Views about free will

Now and then comes the day when you feel that you have said all that might possibly interest anyone somewhere in this waste Universe and even an attempt to think about some problem creates a feeling of deep disgust. I try to escape this depressive mood by meandering around the web in the hope that some colleague or blogger might have written something original. Usually the outcome is a disappointment. Original and not obviously wrong thoughts are as rare as genuine anomalies.

This kind of cheerless wandering around web led me to read some postings and articles about free will. Even some physicists have now accepted "free will" into their vocabulary. The fQXI conference about the nature of time held in some boat sailing from Norway towards Copenhagen las summer had inspired several blog postings. Also I wrote comments about the excellent lecture of David Eagleman about perceived time. This kind of sailing trips cost and it is good if they induce interaction between people with different backgrounds: now at least physicists, neuro-scientists, computer scientists and philosophers were solving both the problem of time and the problems caused by sea sickness at one blow.

I did not find it surprising that I did not find anything surprising in these postings. The common feature of all these articles is that quite too much is assumed. Sean Carroll as a descent reductionist makes especially strong assumptions. All writers have managed to remain unaware of the dramatic distinctions between subjective time and the geometric time of physicist. They also make the same error: in the process of trying to understand free will scientifically their first step is to carefully eliminate conscious mind from the picture. The outcome is free will as something effective and emergent or free will as resulting from deterministic but non-predictable/non-computable process. My humble question is: Why on earth something very complex or non-computable would generate sensation "I decide to do this!"?! A non-deterministic behavior serves as a correlate of free will but non-predictable (but possibly deterministic) behavior does not imply experience of free will.

Every writer grasps something essential but fails to see the essence of the problem and connect it with many related problems like the puzzle of time and the basic paradox of quantum measurement theory. One should not be however too critical since the position of the writers is unrewarding. Being names in blogo-sphere they have been invited to solve the problem of time with minimal background: this is like solving some deep problem in topology with the background given by a couple of hastily read Wikipedia articles.

I was a little bit disappointed but understood that I must also realize that the understanding free will is as difficult as the understanding of the nature of time. It requires a lot of time and a flash of genius: a sea trip from Norway to Copenhagen with National Geographic Explorer - even in a good company - need not be enough to spark this kind of flash. I have been trying to communicate more than 15 years my own flash of genius relating to free will and the relationship between experienced time and the geometric time of physicist but it seems that this has been waste of time. They must discover it themselves! Let us hope better luck during the next cruise! In the following some more detailed comments about articles of the peopled who participated the trip.

Sabine Hossenfelder: Free Will function

Sabine Hossenfelder has a posting titled "Free Will function". I agree with Sabine that the idea about emergent free will is self deception. Free will does not emerge from a deterministic microscopic dynamics.

The believers in emergence say that free will is an effective concept. Not real but useful. If the system in question is complex enough and behaves non-predictably as seen by outsider one can say that it has effective free will. But why the impossibility to predict a deterministic dynamics in practice would generate the experience "I will do this!". There is absolutely no justification for this belief.

A good objection against this identification comes from neuro-science and is described in the article The Brain on Trial by David Eagleman. People suffering Tourette's syndrome, split brain patients, persons with side personalities, and patients with choreic motions behave from the point of view of outsider as they would have free will. Using biblical language: they act as if being possessed. They do not experience free will. Who wills? Who uses the biological body of the patient? Same questions can be made in the situation when people who have done mass murder become conscious and begin to wonder what these bloody hands have done. Who used these hands? Are we merely our brains and bodies? Who uses my biological body? What is this "me"? Is this particular biological body used only by single intentional agent, by single "me" only? I could continue by telling about the notion of magnetic body but will not do it here.

Acidic out-of-topic side remark: Effective theories have become the basic curse of theoretical physics today. No-one can seriously claim that string models say anything about the world of experimental physicists. But there is a loop hole. By postulating effective field theory approach one can build entire landscape of effective theories. This is non-sense but it works. The only honest reaction would be to admit that string models are nice theories but not theories about the world we live in.

Sabine Hossenfelder suggests as a solution something that she calls free will function. Sabine considers a machine spitting out digits of π. This process is fully deterministic but outsider has no means of predicting what the next digit will be and what number the digit sequence represents unless he manages to get the program code. The proposal is that our brain has this kind of free will function. The strange assumption is that the inability to predict would in some mysterious manner generate experience of free will. But Sabine a physicist has learned that one must forget all subjective elements when doing science. In this mental framework the only conceivable goal of a theory of consciousness is to eliminate it. The fruitless searches of "consciousness modules" assumed to reside somewhere in the brain are fruits of similar "consciousness as function" thinking.

Sean Carroll: Free will as real as baseball

Also Sean Carroll has written about free will in his posting "Free will as real as baseball". Sean belongs to the effective theory camp and sees free will as a convenient tool of description just like baseball is seen by a reductionist as a convenient abstraction to describe the dynamics of a condensed matter system.

Sean makes two very strange claims.

1. The first strange claim is that free will is inconsistent with the laws of physics. This is the case only if the experienced time and geometric time of physicists are identified. The are not one and the same thing as even child realizes. Experienced time is irreversible and there is no subjective future. Geometric time is reversible and future and past are in the same position. In general relativity 4-D space-time region becomes the basic entity instead of time=constant snapshot which is the basic entity according to Newtonian thinking. Amusingly, all writers except Scott Aaronson seem to belong to the species of Newtonians as far as their views about time are considered. The first years of scientific education involves really heavy social conditioning and it is really hard to de-learn even the obviously wrong beliefs.

2. The second strange claim of Sean Carroll is that the physics is completely understood in everyday realm! Really! Do we really understand the physics underlying living matter?! I cannot do help it: this kind of loose text book statements irritate me that suddenly the dull depressive mood has gone and I am full of adrenaline!

Interestingly, Sean Carroll notices analogy of poorly understood notion of free will with the poorly understood notion of time. The arrow of time is in conflict with microscopic reversibility but - according to Sean Carroll - physicists do not see this as a problem so that it is not a problem. Continuing in the same spirit: if billions of Chinese believe in communism then marxism is the only correct view about society and is indeed law of Nature! The effective theory solution is simple: also the arrow of time somehow emerges. Exactly how?: this we do not understand but it does not matter.

This is self deception. One should admit this and really try to understand second law. If one does this, the first observation is that Boltzmann's equations are deduced by assuming the occurrence of state function reductions in time scale much shorter than the time scale of observations. State function reduction is what makes quantum physics non-deterministic at the level of single quantum system - also internally in-consistent: the determinism of Schrödinger equation is in blatant conflict with state function reduction if one identifies experienced time with the geometric time of physicist. One should be able to resolve this logical flaw and this requires that the two times are different - something which of course even child knows! If we have two times we have also two independent causalities: the causality of field equations and that of "free will". This would be the first step towards the solution.

Sean Carroll also represents what he calls consequent argument. The argument begins with an innocent looking statement that our past is fixed. Therefore free will obeying field equations is impossible since it would change both future and past. Wrong again: the assumption about fixed past in the geometric sense need not be true. About subjective past it is. Already Wheeler was led to propose that in state function reduction the geometric past changes: see for the Wheeler's delayed choice experiment. Maybe Wheeler's general relativistic background helped him to make this conceptual leap, which leads very near to TGD view about quantum jump.

