Hyper-finite factors and construction of S-matrix

During years I have spent a lot of time and effort to attempts to imagine various options for the construction of S-matrix. Contrary to my original belief, the real problem has not been the lack of my analytic skills but the failure of ordinary QFT based thinking in TGD framework.

Super-conformal symmetries generalized from string model context to TGD framework are symmetries of S-matrix. This is very powerful constraint to S-matrix but useless unless one has precisely defined ontology translated to a rigorous mathematical framework. The zero energy ontology of TGD is now rather well understood but differs dramatically from that of standard quantum field theories. Second deep difference is that path integral formalism is given up and the goal is to construct S-matrix as a generalization of braiding S-matrices with reaction vertices replaced with the replication of number theoretic braids associated with partonic 2-surfaces taking the role of vertices. Also number theoretic universality requiring fusion of real physics and various p-adic physics to single coherent whole is a completely new element.

The most recent vision about S-matrix combines ideas scattered in various chapters of various books and often drowned with details. A very brief summary would be as follows.

1. In TGD framework functional integral formalism is given up. S-matrix should be constructible as a generalization of braiding S-matrix in such a manner that the number theoretic braids assignable to light-like partonic 3-surfaces glued along their ends at 2-dimensional partonic 2-surfaces representing reaction vertices replicate in the vertex. This means a replacement of the free dynamics of point particles of quantum field theories with braiding dynamics associated with partonic 2-surfaces carrying braids and the replacement of particle creation with the creation of partons and replication of braids.

2. The construction of braiding S-matrices assignable to the incoming and outgoing partonic 2-surfaces is not a problem. The problem is to express mathematically what happens in the vertex. Here the observation that the tensor product of hyper-finite factors (HFFs) of type II is HFF of type II is the key observation. Many-parton vertex can be identified as a unitary isomorphism between the tensor product of incoming resp. outgoing HFFs. A reduction to HFF of type II₁ occurs because only a finite-dimensional projection of S-matrix in bosonic degrees of freedom defines a normalizable state. Most importantly, unitarity and non-triviality of S-matrix follows trivially.

3. HFFs of type III could also appear at the level of field operators used to create states but that at the level of quantum states everything reduces to HFFs of type II₁ and their tensor products giving these factors back. If braiding automorphisms reduce to the purely intrinsic unitary automorphisms of HFFs of type III then for certain values of the time parameter of automorphism having interpretation as a scaling parameter these automorphisms are trivial. These time scales could correspond to p-adic time scales so that p-adic length scale hypothesis would emerge at the fundamental level. In this kind of situation the braiding S-matrices associated with the incoming and outgoing partons could be trivial so that everything would reduce to this unitary isomorphism: a counterpart for the elimination of external legs from Feynman diagram in QFT. p-Adic thermodynamics and particle massivation could be also obtained when the time parameter of the automorphism is allowed to be complex as a generalization of thermal QFT.

4. One might hope that all complications related to what happens for space-like 3-surfaces could be eliminated by quantum classical correspondence stating that space-time view about particle reaction is only a space-time correlate for what happens in quantum fluctuating degrees of freedom associated with partonic 2-surfaces. This turns out to be the case only in non-perturbative phase. The reason is that the arguments of n-point function appear as continuous moduli of Kähler function. In non-perturbative phases the dependence of the maximum of Kähler function on the arguments of n-point function cannot be regarded as negligible and Kähler function becomes the key to the understanding of these effects including formation of bound states and color confinement.

5. In this picture light-like 3-surface would take the dual role as a correlate for both state and time evolution of state and this dual role allows to understand why the restriction of time like entanglement to that described by S-matrix must be made. For fixed values of moduli each reaction would correspond to a minimal braid diagram involving exchanges of partons being in one-one correspondence with a maximum of Kähler function. By quantum criticality and the requirement of ideal quantum-classical correspondence only one such diagram would contribute for given values of moduli. Coupling constant evolution would not be however lost: it would be realized as p-adic coupling constant at the level of free states via the log(p) scaling of eigen modes of the modified Dirac operator.

