that you have idea about what complex numbers are and what is imaginary unit. It is a number whose square is -1: i^2=-1. i cannot be a real number, hence "imaginary".

Kahler geometry requires a geometric representation of i. Here some knowledge of tensor analysis and basics of geometry would required. i is represented as geometric operation in tangent space (again something new) of Kahler manifold. In complex plane it is just reflection with respect to x-axis: (x,y) ---(x,-y), which can be visualised. This something concrete, not imaginary anymore.

Kahler Einstein geometry requires also that Einstein tensor which by Einsteins equations would be proportional to energy momentum tensor is proportional to metric. Very conscisely: G= k*g. Standard sphere is Kahler Einstein geometry and also symmetric space: all points are equivalent geometrically as is easy to understand. Also CP_2 is such a geometry.It has also quaternion structure but this is another story.

Hi,

I assume that you have idea about what complex numbers are and what is imaginary unit. It is a number whose square is -1: i^2=-1. i cannot be a real number, hence "imaginary".

Kahler geometry requires a geometric representation of i. Here some knowledge of tensor analysis and basics of geometry would required. i is represented as geometric operation in tangent space (again something new) of Kahler manifold. In complex plane it is just reflection with respect to x-axis: (x,y) ---(x,-y), which can be visualised. This something concrete, not imaginary anymore.

Kahler Einstein geometry requires also that Einstein tensor which by Einsteins equations would be proportional to energy momentum tensor is proportional to metric. Very conscisely: G= k*g. Standard sphere is Kahler Einstein geometry and also symmetric space: all points are equivalent geometrically as is easy to understand. Also CP_2 is such a geometry.It has also quaternion structure but this is another story.

http://en.wikipedia.org/wiki/K%C3%A4hler%E2%80%93Einstein_metric

here are three possibilities, and they are all prooved? http://mathworld.wolfram.com/news/2003-04-15/poincare/

Is this like the em-field with a torus? It has a quaternion structure too. This is also linked to Fano manifolds, hence atoms.

I also have problems understanding the term "preferred". Is that your own term? Calabi only talked of extremals? From what did it come and how is the criticality done exactly? (without big math :)). If the criticality depends on i (Kähler geometry, linked with p-adics?) it forms a circle (octonion) as Baez writes about? Or knot.

When is Kähler geometry used and when Kähler-Einstein geometry? Is it only when we link in another space (i) that we get Kähler geometry? What is a wormhole or Dirac fermion made of as instance?

This is not my field of expertise, so I hope you get the meaning of my questions.

When the text says "a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric" does this mean the same cone can have both Kähler metrics in one end and Einstein-metrics in the other end of the cone/manifold/surface, or is this case always flat, and the i is required to get the curvature?

This is an example of what I mean. http://suess.sdf-eu.org/website/lang/en/research.php

"I became interested in the question of the existence of Kähler-Einstein metrics on Fano T-varieties." see the fig. How is the Zeeman effect realized, and the different 'shells' and sizes of atoms, as seen as ions (momentum change, Cooper-pairs?) etc. It cannot be only an effect of hbar or p-adics seen as hierarchy (maybe the n-shells are a hierarchy?)?

You have talked of cyclotron-frequencies, but I don't quite grasp that...

Maybe these are too complex questions?

http://iopscience.iop.org/0264-9381/25/22/222002/fulltext/ Ricci flows and wormholes. I guess the ERR model is outdated?

To Ulla:

1. Perelman's proof of Poincare conjecture is accepted. It involves volume flow for a metric defined by its Einstein tensor and this leads to

constant curvature spaces asymptotically. The analog was well-known in 2-D case: sphere, torus,etc allows constant curvature metric too.

One could say that the flow as irreversible process does what approach to thermal equilibrium does: all irrelevant details are polished out and the simple constant curvature metric remains.

What is also physically interesting that the constant curvature metric of manifold classifies it topologically and that most 3-manifolds allow this kind of metric. There would be a deep connection between Riemannian geometry and topology. In TGD framework preferred externals - analogs of Bohr orbits- could also define 4-D topological invariants since one can hope that for given space-time topology the preferred extremal is unique.