In TGD framework quantum states are superpositions of classical histories and quantum jumps replace them with new ones and the average geometric past also changes. The finding of Libet that in volitional act neural activity begins a fraction of second before the conscious decisions supports the idea that we are replacing our geometric past with a new one all the subjective time.

Sean Carroll notices also the ethical aspect of problem. If we really believe that free will is illusion, we have no justification for moral rules. The criminal has been doomed to perform his crime at the moment of Big Bang and we cannot therefore accuse him.

Of course, there could be something badly wrong in the brain of the mass murderer and it has indeed become clear that our behavior correlates strongly with biology. This does not however mean that free choices are not possible. Braid disorer only changes the probabilities of different outcomes of the choices. We have the experience of free will as every reader can testify. This we must accept and try to understand the physical correlates of this experience irrespective of whether the free will is real or not.

In fact, neuroscience has led to quite concrete progress in the understanding of the correlations between biology and behavior. This has also practical consequences. Many mass murderers have been victims of child abuse or have suffered from brain tumor. This does not mean that we should allow mass murders to continue with their rare hobby. We can however do our best to prevent child abuse. Also the degeneration of some regions of frontal lobes can lead to highly asocial behaviors when stimuli usually inhibited are not inhibited anymore. One could say that there are competing free wills using the same biological body and the one wanting to perform the mass murder wins.

These issues were discussed already at times of Dostojevski and Turgeniev. The fashionable thinking was that we are nothing but physiology and that we can indeed forget the rules of moral. The people propagating this view and trying to live according to this philosophy were known as nihilists: they were mad but fully logical in their madness. Many people calling themselves skeptics today are surprisingly near to these views. Thanks to God, most of us are to not too strict in their logics and follow their conscience rather than materialistic dogmas.

Scott Aaronson's view

G. Musser has summarized computer scientist Scott Aaronson's talk about free will.

Also Scott Aaronson studies the idea of reducing free will to behavior observed from outside. Aaronson's thought experiment considers a Turing like test allowing to decide whether you have free will. A computer model of you is built using all available data about the initial state of your brain: this of course assumes determinism or at least quantum statistical determinism. If the computer is able to mimic your behavior faithfully, one can say that you have no free will. The proponent of effective free will might say that the longer the needed computer code is, the more you have effective free will. This kind of free-will-meter is of course not possible in practice except with some accuracy so that the whole thing reduces to mere mimicry, kind of parameter fit.

Aaronson represents the non-cloning theorem of quantum theory as a first principle objection against Turing test of free-will-meter. Even in principle it is not possible to construct complete copy of brain state to make a complete simulation possible. This kind of machine would be successful in what Aaronson calls Toddler test but this would be a fake success. Any toddler says completely predictably "No" to any question. We however know that the toddler expresses by behaving irrationally that he/she has discovered his/her free will (but can this kind of free will be effective?)!

Aaronson brings in special relativity and notices that free will means also backward causation if it is to be consistent with the causality of field equations. From this it would be only a short step to the realization that the causality of free will could act in the space of quantum states defined as superposition of solutions of classical field equations consistent with holography in the sense that 3-D section determines the entire space - at least below certain scale! The problem would have been solved! Scott makes a near miss!

To sum up, Aaronson dimly realizes that in general relativity - and in any 4-D Universe obeying general coordinate invariance - we live in a kind of block world consisting of 4-D blocks but the other writers continue in the good old Newtonian style. In TGD zero energy ontology would realize blocks as causal diamonds and would extend free will from a mere choice between given alternatives to creation of new worlds. Sabine Hossenfelder realizes that emergence is self deception: I cannot but agree. Sean Carroll grasps the full meaning of the absence of free will at the level of moral issues. Eagleman describes real life situations, which should be highly valuable for any-one proposing in earnest a theory of consciousness. Also the lecture of Eagleman about perceived time was excellent. To me it seems that physicists and (quantum) computer scientists should be able to forget for a moment their formulas and rhetorics making possible to get rid of a problems they cannot solve, and open their minds for the problem to get settled.

Quantum Adeles as a Golden Road to Number Theoretical Universality?

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.

1. 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.

2. 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.

3. Canonical identification finds a fundamental role in the definition of the norm for both quantum p-adics and quantum adeles.

4. 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.

1. 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.

2. 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.

3. 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 Gl₁ 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 M⁸-M⁴×CP₂ 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".

No stop but maybe cold fusion

The rumors about the detection of stop particle at LHC have been circulating for some time. Here stop is understood in the sense of standard SUSY predicting R-parity conservation so that sparticles are produced only in pairs and that stop is the lightest squark. Missing energy corresponding to lightest - and thus stable - neutral sparticle is the basic decay signature of stop in this sense.

For those who took these rumors as more than wishful thinking, the ATLAS collaboration produced a dissappointment: the analysis of integrated luminosity 2.05/fb shows no significant excess. The new limits tell that gluino mass must be above 650 GeV stop mass above 450 GeV.

Also in TGD framework both Higgs and SUSY are also in TGD are creators of tension. It would be nice to have a computer program listing the predictions of the theory but the situation is not so simple. Developing and interpreting the theory is a complex process requiring a continual interaction with experiment making educated guesses. Even in the case of Higgs the situation in TGD is still not closed. Higgs is not needed in TGD and no-Higgs option is the most elegant one: but does Nature think in the same manner as I happen to do just now?

SUSY in TGD sense means that sfermion is obtained by adding right handed neutrino to a wormhole throat carrying quantum numbers of fermion. R-parity as well as B and L are conserved and spartners are created in pairs. The simplest option is that the right-handed neutrino corresponds to a covariantly constant spinor in CP₂ degrees of freedom. More complex option possibly allowed by super-conformal symmetry is that right-handed neutrinos appear as color octets.

LHC tells us that sfermions and gluinos are heavy very (TeV mass scale) if they obey standard SUSY. The conclusion comes from the missing missing energy. This conclusion might be circumvented in TGD.

1. Squarks are colored and interact strongly. This allows them to fuse together to form shadrons: say smesons formed from squark pair. This could be the dominating decay channel leading eventually to ordinary hadrons.

2. For covariantly constant right handed neutrino this however leaves the decays of squarks to quarks and electroweak gauginos proceeding with a rate fixed by electro-weak gauge symmetry. The situation seems to be like that in standard SUSY. Gauginos would decay eventually produce missing energy seen as righthanded neutrinos which mix with left handed components. It might well be that LHC already kills this option unless one assumes short enough p-adic length scales for squarks which is of course possible.