6. A completely unexpected prediction deserving a special emphasis is that number theoretic braids replicate in vertices. This is of course the braid counterpart for the introduction of annihilation and creation of particles in the transition from free QFT to an interacting one. This means classical replication of the number theoretic information carried by them. This allows to interpret one of the TGD inspired models of genetic code in terms of number theoretic braids representing at deeper level the information carried by DNA. This picture provides also further support for the proposal that DNA acts as topological quantum computer utilizing braids associated with partonic light-like 3-surfaces (which can have arbitrary size). In the reverse direction one must conclude that even elementary particles could be information processing and communicating entities in TGD Universe.

To sum up, my personal feeling is that the constraints identified hitherto might lead to a more or less unique final result and I can only hope that some young analytically blessed brain would bother to transform this picture to concrete calculational recipes.

For more details see the new chapter Hyper-Finite Factors and Construction of S-matrix of "Towards S-matrix".

About dogmas and world view as a disease

The PEAR Lab (Princeton Engineering Anomalies Research Lab) will be closing at the end of February of 2007. It is regrettable that the experimental research challenging the cherished dogmas of our scientific world view is not allowed to continue.

About PEAR

The research group was directed by Robert G. Jahn and studied both machine mind interactions and remote perception. Reader can find a brief description of these experiments in Wikipedia but just to make clear for myself what is involved I see the trouble of reproducing the description of machine mind interaction experiments here.

REG (Random Event Generator) experiment serves as a prototype for machine-mind interaction experiment.

1. Random noise was sampled with given frequency which varied from experiment to experiment and the outcome was coded into a bit.

2. The experiment involved three different intentions. Intention to produce bits 1 ("high"), bits 0 ("low"), and observing the data generation without any effort to affect the outcome.

3. The operator reported in the beginning of each trial her intention. The result of a particular trial was r= 200N(bit=1)/N(bits). In the absence of any effect the result should have been r= 100. The result of the run involving 8×10^5 trials per intention with 200 bits per trial was r(high)=100.026 and r(low)=99.984. The difference corresponds to 3.8 or 3.8 standard deviations. The deviation is 3.8 times large than the expected margin of error in the measurement and can be regarded as statistically significant.

This experiment served as a template for several other experiments such as remote experiment in which the device was influenced from distance; pseudo experiments in which operator was replaced with random analog noise source; a random mechanical cascade in which experimenter tried to affect the trajectories of macroscopic polystyrene balls falling through an array of pegs; a pendulum experiment in which operator tried to affect to motion of pendulum.

The conclusions were following. Human mind can affect random physical processes to a small but statistically significant degree. The effect seems to disappear when genuinely random sources are replaced with deterministic one (pseudo random sources). Different individuals produce different results. The effect shows long term fluctuations, which can be partly but not completely explained by changes in the operator pool.

Could quantum critical systems be more interesting than machines?

From the point of view of TGD inspired theory of consciousness, the attempt to affect random noise is certainly not the optimal experimental situation if one wants to detect strong effects. The optimal choice of system to be affected by intentional action would be a macroscopic quantum critical system. In TGD Universe high temperature super-conductors would be one example of such systems. Another system of this kind would be capacitor very near to the voltage at which di-electric breakdown occurs. Cell membrane provide one example of this kind of system.TGD also predicts that dark matter corresponds to a hierarchy of macroscopic quantum phases with increasing value of Planck constant responsible also for the very special properties of living matter.

The power of dogma

In TGD Universe, my own biological body would be a quantum critical system and I could argue that I experience the effects of intentions on living matter every day! I think that many readers would agree with me. Why should then people be burned on stake for suggesting that our intentions might have small effects on the material world outside our biological body? One can understand this irrational behavior only by realizing how enormous is the power of dogma.

So what happens when I raise my hand according to scientific explanation? "I decide to raise my hand and it raises!" would be the spontaneous answer of an innocent layman. Wrong! According to the belief system of an orthodox materialistic scientist wishing to keep his job, there is no intentional action involved. The scientist admits that there is some small quantum mechanical non-determinism in atomic length scales and below but in human scales all these effects give just random background noise. The initial conditions at the moment of big bang just happened to be such that my hand raises and I experience the illusion of having an intention to raise my hand. The reason why I have this illusion of free will and intention is not completely clear to the materialist. My complex initial value sensitive system and for some funny reason initial value sensitive systems have a tendency to create this kind of illusions. But not intentionally of course!