[It can actually happen, that the preferred externals are not quite unique but this is a delicacy.]

Constant curvature hyperbolic 3-manifolds might be relevant for TGD too. One can take the proper time constant hyperboloid of M^4: t^2-x^2y^2-z^2=a^2, choose one, call it G, of the infinite number of discrete subgroups of Lorentz group L acting in this space and form a coset space L/G.

The resulting space is hyperbolic constant curvature manifold. The subgroups of Lorentz group acting in 3-D hyperbolic space could be completely analogous to lattice groups in Euclidian 3-space. The constant curvature spaces would be analogous to cubic lattice cells.

The Lorentz boosts in the subgroup of Lorentz group in turn would be analogous to the discrete lattice of translations. There are indications that

cosmic redshifts are quantized: my proposal is that the allowed redshifts correspond to the analog of lattice defined this subgroup. The distant objects would recede with quantized velocities. Astrophysical objects would form lattice like structure. This is a testable proposal.

2. Preferred extremal is purely TGD based notion.

It is analogous to Bohr orbit and forced by the condition that the WCW Kahler metric in the space of 3-surfaces has general coordinate invariant in 4-D sense.

This is possible only if the very definition of Kahler function assigns to 3-surface a unique 4-surface. Kahler function is the Kahler action for a preferred extremal (space-time surface) connecting 3-surfaces at the ends of causal diamond. Kahler action in turn has direct physical interpretation so that classical physics is coded by WCW Kahler function.

3. That manifold has both Kahler metric and Einstein metric expresses only the fact that Kahler manifold as Ricci tensor proportional to metric tensor. This is true for instance for constant curvature spaces/symmetric spaces.

4. I do not believe that http://suess.sdf-eu.org/website/lang/en/research.php gas anything to do with Zeeman effect or physics.

This high specialised and technical algebraic geometry: a luxury to which physicists trying to achieve something during this lifetime cannot afford;-).

To sum up, most of the questions have basically rather trivial answer: the problem is the lack of context which makes it very difficult to explain what it involved in an understandable manner: you see that I am forced to introduce additional notions which are standard but would require a further explanation so that the explanation would grow to a two-semester course in basic mathematics;-). To answer understandably without the context would be a Muenchausen trick;-).

Matti, I "understood" all of that... nice work... I am continually amazed by your

, but I am also a bit dismayed that I understand because it seems useless to understand if no else does? or not many rather. like, it is deemed "weird" to understand ... but that's more of a social issue I guess. Peace' can u summarize the current set of open questions in TGD? also yes... preferred... lol... by who...

Hi,

it is lonely at the top;-). Personally I am driven by the passion to understand. Basically it does not matter much to me whether I am the only one who understands my life work. I am of course happy that there are at least few who have some idea about the importance of my work. I do my best to share my work.

It is really wonderful to develop a new world view and know that it will be the world view of future.

What makes me sad is that recent academic science is to high extent CV production. But what else Big Science could be. The degeneration of our society is taking place both in economy, academies, and society in general, and one can make only guesses about how long the new Middle Age lasts. People like me are for science what mystics are for religion. We keep it alive over the dark periods.

Concerning your questions. One class of open questions relates to the mathematical side. All this is very technical and personally I can develop only overall view. I develops slowly.

*More detailed understanding of WCW geometry would be needed: I am now going through the entire existing material to update it and considerable progress has occurred.

*I have a handful of characterisations of preferred extremals. Are they equivalent?

*Adelic formulation of the theory seems to be very promising and here also much could be done. But people handling the needed enormously complex mathematics would be needed. I can provide only the physical insight.

*More detailed understanding of stringy twistor approach to TGD would be needed: here I am too old to do anything detailed: just the understanding of the general vision would be enough for me.

Very fascinating result is that M^4 and CP_2 are the only 4-D manifolds allowing twistor space with Kahler structure. What does this imply?

*One should go through p-adic thermodynamics by applying the recent model of elementary particles as pairs of wormhole contacts. I am too old to carry out the calculations again.

Concerning particle physics.

* LHC will probably provide new interesting findings and here M_89 hadron physics is the

most interesting issue.