3. If right-handed neutrino is in color octet partial wave, the situation changes. Shadrons are the only final states by color confinement and quarks and squarks could have even same p-adic mass scale for both ordinary and M₈₉ hadron physics. Fuel for the speculations with this option comes from so called X and Y bosons, which are charmonium like states which should not be there: are they scharmoniums? There are also two other anomalies discussed in previous posting suggesting that mesons have what I call IR Regge trajectories with mass scale of 38 MeV. They are very natural in TGD framework in which hadrons are accompanied by color magnetic flux tubes behaving like string-like objects and thus contributing to to hadron mass a stringy contribution with a small string tension. Is TGD SUSY needed to explain X and Y boson or could also IR Regge trajectories do the same (probably not): this is the first thing to check. Quite often I feel that this endless questioning rather frustrating. Life would be so easy if I could just believe.

Blind believing makes things simple but eventually it leads to painful conflicts with facts. Lubos has been especially strong believer of stop rumours and it is a pity that he is wrong again with so much authority a (big names such as Gell Mann) behind his arguments;-).

This is hard time for Lubos also otherwise;-): Lubos has used all tools of bad rhetorics to attack cold fusion but demonstrations continue to generate support for the effect. The progress of physics understood as a reductionistic (and highly imperialistic;-)) enterprise proceeding to shorter and shorter length scales has perhaps been quite not so successful as we have been taught. There are a lot of bridges of belief on the road of reductionism and this particular bridge - the belief that there is no interaction between atomic and nuclear length scales - might be collapsing under merciless pressures of cold fusion researchers whom Lubos does not want to count as scientists at all. It might be that we do not understand nuclear physics properly, and this mis-understanding - if it continues- can have profound impact on the future of our civilization.

Even worse, there will be a cold fusion colloqium - and believe or not - at CERN! On Thursday, March 22nd. I have written some postings earlier debunking the cold fusion debunkings of Lubos (see for instance this). I admit that I have to make a conscious effort to keep a fully serious face;-). Here is the rant of Lubos inspired by cold fusion colloqium at CERN. Lubos is learning- or at least he should finally learn - that authority means absolutely nothing for Nature.

More evidence for IR Regge trajectories

TGD based view about non-perturbative aspects of hadron physics (see this) relies on the notion of color magnetic flux tubes. These flux tubes are string like objects and it would not be surprising if the outcome would be satellite states of hadrons with string tension below the pion mass scale. One would have kind of infared Regge trajectories satisfying in a reasonable approximation a mass formula analogous to string mass fomula. What is amazing that this phenomenon could allow new interpretation for the claims for a signal interpreted as Higgs at several masses (115 GeV by ATLAS, at 125 GeV by ATLAS and CMS, and at 145 GeV by CDF).

Consider first the mass formula for the hadrons at IR Regge trajectories.

1. There are two options depending on whether the mass squared or mass for hadron and for the flux tubes are assumed to be additive. p-Adic physics would suggest that if the p-adic primes characterizing the flux tubes associated with hadron and hadron proper are different then mass is additive. If the p-adic prime is same, the mass squared is additive.

2. The simplest guess is that the IR stringy spectrum is universal in the sense that m₀ does not depend on hadron at all. This is the case if the flux tubes in question correspond to hadronic space-time sheets characterized by p-adic prime M₁₀₇ in the case of ordinary hadron physics. This would give for the IR contribution to mass the expression

   m²=(m₀²+ nm₁²)^1/2 .

3. The net mass of hadron results from the contribution of the "core" hadron and the stringy contribution. If mass squared is additive, one obtains

   m(H_n)= [m²(H₀) +m₀²+ nm₁²]^1/2,

   where H₀ denotes hadron ground state and H_n its excitation assignable to magnetic flux tube. For heavy hadrons this would give the approximate spectrum

   m(H_n)≈ m(H₀)+ [m₀²+nm₁²]/2m(H₀) .

   The mass unit for the excitations decreases with the mass of the hadron.

4. If mass is additive as one indeed expects since the p-adic primes characterizing heavy quarks are smaller than hadronic p-adic prime, one obtains

   m(H_n)= m(H₀)+ (m₀²+ nm₁²)^1/2 .

   For m₀²>> m₁² one has

   m(H_n)= m(H₀)+ m₀+ nm₁²/2m₀ .

   If the flux tubes correspond to p-adic prime. This would give linear spectrum which is same for all hadrons.

There is evidence for this kind of states.

1. Tatischeff and Tomasi-Gustafsson claim the existence of states analogous to ordinary pion with masses 60, 80, 100, 140,.... MeV. Also nucleons have this kind of satellite states.

2. Second piece of evidence comes from two articles by Eef van Beveren and George Rupp. The first article is titled First indications of the existence of a 38 MeV light scalar boson. Second article has title Material evidence of a 38 MeV boson . The basic observations are following. The rate for the annihilation e⁺+e^-→ uubar assignable to the reaction e⁺+e^-→ π⁺π^- has a small periodic oscillation with a period of 78+/- 2 MeV and amplitude of about 5 per cent. The rate for the annihilation e⁺+e^-→ bbbar, assignable to the reaction e⁺+e^-→ Υπ⁺π^- has similar oscillatory behavior with a period of 73+/- 3 MeV and amplitude about 12.5 per cent. The rate for the annihilation ppbar→ cbbar assignable to the reaction e⁺+e^-→ J/Ψπ⁺π^- has similar oscillatory behavior with period of 79+/- 5 MeV and amplitude .75 per cent.

   In these examples universal Regge slope is consistent with the experimental findings and supports additive mass formula and the assignment of IR Regge trajectories to hadronic flux tubes with fixed p-adic length scale.

What does one obtain if one scales up the IR Regge trajectories to the M₈₉ which replaces Higgs in TGD framework?

1. In the case of M₈₉ pion the mass differences 20 MeV and 40 MeV appearing in the IR Regge trajectories of pion would scale up to 10 GeV and 20 GeV respectively. This would suggest the spectrum of pion like states with masses 115, 125, 145, 165 GeV. What makes this interesting that ATLAS reported during last year evidence for a signal at 115 GeV taken as evidence for Higgs and CDF reported before this signal taken as evidence for Higgs around 145 GeV! 125 GeV is the mass of the the most recent Higgs candidate. Could it be that all these reported signals have been genuine signals - not for Higgs- but for M₈₉ pion and corresponding spion consisting of squark pair and its IR satellites?

2. I the case of M₈₉ hadron physics the naive scaling of the parameters m₀ and m₁ by factor 512 would scale 38 MeV to 19.5 GeV.

Indeed! Is it really Higgs?

Jester comments the latest release of results from LHC relating to the signal interpreted by all fashionable and well-informed physics bloggers as Higgs.

Additional support for a resonance at 125 GeV is emerging. What is new are two events which are interpreted as fusion of two W bosons to Higgs. This is very nice. The only problem is that the predicted rate for these events is so small for standard model Higgs that they would not have been observed. Second not anymore pleasant surprise is that Higgs candidates are indeed produced but with a rate twice than the predicted rate.

Hitherto these signals which are too strong to allow interpretation as standard model Higgs have been interpreted by saying that both CMS and ATLAS have been "lucky". I warned already in the previous Higgs posting that if this good luck continues, it turns to a serious problem. And as Jester mentions, already now people are beginning to suspect that this Higgs is not quite the standard model Higgs. The next step will come sooner or later and will be a cautious proposal spoiling the euphoric mood of co-bloggers: perhaps it is not Higgs at all!