This example should demonstrate how difficult the challenge of proving that our intentions can affect the world outside us really is. I would actually talk abot mission impossible. Materialistic scientist can always fabulate a story explaining the outcome of any intentional action, whether it affects his own body or external world, as resulting from a deterministic laws of physics. They can always claim that statistical methods used have some flaw and it is always possible to say there is fraud involved.

The dogmas are what really matter for the average scientists as any average person willing to survive socially. Materialistic and reductionistic dogmas declare that in length scales above atomic length scale quantum effects are negligible and the world is in practice deterministic. Quantum world in macro scales is a random soup of matter with no long range quantal correlations otherwise made possible by quantum entanglement. Who argues something different is a crackpot (although I have heard this word so many times in physics blogs it still makes me almost puke!).

Dogmas and mathematics

The basic dogmas materialize themselves also mathematically. In standard quantum mechanics based on von Neumann algebras known as factors of type I. These algebras apply to the quantum theory of simple systems with finite number of degrees of freedom. Hydrogen atom is the classical example.

This is however not the only mathematical possibility. Von Neumann algebras known as hyper-finite factors of type II₁ about which I have been talking a lot during last two years are the mathematics for a quantum universe which behaves as a single cosmic organism. Quantum entanglement over arbitrarily long spatial (and temporal!) scale is always present and can be reduced only partially. That Planck constant can have arbitrarily large values realizes this as a new element of quantum physics. Everything is connected with everything. One of the basic implications at the level of consciousness theory are sharing and fusion of mental effects and collective pool of mental images. All this disgusting new age stuff is realized mathematically and even worse: the person responsible for all this scandalous mathematics is the father of classical computer architecture: you cannot trust anyone nowadays!

Theory is regarded as successful if it explains empirical facts. Eastern meditation practices are the empirical study of consciousness and this picture indeed confirms to surprising degree with the views about consciousness provided by these practices. Of course, this picture explains the facts about brain produced by western science: in particular a successful model for EEG emerges.

Dogmas materialize themselves even in the notion of number used. For physicists basically only the magnitude of the number matters, not its number theoretic anatomy. Numbers tell what some things weighs, nothing else. In fact that number theory has become basic element of TGD inspired theory of consciousness and number theoretical complexity becomes a quantitative measure for cognitive level.

Beliefs in crisis

All is after all about beliefs: which beliefs we raise to dogmas as we try to make sense of the world around us. Beliefs have however a finite life span. The explanatory power is what matters in the long run. Entire societies fall down when everyone knows that everyone knows that dogma is wrong. I strongly feel that materialistic and reductionistic dogma has now reached the end of its life span.

During last three decades the materialistic dogmatics combined with American pragmatism (kind of analog of quartal economy in science) has led to the deepest crisis that theoretical physics has ever experienced. The fate of super string theorists was to devote their lifetime for a theory which was not a beginning of something fantastic but an unavoidable culmination for the world view based on dogmas which do not work anymore. Super-string gurus cannot but continue to declare that tiny little strings of size Planck length (reductionism!) are the ultimate building blocks of matter. It does not matter that in its recent form this theory cannot even predict the dimension of space-time to say nothing about what we observe in laboratory! The reaction of the most fanatic gurus is that since this theory is the only possible one (by some misty arguments involving usually big names) we must give up the idea that physics can predict something. The reader has probably detected the deep irony here: after all, the materialistic dogma was based on the idea of complete predictability!

During these 28 years as out-of-law in science I have pondered many times how this kind of incredibly irrational behavior is possible. The people behind these prejudices do not look like lunatics, fanatics, or series killers. The only explanation I can imagine is that we are not able to tolerate social pressures. That ordinary decent people can by forced by social pressures to believe and even do almost everything has been demonstrated in many experiments. As an example consider the following social experiment. There is a group of subject persons. As a matter fact, some of them are actors and take the role of an influental social leader. These people are asked to tell what 1+1 is but in such manner that everyone knowns the answers of others. Actors tell first their opinion which is 1+1 =3. Surprisingly many of participants cannot but agree after a painful internal battle!