*Massivation of gauge bosons and Higgs should be understood better. The basic difference between TGD is that Higgs is there but Higgs mechanism is replaced with p-adic thermodynamics and Higgs couplings to fermions are just gradient couplings: this gives naturalness automatically.

*TGD view about SUSY is something that I do not

understand well enough. Progress probably requires rather new insights which as such could be rather trivial. All revolves around right handed neutrino: I have pieces but I am unable to put them together.

In consciousness theory and quantum biology I am rather happy with the recent situation. The hardest problem has been the understanding of time and now I can safely say that it is understood. The recent model for microtubuli gave fantastic insights to how topological quantum computation (TQC) like processes form the core of quantum biology. Even understanding of the origin of genetic code might be possible. One new element is generalisation TQC using 1-braids to that using 2-braids: string world sheets are knotted in 4-D space-time!

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I have studied TGD in so many years, and I thought now, after reading about other models, I would finally be ready to understand it a bit better. I am sorry if my questions are not easily grasped.

I want to have a more clear view of the analog to Bohr model. The preferred extremals come from it. Are in your texts a good description of this analog. I have searched but when there is so much text.

Also about how the criticality and stability is done in TGD I would be happy to have a better text.

Bosonic strings are massless. Seen in a wormhole or Dirac fermione, how is that mechanism between fermions and bosons made?

You know I am no enemy, at least you should know. I have helped you what I can.

To Ulla:

I am sorry that my answers are not easy to grasp. This not my or anyone else's fault. There is so much context lacking and I have only words which have strongly context dependent meaning.

Bosons in the simplest - not yet quite correct - description is pair of massless fermion and antifermion which however do not move quite parallel so that boson can become massive. What is important are not details, but the idea of bosonic emergence. Only fundamental fermions exist. From these one can engineer observed fermions and bosons.

Strings as such are not massless: dominant part of weak boson mass would come from the stringy contribution to mass.

Detailed understanding of weak boson masses and Higgs has been the most difficult exercise in application of TGD, which are of course amateurish when seen from the Godly CERN perspective.

My views have been fluctuation between all imaginable alternatives. Thank for LHC making end of this fluctuation! Theoreticians would be totally helpless without the helping had of experimentalist!

"What is important are not details, but the idea of bosonic emergence. Only fundamental fermions exist. From these one can engineer observed fermions and bosons."

Charge conservation when bosons are carriers for the gauge field???? This means that the charge conservation IS false dogma?

To Ulla:

Certainly not. I cannot why this should be the case.

Dear Matti,

Suppose there is an electron in a system. In standard QFT, in really there is not just this electron, but there are a lot of particles when one goes to higher and higher energies.

In TGD, there is space of 3-surfaces. But is there any different between high and low energy? When 3-surface like particle is replaced with point like particle, this is similar to high energy QFT, that there are a lot of point like particles?

In macroscopic, when there is a system of gas with atoms of gas as smaller 3-surfaces glued to space-time sheet of the system, one can say in the viewpoint of TGD, there is space of 3-surfaces. These 3-surfaces corresponded to atoms of the gas.

Now!, Let’s we pick up the particles of the object one after one. Hence the number of 3-surfaces in the WCW reduced in the process.

But I think the number of 3-surfaces in WCW, is a mathematical property of WCW and is not related to how many atoms of gas there is in the system in the external world.

What is my misunderstanding?

I think that the misunderstanding is following.

In ZEO WCW is space of all 3-surfaces: consisting of pairs of 3-surfaces at boundaries of given CD.

At each boundary 3 surface as several disjoint components. This kind of multicomponent 3-surface represents a single point in WCW.

WCW decomposes to sectors labelled by the numbers of 3-surfaces at the two boundaries of CD and the moduli characterising the CD. Your thought experiments means only moving from a sector of WCW to another one and having different numbers of 3-surfaces at the boundaries of CD.

Locally the given sector of WCW is Cartesian product of sectors containing only single 3-surface. Just as in many particle physics one effectively has configuration spar (E^3)^n. This approximation fails when the 3-surfaces touch.

This is however improbable since 3-surfaces in 7-D space delta M^4_+xCP_2 are in question. Intersection disappears with arbitrary small deformation.

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