But things go slowly. Colleagues are rather conformistic and remarkably slow as thinkers. There are even those who are still making bets for standard SUSY;-)! I can however hope that after this step colleagures would be finally psychologically mature to consider the TGD prediction for M₈₉ hadron physics as an alternative to Higgs. Accepting this hypothesis as something worth of testing would mean enormous progress on both the theoretical and the experimental side.

One more good reason for p-adic cognition

One can present several justifications for why p-adic numbers are natural correlates of cognition and why p-adic topology is tailor-made for computation. One possible justification derives from the ultrametricity of p-adic norm stating that the p-adic norm is never larger than the maximum of the norms of summands.

If one forms functions of real arguments, a cutoff in decimal or more general expansion of arguments introduces a cumulating error, and in principle one must perform calculation assuming that the number of digits for the arguments of function is higher than the number digits required by the cutoff, and drop the surplus digits at the end of the calculations.

In p-adic case the situation is different. The sum for the errors resulting from cutoffs is never p-adically larger than the largest individual error so that there is no cumulation of errors , and therefore no need for surplus pinary digits for the arguments of the function. In practical computations this need not have great significance unless they involve very many steps but in cognitive processing the situation might be different.

Bullying as a national disease

We have a presidential election in Finland. The two main candidates are Sauli Niinistö and Pekka Haavisto. Niinistö can be said to represent the old world order in which economical values dictate everything. Haavisto is a representative of the new world order in which humanity, freedom, equality, and environment represent the most important values. For me the choice between these options is easy although I have nothing against Niinistö personally.

Haavisto crystallized something very essential about Finland as a nation as he said that bullying is the national disease of Finland. Teasing begins already in elementary schools and continues in various educational establishments and eventually it continues at working places. Web has become also an arena of bullying providing completely new opportunities. Now and then some-one gets enough. The two mass murders that took place in educational establishments for few years ago are just two sad examples of what "enough is enough" really means.

Personally I belong to the victims of academic bullying. The terror began for 34 years ago and has continued since then. I have lost my academic human rights and have been unemployed most of the time after I began to write my thesis 1977. I will remain so until I get to the age of 63 (only two years of this humiliation anymore!) and start to receive a minimal pension. I have done impressive life work: 15 books making about 12 thousand pages and a lot of articles. This does not means anything since so called "evaluation by equals" (direct translation for "vertaisarviointi"), a scientific equivalent of inquisition, can be used to label my work as crackpottery. There are many people out-broad and also in Finland who appreciate my work and they have made attempts to inform about my work in Wikipedia but (very probably finnish) censors have reacted immediately and vandalized the attempts.

I have tried to understand what drives people to this kind of sadistic behaviors in which human life is literally destroyed. As far as I know, these people are quite descent human beings as individuals. But as members of collective they become sadistic beasts. Or some of them. The others remain completely passive and this is probably the core problem. We do not have the courage to say no when some sociopath starts the cruel game. I am of course just one of the many victims of this national hobby and I sincerely hope that Finland as a nation could heal from it. Haavisto is certainly experienced as a symbol of this healing and my sincere hope is that he wins.

Quantum p-adic deformations of space-time surfaces as a representation of finite measurement resolution?

A mathematically - and also physically - fascinating question is whether one could use quantum arithmetics as a tool to build quantum deformations of partonic 2-surfaces or even of space-time surfaces and how could one achieve this. These quantum space-times would be commutative and therefore not like non-commutative geometries assigned with quantum groups. Perhaps one could see them as commutative semiclassical counterparts of non-commutative quantum geometries just as the commutative quantum groups (see this) could be seen commutative counterparts of quantum groups.

As one tries to develop a new mathematical notion and interpret it, one tends to forget the motivations for the notion. It is however extremely important to remember why the new notion is needed.

1. In the case of quantum arithmetics Shnoll effect is one excellent experimental motivation. The understanding of canonical identification and realization of number theoretical universality are also good motivations coming already from p-adic mass calculations. A further motivation comes from a need to solve a mathematical problem: canonical identification for ordinary p-adic numbers does not commute with symmetries.

2. There are also good e motivations for p-adic numbers? p-Adic numbers and quantum phases can be assigned to finite measurement resolution in length measurement and in angle measurement. This with a good reason since finite measurement resolution means the loss of ordering of points of real axis in short scales and this is certainly one outcome of a finite measurement resolution. This is also assumed to relate to the fact that cognition organizes the world to objects defined by clumps of matter and with the lumps ordering of points does not matter.

3. Why quantum deformations of partonic 2-surfaces (or more ambitiously: space-time surfaces) would be needed? Could they represent convenient representatives for partonic 2-surfaces (space-time surfaces) within finite measurement resolution?
   1. If this is accepted there is not compelling need to assume that this kind of space-time surfaces are preferred extremals of Kähler action.

   2. The notion of quantum arithmetics and the interpretation of p-adic topology in terms of finite measurement resolution however suggest that they might obey field equations in preferred coordinates but not in the real differentiable structure but in what might be called quantum p-adic differentiable structure associated with prime p.

   3. Canonical identification would map these quantum p-adic partonic (space-time surfaces) to their real counterparts in a unique a continuous manner and the image would be real space-time surface in finite measurement resolution. It would be continuous but not differentiable and would not of course satisfy field equations for Kähler action anymore. What is nice is that the inverse of the canonical identification which is two-valued for finite number of pinary digits would not be needed in the correspondence.

   4. This description might be relevant also to quantum field theories (QFTs). One usually assumes that minima obey partial differential equations although the local interactions in QFTs are highly singular so that the quantum average field configuration might not even possess differentiable structure in the ordinary sense! Therefore quantum p-adicity might be more appropriate for the minima of effective action.

   The conclusion would be that commutative quantum deformations of space-time surfaces indeed have a useful function in TGD Universe.

Consider now in more detail the identification of the quantum deformations of space-time surfaces.

1. Rationals are in the intersection of real and p-adic number fields and the representation of numbers as rationals r=m/n is the essence of quantum arithmetics. This means that m and n are expanded to series in powers of p and coefficients of the powers of p which are smaller than p are replaced by the quantum counterparts. They are quantum quantum counterparts of integers smaller than p. This restriction is essential for the uniqueness of the map assigning to a give rational quantum rationals.

2. One must get also quantum p-adics and the idea is simple: if the pinary expansions of m and n in positive powers of p are allowed o become infinite, one obtains a continuum very much analogous to that of ordinary p-adic integers with exactly the same arithmetics. This continuum can be mapped to reals by canonical identification. The possibility to work with numbers which are formally rationals is utmost importance for achieving the correct map to reals. It is possible to use the counterparts of ordinary pinary expansions in p-adic arithmetics.

3. One can defined quantum p-adic derivatives and the rules are familiar to anyone. Quantum p-adic variants of field equations for Kähler action make sense.
   1. One can take a solution of p-adic field equations and by the commutativity of the map r=m/n→ r_q=m_q/n_q and of arithmetic operations replace p-adic rationals with their quantum counterparts in the expressions of quantum p-adic imbedding space coordinates h^k in terms of space-time coordinates x^α.