Spiritual IQ and world view as a disease

Some of us are more able to resist social pressures. How do they differ from the rest of us? I suggest a simple three-letter answer: SIQ, Spiritual Intelligence Quotient. At the level of individual the growth of SIQ often occurs through a turning point experiences in which previous world view is dramatically transformed and one realizes that old certainties are not much more than a result of a need to gain social acceptance. After the great change one central theme often rules the life of these people: the puzzle of consciousness. Just the mere attempt to understand this mystery can give a full meaning to the life and the realization that there are things larger than biological life gradually gives the courage to insist also the magic power of social pressures. Almost as a rule these people gain the respect of the people who know them personally.

Just to see how SIQ manifests itself in social situations go to a typical physics blog (if you want almost physical violence choose the blog from US: you can start from Not-Even-Wrong or Reference Frame and follow the links to other blogs), and follow the discussion or postings. You will find that a considerable portion of debate consists of exchanges of direct personal insults. You soon realize that many of the participants cannot be very happy: there is too much frustration, aggression, and arrogance. Of course, this kind of direct violence is not possible in everyday academic life but from personal experience I can tell that academic people are masters of the refined forms of implicit violence.

My explanation is that wrong world view is the disease that these people are suffering. The universe of materialistic and reductionistic science lacks both purpose and meaning. In this kind of world human life span is not more than a random spark between two darknesses. No wonder that a person believing that he really lives in this kind of world becomes sick.

To test this hypothesis you can make a comparison with a blog where SIQ should be higher at least if the interest in the mystery of consciousness is accepted as a rough criterion. For instance, you can go to the blog Conscious Entities. You find nice articles pondering various ideas about consciousness; discussions are civilized; there are no sudden bursts of violence; there are no ad hominem attacks. It seems that on the average these people are happy; they have a passion in their life but they are not fanatics; and they do not have any need to tell to the rest of the world that they belong to some kind of super-species and all those who think differently are crackpots (or "social waste", one of the newest verbal fruits in physics blog discussions!).

Jones inclusions and construction of S-matrix and U matrix

TGD leads naturally to zero energy ontology which reduces to the positive energy ontology of the standard model only as a limiting case. In this framework one must distinguish between the U-matrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the S-matrix defining time-like entanglement between positive and negative energy parts of the zero energy state and coding the rates for particle reactions which in TGD framework correspond to quantum measurements reducing time-like entanglement.

1. S-matrix

In zero energy ontology S-matrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS⁺)=Tr(Id)=1 holds true). S-matrix would code for transition rates measured in particle physics experiments with particle reactions interpreted as quantum measurements reducing time like entanglement. In TGD inspired quantum measurement theory measurement resolution is characterized by Jones inclusion (the group G defines the measured quantum numbers), N subset M takes the role of complex numbers, and state function reduction leads to N ray in the space M/N regarded as N module and thus from a factor to a sub-factor.

The finite number theoretic braid having Galois group G as its symmetries is the space-time correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finite-dimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electro-weak quantum numbers in spinorial degrees of freedom. McKay correspondence would allow to assign to G also compact ADE type Lie group so that also Lie group type quantum numbers could be included in the repertoire.

Galois group G would characterize sub-spaces of the configuration space ("world of classical worlds") number theoretically in a manner analogous to the rough characterization of physical states by using topological quantum numbers. Each braid associated with a given partonic 2-surface would correspond to a particular G that the state would be characterized by a collection of groups G. G would act as symmetries of zero energy states and thus of S-matrix. S-matrix would reduce to a direct integral of S-matrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2-surfaces. It is not difficult to criticize this picture.

1. Why time like entanglement should be always characterized by a unitary S-matrix? Why not some more general matrix? If one allows more general time like entanglement, the description of particle reaction rates in terms of a unitary S-matrix must be replaced with something more general and would require a profound revision of the vision about the relationship between experiment and theory. Also the consistency of the zero energy ontology with positive energy ontology in time scales shorter than the time scale determined by the geometric time interval between positive and negative energy parts of the zero energy state would be lost. Hence the easy way to proceed is to postulate that the universe is self-referential in the sense that quantum states represent the laws of physics by coding S-matrix as entanglement coefficients.