   2. After this one can map the quantum p-adic surface to a continuous real surface by using the replacement p→ 1/p for every quantum rational. This space-time surface does not anymore satisfy the field equations since canonical identification is not even differentiable. This surface - or rather its quantum p-adic pre-image - would represent a space-time surface within measurement resolution. One can however map the induced metric and induced gauge fields to their real counterparts using canonical identification to get something which is continuous but non-differentiable.

4. This construction works nicely if in the preferred coordinates for imbedding space and partonic (space-time) surface itself the imbedding space coordinates are rational functions of space-time coordinates with rational coefficients of polynomials (also Taylor and Laurent series with rational coefficients could be considered as limits). This kind of assumption is very restrictive but in accordance with the fact that the measurement resolution is finite and that the representative for the space-time surface in finite measurement resolution is to some extent a convention. The use of rational coefficients for the polynomials involved implies that for polynomials of finite degree WCW reduces to a discrete set so that finite measurement resolution has been indeed realized quite concretely!

Consider now how the notion of finite measurement resolution allows to circumvent the objections against the construction.

1. Manifest GCI is lost because the expression for space-time coordinates as quantum rationals is not general coordinate invariant notion unless one restricts the consideration to rational maps and because the real counterpart of the quantum p-adic space-time surface depends on the choice of coordinates. The condition that the space-time surface is represented in terms of rational functions is a strong constraint but not enough to fix the choice of coordinates. Rational maps of both imbedding space and space-time produce new coordinates similar to these provided the coefficients are rational.

2. Different choices for imbedding space and space-time surface lead to different quantum p-adic space-time surface and its real counterpart. This is an outcome of finite measurement resolution. Since one cannot order the space-time points below the measurement resolution, one cannot fix uniquely the space-time surface nor uniquely fix the coordinates used. This implies the loss of manifest general coordinate invariance and also the non-uniqueness of quantum real space-time surface. The choice of coordinates is analogous to gauge choice and quantum real space-time surface preserves the information about the gauge.

For background see chapter Quantum Arithmetics of "Physics as Generalized Number Theory".

The anatomy of quantum jump in zero energy ontology

The understanding of the anatomy of quantum jump identified as a moment of consciousness in the framework of Zero energy ontology (ZEO) is gradually getting more detailed and the following is the summary of the recent understanding. The general vision about quantum jump in zero energy ontology generalizes the ordinary auantum measurement theory bringing in also the selection of maximal set of mutually commuting set of observables. Also the connection with the breaking of time reversal invariance at the level of zero energy states as a necessary condition for the non-triviality of U-matrix is new.

1. Quantum jump begins with unitary process U described by unitary matrix assigning to a given zero energy state a quantum superposition of zero energy states. This would represent the creative aspect of quantum jump - generation of superposition of alternatives.

2. The next step is a cascade of state function reductions proceeding from long to short scales. It starts from some CD and proceeds downwards to sub-CDs to their sub-CDs to ...... At a given step it induces a measurement of the quantum numbers of either positive or negative energy part of the quantum state. This step would represent the measurement aspect of quantum jump - selection among alternatives.

3. The basic variational principle is Negentropy Maximization Principle(NMP) stating that the reduction of entanglement entropy in given quantum jump between two subsystems of CD assigned to sub-CDs is maximal. Mathematically NMP is very similar to the second law although states just the opposite but for individual quantum system rather than ensemble. NMP actually implies second law at the level of ensembles as a trivial consequence of the fact that the outcome of quantum jump is not deterministic.

   For ordinary definition of entanglement entropy this leads to a pure state resulting in the measurement of the density matrix assignable to the pair of CDs. For hyper-finite factors of type II₁ (HFFs) state function reduction cannot give rise to a pure state and in this case one can speak about quantum states defined modulo finite measurement resolution and the notion of quantum spinor emerges naturally. One can assign a number theoretic entanglement entropy to entanglement characterized by rational (or even algebraic) entanglement probabilities and this entropy can be negative. Negentropic entanglement can be stable and even more negentropic entanglement can be generated in the state function reduction cascade.

The irreversibility is realized as a property of zero energy states (for ordinary positive energy ontology it is realized at the level of dynamics) and is necessary in order to obtain non-trivial U-matrix. State function reduction should involve several parts. First of all it should select the density matrix or rather its Hermitian square root. After this choice it should lead to a state which prepared either at the upper or lower boundary of CD but not both since this would be in conflict with the counterpart for the determinism of quantum time evolution.

Generalization of S-matrix

ZEO forces the generalization of S-matrix with a triplet formed by U-matrix, M-matrix, and S-matrix. The basic vision is that quantum theory is at mathematical level a complex square roots of thermodynamics. What happens in quantum jump was already discussed.

1. U-matrix as has its rows M-matrices , which are matrices between positive and negative energy parts of the zero energy state and correspond to the ordinary S-matrix. M-matrix is a product of a hermitian square root - call it H - of density matrix ρ and universal S-matrix S commuting with H: [S,H]=0. There is infinite number of different Hermitian square roots H_i of density matrices which are assumed to define orthogonal matrices with respect to the inner product defined by the trace: Tr(H_iH_j)=0. Also the columns of U-matrix are orthogonal. One can interpret square roots of the density matrices as a Lie algebra acting as symmetries of the S-matrix.

2. One can consider generalization of M-matrices so that they would be analogous to the elements of Kac-Moody algebra. These M-matrices would involve all powers of S.

   1. The orthogonality with respect to the inner product defined by < A| B> = Tr(AB) requires the conditions Tr(H₁H₂Sⁿ)=0 for n≠ 0 and H_i are Hermitian matrices appearing as square root of density matrix. H₁H₂ is hermitian if the commutator [H₁,H₂] vanishes. It would be natural to assign n:th power of S to the CD for which the scale is n times the CP₂ scale.

   2. Trace - possibly quantum trace for hyper-finite factors of type II₁) is the analog of integration and the formula would be a non-commutative analog of the identity ∈t_S¹ exp(inφ) dφ=0 and pose an additional condition to the algebra of M-matrices. Since H=H₁H₂ commutes with S-matrix the trace can be expressed as the sum

      ∑_i,jh_is_j(i)= ∑_i,j h_i(j)s_j

      of products of correspondence eigenvalues and the simplest condition is that one has either ∑_j s_j(i)=0 for each i or ∑_i h_i(j)=0 for each j.

   3. It might be that one must restrict M matrices to a Cartan algebra for a given U-matrix and also this choice would be a process analogous to state function reduction. Since density matrix becomes an observable in TGD Universe, this choice could be seen as a direct counterpart for the choice of a maximal number of commuting observables which would be now hermitian square roots of density matrices. Therefore ZEO gives good hopes of reducing basic quantum measurement theory to infinite-dimensional Lie-algebra.

Unitary process and choice of the density matrix

Consider first unitary process followed by the choice of the density matrix.

1. There are two natural state basis for zero energy states. The states of these state basis are prepared at the upper or lower boundary of CD respectively and correspond to various M-matrices M_K⁺ and M_L^-. U-process is simply a change of state basis meaning a representation of the zero energy state M_K^+/- in zero energy basis M_K^-/+ followed by a state preparation to zero energy state M^+/-_K with the state at second end fixed in turn followed by a reduction to M_L^-/+ to its time reverse, which is of same type as the initial zero energy state.

   The state function reduction to a given M-matrix M_K^+/- produces a state for the state is superposition of states which are prepared at either lower or upper boundary of CD. It does not yet produce a prepared state on the ordinary sense since it only selects the density matrix.

2. The matrix elements of U-matrix are obtained by acting with the representation of identity matrix in the space of zero energy states as

   I= ∑_K | K⁺> < K⁺|

   on the zero energy state | K^-> (the action on | K⁺> is trivial!) and gives

   U⁺_KL= Tr(M⁺_KM⁺_L) .

   In the similar manner one has

   U^-_KL=(U^+†)_KL= Tr(M^-_LM^-_K) = (U⁺_LK)^* .

   These matrices are Hermitian conjugates of each other as matrices between states labelled by positive or negative energy states. The interpretation is that two unitary processes are possible and are time reversals of each other. The unitary process produces a new state only if its time arrow is different from that for the initial state. The probabilities for transitions |K₊> → |K_-> are given by

   p_mn= |Tr(M_K⁺ M_L⁺)|².

State function preparation

Consider next the counterpart of the ordinary state preparation process.

1. The ordinary state function process can act either at the upper or lower boundary of CD and its action is thus on positive or negative energy part of the zero energy state. At the lower boundary of CD this process selects one particular prepared states. At the upper boundary it selects one particular final state of the scattering process.

2. Restrict for definiteness the consideration to the lower boundary of CD. Denote also M_K by M. At the lower boundary of CD the selection of prepared state - that is preparation process- means the reduction

   ∑_m⁺n^-M^+/-_m⁺n^-| m⁺> | n^-> → ∑_n^-M^+/-_m⁺n^-| m⁺> | n^-> .

   The reduction probability is given by

   p_{m= ∑n^- | Mm⁺n^-|² = ρm⁺m⁺ .}

   For this state the lower boundary carries a prepared state with the quantum numbers of state | m₊> . For density matrix which is unit matrix (this option giving pure state might not be possible) one has p_m=1.

State function reduction process

The process which is the analog of measuring the final state of the scattering process is also needed and would mean state function reduction at the upper end of CD - to state | n^-> now.

1. It is impossible to reduce to arbitrary state | m₊> | n_-> and the reduction must at the upper end of CD must mean a loss of preparation at the lower end of CD so that one would have kind of time flip-flop!

2. The reduction probability for the process

   | m₊ >== ∑_n^-M_m⁺n^-| m⁺> | n^-> → n_->= ∑_m⁺M_m⁺n^-| m⁺> | n^->

   would be

   p_{mn =| Mmn|² .}

   This is just what one would expect. The final outcome would be therefore a state of type | n^-> and - this is very important- of the same type as the state from which the process began so that the next process is also of type U⁺ and one can say that a definite arrow of time prevails.

3. Both the preparation and reduction process involves also a cascade of state function reductions leading to a choice of state basis corresponding to eigenstates of density matrices between subsystems.

Can the arrow of geometric time change?

A highly interesting question is what happens if the first state preparation leading to a state | K⁺> is followed by a U-process of type U^- rather than by the state function reduction process |K⁺> → |L^->. Does this mean that the arrow of geometric time changes? Could this change of the arrow of geometric time take place in living matter? Could processes like molecular self assembly be entropy producing processes but with non-standard arrow of geometric time? Or are they processes in which negentropy increases by the fusion of negentropic parts to larger ones? Could the variability relate to sleep-awake cycle and to the fact that during dreams we are often in our childhood and youth. Old people are often said to return to their childhood. Could this have more than a metaphoric meaning? Could biological death mean return to childhood at the level of conscious experience? I have explained the recent views about the arrow of time here .

For background see new chapter Construction of Quantum Theory: More About Matrices of "Towards M-matrix" .

How it went?

Mark McWilliams requested some kind of summary about the development of TGD, and I decided to write an article about the the history of TGD. I could not avoid telling also about turning points of my personal life since my work and life are to high extent one and the same thing.

I have tried to represent the development chronologically but I must confess that I have forgotten precise dates so that the chronology is not exact. Very probably I have also forgotten many important ideas and many side tracks which led nowhere. Indeed, the study of the tables of contents of books and old blog postings and What's New articles at the homepage forces me to wonder how I can forget something so totally.

The article should help a novice to get an overall view about the basic ideas of TGD are their evolution during these 34 years. To myself a real surprise was to see how many deep ideas have emerged after 2005: one can really speak about a burst of new ideas. Most of them relate to the evolution of the mathematical aspects of TGD and to their physical interpretation but also the experimental input from LHC, Fermilab, and elsewhere has played a decisive role in stimulating ideas about the interpretation of the theory.

Unavoidably the emphasis is on the latest ideas and there is of course the risk that some of them are not here to stay. Even during writing process some ideas developed into more concrete form. A good example is the vision about what happens in quantum jump and what the unitarity of U-matrix really means, how M-matrices generalize to form Kac-Moody type algebra, and how the notion of quantum jump in zero energy ontology (ZEO) reproduces the basic aspects of quantum measurement theory. Also a slight generalization of quantum arithmetics suggested itself during the preparation of the article.

I gave to the article a title which is easy to guess: "Evolution of TGD". It can be found at my homepage which is now living at webhotel with address http://tgdtheory.com/.

Note: The links of old postings to my homepage do not work anymore. Apologies. To get to the link work one can replace "http://tgd.wippiespace.com/" with "http://tgdtheory.com/", and if this does not work, with "http://tgdtheory.com/public_html/".

Does 2-adic quantum arithmetics explain p-adic length scale hypothesis?

For p=2 quantum arithmetics looks singular at the first glance. This is actually not the case since odd quantum integers are equal to their ordinary counterparts in this case. This applies also to powers of two interpreted as 2-adic integers. The real counterparts of these are mapped to their inverses in canonical identification.

Clearly, odd 2-adic quantum quantum rationals are very special mathematically since they correspond to ordinary rationals. It is fair to call them "classical" rationals. This special role might relate to the fact that primes near powers of 2 are physically preferred. CDs with n=2^k would be in a unique position number theoretically. This would conform with the original - and as such wrong - hypothesis that only these time scales are possible for CDs. The preferred role of powers of two supports also p-adic length scale hypothesis.

The discussion of the role of quantum arithmetics in the construction of generalized Feynman diagrams allows to understand how for a quantum arithmetics based on particular prime p particle mass squared - equal to conformal weight in suitable mass units - divisible by p appears as an effective propagator pole for large values of p. In p-adic mass calculations real mass squared is obtained by canonical identification from the p-adic one. The construction of generalized Feynman diagrams allows to understand this strange sounding rule as a direct implication of the number theoretical universality realized in terms of quantum arithmetics.

Witten about mass gap

Witten has a nice talk about mass gap problem in 3-D (mostly) and 4-D gauge theories demonstrating how enormous his understanding and knowledge about mathematical physics is. Both Peter Woit and Kea have commented it.