2. Second objection is that there might a huge number of unitary S-matrices so that it would not be possible to speak about quantum laws of physics anymore. This need not be the case since super-conformal symmetries and number theoretic universality pose extremely powerful constraints on S-matrix. A highly attractive additional assumption is that S-matrix is universal in the sense that it is invariant under the inclusion sequences defined by Galois groups G associated with partonic 2-surfaces. Various constraints on S-matrix might actually imply the inclusion invariance.

3. One can of course ask why S-matrix should be invariant under inclusion. One might argue that zero energy states for which time-like entanglement is characterized by S-matrix invariant in the inclusion correspond to asymptotic self-organization patterns for which U-process and state function reduction do not affect the S-matrix in the relabelled basis. The analogy with a fractal asymptotic self-organization pattern is obvious.

2. U-matrix

In a well-defined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that U-matrix means a quantum transition in which a factor becomes a sub-factor whereas state function reduction would lead from a factor to a sub-factor.

Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing S-matrices for integrable quantum field theories in 2-dimensional Minkowski space. For these S-matrices particle scattering would mean only a permutation of momenta in momentum space. If S-matrix is invariant under inclusion then U matrix should be in a well-defined sense almost trivial apart from a dispersion in zero modes leading to a superpositions of states characterized by different collections of Galois groups.

3. Relation to TGD inspired theory of consciousness

U-matrix could be almost trivial with respect to the transitions which are diagonal with respect to the number field. What would however make U highly interesting is that it would predict the rates for the transitions representing a transformation of intention to action identified as a p-adic-to-real transition. In this context almost triviality would translate to a precise correlation between intention and action.

The general vision about the dynamics of quantum jumps suggests that the extension of a sub-factor to a factor is followed by a reduction to a sub-factor which is not necessarily the same. Breathing would be an excellent metaphor for the process. Breathing is also a metaphor for consciousness and life. Perhaps the essence of living systems distinguishing them from sub-systems with a fixed state space could be cyclic breathing like process N→ M supset N → N₁ subset M→ .. extending and reducing the state space of the sub-system by entanglement followed by de-entanglement.

One could even ask whether the unique role of breathing exercise in meditation practices relates directly to this basic dynamics of living systems and whether the effect of these practices is to increase the value of M:N and thus the order of Galois group G describing the algebraic complexity of "partonic" 2-surfaces involved (they can have arbitrarily large sizes). The basic hypothesis of TGD inspired theory of cognition indeed is that cognitive evolution corresponds to the growth of the dimension of the algebraic extension of p-adic numbers involved.

If one is willing to consider generalizations of the existing picture about quantum jump, one can imagine that unitary process can occur arbitrary number of times before it is followed by state function reduction. Unitary process and state function reduction could compete in this kind of situation.

4. Fractality of S-matrix and translational invariance in the lattice defined by sub-factors

Fractality realized as the invariance of the S-matrix in Jones inclusion means that the S-matrices of N and M relate by the projection P: M→N as S_N=PS_MP. S_N should be equivalent with S_M with a trivial re-labelling of strands of infinite braid.

Inclusion invariance would mean translational invariance of the S-matrix with respect to the index n labelling strands of braid defined by the projectors e_i. Translations would act only as a semigroup and S-matrix elements would depend on the difference m-n only. Transitions can occur only for m-n≥ 0, that is to the direction of increasing label of strand. The group G leaving N element-wise invariant would define the analog of a unit cell in lattice like condensed matter systems so that translational invariance would be obtained only for translations m→ m+ nk, where one has n≥ 0 and k is the number of M(2,C) factors defining the unit cell. As a matter fact, this picture might apply also to ordinary condensed matter systems.

For more details see the chapters Construction of Quantum Theory: S-matrix and Was von Neumann Right After All? of "Towards S-matrix".

Langlands Program and TGD

Number theoretic Langlands program can be seen as an attempt to unify number theory on one hand and theory of representations of reductive Lie groups on the other hand. So called automorphic functions to which various zeta functions are closely related define the common denominator. Geometric Langlands program tries to achieve a similar conceptual unification in the case of function fields. This program has caught the interest of physicists during last years.

TGD can be seen as an attempt to reduce physics to infinite-dimensional Kähler geometry and spinor structure of the "world of classical worlds" (WCW). Since TGD ce be regarded also as a generalized number theory, it is difficult to escape the idea that the interaction of Langlands program with TGD could be fruitful.