In 3-D case the coupling strength g² has the dimension of inverse length and therefore it would not be surprising if mass gap would emerge. Witten argues that by adding to the theory a Chern-Simons term the theory could reduce in long length scales to non-trivial topological QFT at the IR limit. This would be also a nice manner to resolve the IR difficulties of 3-D gauge theories. Could one imagine effective reduction to topological QFT in long length scales also in 4-D case as a solution to IR divergences?

In D=4 the situation is much more difficult since the gauge coupling is dimensionless. My un-educated is opinion is that the proper question is whether the theory actually exists mathematically and my equally un-educated guess is that it does not - unless one brings in the mass scale somehow by hand. The standard Muenchausen trick to bring the scales in perturbation theory is via UV and IR cutoffs. This is going outside what one means with gauge theory strictly mathematically. In order to make progress, one must bring in the new physics and mathematics. A rigorous mathematical formulation of 4-D gauge theory is not enough: it simply does not exist since something very important is missing.

TGD view about the mass gap problem

TGD is one proposal for what this new physics and mathematics could be. I do not try to re-explain in any detail what this new physics and mathematics might be since I have done this explaining for 6 years in this blog. The basic statement is however that the fundamental UV length scale must be present explicitly in the definition of the theory and must have concrete geometric interpretation rather than being a dimensional number like string tension. In TGD framework it corresponds to the "radius" of CP₂, which is fixed from simple symmetry arguments as the only possible choice. This scale is not an outcome of some conceptually highly questionable procedure like spontaneous compactification, which has paralyzed theoretical physics for more than two decades and led to the landscape problem and the proposal to bring anthropic principle to physics - something extremely uninviting for anyone who has spent few minutes by trying to understand what one can say about consciousness as a physicist and mathematician.

To my rebellionary view the mathematics of standard gauge theories is not enough.

1. Quite a far reaching generalization is needed besides the replacement of the recent view about space-time with the identification of space-times as 4-surfaces.

2. The usual positive energy ontology having its roots in Newtonian mechanics based on absolute time (Hamiltonian approach especially) must be replaced with zero energy ontology which is natural in the relativistic context.

3. A further generalization is number theoretical universality requiring that the physics in different number fields must be unified to single coherent whole.

4. In some of the latest postings I have explained how the number theoretical universality would be realized in terms of quantum arithmetics - something also missing from ordinary gauge theory but for the existence of which there exist indications (quantum groups, inclusions of hyper-finite factors, and Shnoll effect on experimental side). In particular, the size scales of CDs coming as powers of 2 correspond to p=2 quantum arithmetics, which is very special in the sense that for odd integers it is just the ordinary arithmetics. For other primes p the corresponding p-adic length scale is in preferred position since the states with mass squared proportional to p as almost massless states giving an almost pole to the propagator which is given in terms of M² momentum. I dare to hope that these observations finally answer the question why p-adic primes near powers of two are favored physically.

Further comments about mass gap

I cannot avoid the temptation to represent some further comments related to how the mass gap - or rather a hierarchy of mass gaps defined by p-adic mass scales in turn expressible in terms of p-adic prime and the fundamental mass scale defined by CP₂ mass - emerges in TGD.

1. One of the surprises of zero energy ontology was that that all - including those associated with virtual particles - braid strands carrying fermion numbers are on massless on mass shell states with possibly negative sign of energy so that wormhole contact can have space-like virtual net momentum. This leads to extremely powerful restrictions of loops integrals and guarantees finiteness and with certain additional natural assumptions deriving from ZEO the number of contributing diagrams is finite (discussed in recent postings: see this and this), and therefore guarantees algebraic universality (sum of infinite number of rationals (rational functions) need not be rational (rational function)!).

2. Quite recently I have learned finally to accept that for generalized Feynman diagrams the presence of preferred M²⊂M⁴ having interpretation in terms of quantization axis of energy and spin is unavoidable (see for instance this). Of course, also propagators for on mass shell massless states are literally infinite unless one restricts the momentum in propagator to its M² projection. There is integral over different choices of M² so that Poincare invariance is not lost. Also number theoretic vision forces M² with an interpretation as commutative subspace of complexified octonions. The last posting about Very Special Relativity and TGD gave one additional justification M².

3. Witten talks about 3-D gauge theories with emphasis on Chern-Simons term and the idea that in long length scales one obtains a non-trivial topological QFT for non-trivial mass gap. In TGD framework effective 2-dimensionality - or strong form of holography - follows from strong form of General Coordinate Invariance and for preferred extremals of Kähler action the action reduces to Chern-Simons terms if weak form of electric-magnetic duality holds true at the space-like 3-D surfaces at the ends of space-time sheet and at wormhole throats.

   The special features of light-like 3-surfaces and boundaries of CDs is that they allow an extension of 2-D conformal invariance by their metric 2-dimensionality: this actually raises 4-D space-time and 4-D Minkowski space in completely unique position mathematically. An extremely simple and profound discovery, whose communication has turned out to be impossible- I think that even my cat is able to understand its significance-: what is wrong with these bright-minded colleagues in their academies;-)? An interesting question raised by Witten's talk is whether also TGD as almost topological TGD in some sense reduces to topological QFT at long length scales for given p-adic length scale. Exponential decrease of correlation function as function of distance might imply this but what happens on light-like boundaries of CDs?

4. There are also open questions. For instance, should one assign different M² to each sub-CD of CD or to each propagator line connecting the 3-vertices? One can be even more general and also consider a local choices of M² defined by an integrable distribution of M^{2⊂ M4 defining the analog of string world sheet.}

The special role of M²⊂M⁴ in relation to mass gap

The special role of M²⊂M⁴ in the construction of generalized Feynman diagrams deserves additional comments.

1. What is remarkable that gauge conditions generalize in the sense that it is M² momentum that appears in gauge conditions so that also the third polarization for gauge bosons creeps into the spectrum and even photon, gluons, and gravitons would receive small mass given in terms of IR cutoff given by the largest causal diamond in the hierarchy of causal diamonds defining the experimental range of length scales about which experimentalist can gain information. This is of course extremely natural from the view point of experimentalist.

2. Physical particles are bound states of massless states with parallel M² momenta assigned with the wormhole throats of the same wormhole contact. Also this bring in the overall important IR cutoff - and thus mass gap - not present in gauge theories. UV cutoff given by the size of the smallest CD gives UV cutoff. As already mentioned, there is no breaking of Poincare invariance.