More concretely, TGD leads to a generalization of number concept based on the fusion of reals and various p-adic number fields and their extensions implying also generalization of manifold concept, which inspires the notion of number theoretic braid crucial for the formulation of quantum TGD. TGD leads also naturally to the notion of infinite primes and rationals. The identification of Clifford algebra of WCW as a hyper-finite factors of type II₁ in turn inspires further generalization of the notion of imbedding space and the idea that quantum TGD as a whole emerges from number theory. The ensuing generalization of the notion of imbedding space predicts a hierarchy of macroscopic quantum phases characterized by finite subgroups of SU(2) and by quantized Planck constant. All these new elements serve as potential sources of fresh insights.

1. The Galois group for the algebraic closure of rationals as infinite symmetric group?

The naive identification of the Galois groups for the algebraic closure of rationals would be as infinite symmetric group S_∞ consisting of finite permutations of the roots of a polynomial of infinite degree having infinite number of roots. What puts bells ringing is that the corresponding group algebra is nothing but the hyper-finite factor of type II₁ (HFF). One of the many avatars of this algebra is infinite-dimensional Clifford algebra playing key role in Quantum TGD. The projective representations of this algebra can be interpreted as representations of braid algebra B_∞ meaning a connection with the notion of number theoretical braid.

2. Representations of finite subgroups of S_∞ as outer automorphisms of HFFs

Finite-dimensional representations of Gal(\overline{Q}/Q) are crucial for Langlands program. Apart from one-dimensional representations complex finite-dimensional representations are not possible if S_∞ identification is accepted (there might exist finite-dimensional l-adic representations). This suggests that the finite-dimensional representations correspond to those for finite Galois groups and result through some kind of spontaneous breaking of S_∞ symmetry.

1. Sub-factors determined by finite groups G can be interpreted as representations of Galois groups or, rather infinite diagonal imbeddings of Galois groups to an infinite Cartesian power of S_n acting as outer automorphisms in HFF. These transformations are counterparts of global gauge transformations and determine the measured quantum numbers of gauge multiplets and thus measurement resolution. All the finite approximations of the representations are inner automorphisms but the limit does not belong to S_∞ and is therefore outer. An analogous picture applies in the case of infinite-dimensional Clifford algebra.

2. The physical interpretation is as a spontaneous breaking of S_∞ to a finite Galois group. One decomposes infinite braid to a series of n-braids such that finite Galois group acts in each n-braid in identical manner. Finite value of n corresponds to IR cutoff in physics in the sense that longer wave length quantum fluctuations are cut off. Finite measurement resolution is crucial. Now it applies to braid and corresponds in the language of new quantum measurement theory to a sub-factor N subset M determined by the finite Galois group G implying non-commutative physics with complex rays replaced by N rays. Braids give a connection to topological quantum field theories, conformal field theories (TGD is almost topological quantum field theory at parton level), knots, etc...

3. TGD based space-time correlate for the action of finite Galois groups on braids and for the cutoff is in terms of the number theoretic braids obtained as the intersection of real partonic 2-surface and its p-adic counterpart. The value of the p-adic prime p associated with the parton is fixed by the scaling of the eigenvalue spectrum of the modified Dirac operator (note that renormalization group evolution of coupling constants is characterized at the level free theory since p-adic prime characterizes the p-adic length scale). The roots of the polynomial would determine the positions of braid strands so that Galois group emerges naturally. As a matter fact, partonic 2-surface decomposes into regions, one for each braid transforming independently under its own Galois group. Entire quantum state is modular invariant, which brings in additional constraints.

   Braiding brings in homotopy group aspect crucial for geometric Langlands program. Another global aspect is related to the modular degrees of freedom of the partonic 2-surface, or more precisely to the regions of partonic 2-surface associated with braids. Sp(2g,R) (g is handle number) can act as transformations in modular degrees of freedom whereas its Langlands dual would act in spinorial degrees of freedom. The outcome would be a coupling between purely local and and global aspects which is necessary since otherwise all information about partonic 2-surfaces as basic objects would be lost. Interesting ramifications of the basic picture about why only three lowest genera correspond to the observed fermion families emerge.

3. Correspondence between finite groups and Lie groups

The correspondence between finite and Lie group is a basic aspect of Langlands.

1. Any amenable group gives rise to a unique sub-factor (in particular, compact Lie groups are amenable). These groups act as genuine outer automorphisms of the group algebra of S_∞ rather than being induced from S_∞ outer automorphism. If one gives up uniqueness, it seems that practically any group G can define a sub-factor: G would define measurement resolution by fixing the quantum numbers which are measured. Finite Galois group G and Lie group containing it and related to it by Langlands correspondence would act in the same representation space: the group algebra of S_∞, or equivalently configuration space spinors. The concrete realization for the correspondence might transform a large number of speculations to theorems.

2. There is a natural connection with McKay correspondence which also relates finite and Lie groups. The simplest variant of McKay correspondence relates discrete groups Gsubset SU(2) to ADE type groups. Similar correspondence is found for Jones inclusions with index M:N≤ 4. The challenge is to understand this correspondence.
   1. The basic observation is that ADE type compact Lie algebras with n-dimensional Cartan algebra can be seen as deformations for a direct sum of n SU(2) Lie algebras since SU(2) Lie algebras appear as a minimal set of generators for general ADE type Lie algebra. The algebra results by a modification of Cartan matrix. It is also natural to extend the representations of finite groups Gsubset SU(2) to those of SU(2).

   2. The idea would that is that n-fold Connes tensor power transforms the direct sum of n SU(2) Lie algebras by a kind of deformation to a ADE type Lie algebra with n-dimensional Cartan Lie algebra. The deformation would be induced by non-commutativity. Same would occur also for the Kac-Moody variants of these algebras for which the set of generators contains only scaling operator L₀ as an additional generator. Quantum deformation would result from the replacement of complex rays with N rays, where N is the sub-factor.

   3. The concrete interpretation for the Connes tensor power would be in terms of the fiber bundle structure H=M⁴_+/-× CP₂→ H/G_a× G_b, G_a× G_b subset SU(2)× SU(2)subset SL(2,C)× SU(3), which provides the proper formulation for the hierarchy of macroscopic quantum phases with a quantized value of Planck constant. Each sheet of the singular covering would represent single factor in Connes tensor power and single direct SU(2) summand. This picture has an analogy with brane constructions of M-theory.

4. Could there exist a universal rational function giving rise to the algebraic closure of rationals?

One could wonder whether there exists a universal generalized rational function having all units of the algebraic closure of rationals as roots so that S_∞ would permute these roots. Most naturally it would be a ratio of infinite-degree polynomials.

With motivations coming from physics I have proposed that zeros of zeta and also the factors of zeta in product expansion of zeta are algebraic numbers. Complete story might be that non-trivial zeros of Zeta define the closure of rationals. A good candidate for this function is given by (ξ(s)/ξ(1-s))× (s-1)/s), where ξ(s)= ξ(1-s) is the symmetrized variant of zeta function having same zeros. It has zeros of zeta as its zeros and poles and product expansion in terms of ratios (s-s_n)/(1-s+s_n) converges everywhere. Of course, this might be too simplistic and might give only the algebraic extension involving the roots of unity given by exp(iπ/n). Also products of these functions with shifts in real argument might be considered and one could consider some limiting procedure containing very many factors in the product of shifted zeta functions yielding the universal rational function giving the closure.

5. What does one mean with S_∞?

There is also the question about the meaning of S_∞. The hierarchy of infinite primes suggests that there is entire infinity of infinities in number theoretical sense. Any group can be formally regarded as a permutation group. A possible interpretation would be in terms of algebraic closure of rationals and algebraic closures for an infinite hierarchy of polynomials to which infinite primes can be mapped. The question concerns the interpretation of these higher Galois groups and HFF:s. Could one regard these as local variants of S_∞ and does this hierarchy give all algebraic groups, in particular algebraic subgroups of Lie groups, as Galois groups so that almost all of group theory would reduce to number theory even at this level?

Be it as it may, the expressive power of HFFs seem to be absolutely marvellous. Together with the notion of infinite rational and generalization of number concept they might unify both mathematics and physics!

For more details see the new chapter TGD and Langlands Program of "TGD as a Generalized Number Theory".