3. The highly non-trivial question is how the p-adic mass calculations can be consistent with the massless of braid strands. How can wormhole throat satisfy stringy mass formula if it is massless? One of the latest realizations is that that it is not the full M⁴mass squared but the longitudinal M² mass squared, which is quantized by the stringy mass formula! The modified Dirac equation indeed strongly suggests that M² momenta have integer valued components. I could not however decide whether only hyper-complex primes should be accepted: it now seems that integers coming as multiple of given hyper-complex integer, whose modulus square is prime, must be allowed. Particles would get longitudinal mass squared by p-adic thermodynamics and this mass would be the observed mass. Mass gap again but only in longitudinal degrees of freedom

4. There is also experimental support for the necessity of introducing M². In QCD one characterizes partons with M² momentum and this again brings to gauge theory as a purely mathematical construct - something which really is not there! The great experimental question is whether Higgs exists or not. In TGD Higgs mechanism is replaced by a microscopic mechanism based on p-adic thermodynamics and identification of the mass squared as longitudinal mass squared (in Lorentz invariance manner since one averages over different M²:s). The natural prediction is that instead of Higgs there is entire M⁸⁹ hadron physics to be discovered. If Higgs really is there as some bloggers have already revealed to us;-), profound re-interpretation of TGD is necessary.

New physics in non-perturbative sector

Asymptotically free gauge theories can handle the UV divergences by using renormalization group approach bringing in a scale analogous to QCD Lambda. Lambda defines the IR scale identified as the length scale associated with hadronization and confinement. One gets rid of UV scale altogether (but not from the mathematically tedious and ugly procedures removing UV infinities). In perturbative gauge theories IR remains however a source of difficulties since one really does not know how to calculate anything since the proposed expression for IR scale is non-analytic function of coupling constant strength (expressible in terms of exp(-8π²ℏ/g²), I hope I remember correctly). Also twistor approach is plagued by IR divergences. These difficulties are of course the reason for arranging a conference about mass gap problem! I do not however believe that the mass gap is a mathematical problem within the framework of 4-D gauge theories. One must go outside the system.

In TGD framework magnetic flux tubes are the concrete classical space-time correlate for non-perturbative aspects of quantum theory and appear in all applications from primordial cosmology to biology to elementary particle physics. They are not present in gauge theories. They are obtained as deformations of what I call cosmic strings, which are Cartesian products of string world sheets in M² with 2-D complex sub-manifolds of CP². In this case one cannot anymore speak about space-time as a small deformation of Minkowski space. The quantized size scale of the complex sub-manifolds brings in the scale via string tension. Wormhole contacts themselves are magnetic monopoles and thus homologically non-trivial surfaces of CP₂ with quantized area so that again the fundamental mass scale creeps in. Note that the Kähler action for the magnetic flux tubes and also for deformations CP₂ vacuum extremals contains a power of exp(-8π²ℏ/g²) giving rise to non-analyticity in g. In gauge theories classical action for instantons would give rise kind of factor.

Note: The address of my homepage has changed and the links to my homepage from the earlier postings will fail. The cure of the problem is the replacement of tgd.wippiespace or tgdq.wippiespace in the address with tgdtheory.

Very Special relativity and preferred role of M2 for generalized Feynman graphs

The preferred role of M² in the construction of generalized Feynman diagrams could be used as a criticism. Poincare invariance is lost. The first answer to the criticism is that one integrates of the choices of M² so that Poincare invariance is lost. One can however defend this assumption also from different view point. Actually Glashow and Cohen did this in their Very Special Relativity proposal! While scanning old files, I found an old text about Very Special Relativity of Glashow and Cohen, and realized that it relates very closely to the special role of M² in the construction of generalized Feynman diagrams. There is article Very Special Relativity and TGD at my homepage but for some reason the text has disappeared from the book that contained it. I add the article more or less as such here.

Configuration space ("world of classical worlds", WCW) decomposes into a union of sub-configuration spaces associated with future and past light-cones and these in turn decompose to sub-sub-configuration spaces characterized by selection of quantization axes of spin and color quantum numbers. At this level Poincare and even Lorentz group are reduced. The possibility that this kind of breaking might be directly relevant for physics is discussed below.

One might think that Poincare symmetry is something thoroughly understood but the Very Special Relativity proposed by nobelist Sheldon Glashow and Andrew Cohen suggests that this might belief might be wrong. Glashow and Cohen propose that instead of Poincare group, call it P, some subgroup of P might be physically more relevant than the whole P. To not lose four-momentum one must assume that this group is obtained as a semi-direct product of some subgroup of Lorentz group with translations. The smallest subgroup, call it L₂, is a 2-dimensional Abelian group generated by K_x+J_y and K_y-J_x. Here K refers to Lorentz boosts and J to rotations. This group leaves invariant light-like momentum in z direction. By adding J_z acting in L₂ like rotations in plane, one obtains L₃, the maximal subgroup leaving invariant light-like momentum in z direction. By adding also K_z one obtains the scalings of light-like momentum or equivalently, the isotropy group L₄ of a light-like ray.

The reasons why Glashow and Cohen regard these groups so interesting are following.

1. All kinematical tests of Lorentz invariance are consistent with the reduction of Lorentz invariance to these symmetries.

2. The representations of group L₃ are one-dimensional in both massive and massless case (the latter is familiar from massless representations of Poincare group where particle states are characterized by helicity). The mass is invariant only under the smaller group. This might allow to have left-handed massive neutrinos as well as massive fermions with spin dependent mass.

3. The requirement of CP invariance extends all these reduced symmetry groups to the full Poincare group. The observed very small breaking of CP symmetry might correlate with a small breaking of Lorentz symmetry. Matter antimatter asymmetry might relate to the reduced Lorentz invariance.

The idea is highly interesting from TGD point of view. The groups L₃ and L₄ indeed play a very prominent role in TGD.

1. The full Lorentz invariance is obtained in TGD only at the level of the entire configuration space ("world of classical worlds", WCW) which is union over sub-configuration spaces associated with what I call causal diamonds (see this). These sub-configuration spaces decompose further into a union of sub-sub-configuration spaces for which a choice of quantization axes of spin reflects itself at the level of generalized geometry of the imbedding space (quantum classical correspondence requires that the choice of quantization axes has imbedding space and space-time correlates) (see this). The construction of the geometry for these sub-worlds of classical worlds reduces to light-cone boundary so that the little group L₃ leaving a given point of light-cone boundary invariant is in a special role in TGD framework.

2. The selection of a preferred light-like momentum direction at light-cone boundary corresponds to the selection of quantization axis for angular momentum playing a key role in TGD view about hierarchy of Planck constants associated with a hierarchy of Jones inclusions implying a breaking of Lorentz invariance induced by the selection of quantization axis. The number theoretic vision about quantum TGD implies a selection of two preferred axes corresponding to time-like and space-like direction corresponding to real and preferred imaginary unit for hyper-octonions (see this). In both cases L₄ emerges naturally.

3. The TGD based identification of Kac-Moody symmetries as local isometries of the imbedding space acting on 3-D light-like orbits of partonic 2-surfaces involves a selection of a preferred light-like direction and thus the selection of L₄.

4. Also the so called massless extremals representing a precisely targeted propagation of patterns of classical gauge fields with light velocity along typically cylindrical tubes without a change in the shape involve L₄. A very general solution ansatz to classical field equations involves a local decomposition of M⁴ to longitudinal and transversal spaces and selection of a light-like direction (see this).

5. The parton model of hadrons assumes a preferred longitudinal direction of momentum and mass squared decomposes naturally to longitudinal and transversal mass squared. Also p-adic mass calculations rely heavily on this picture and thermodynamics mass squared might be regarded as a longitudinal mass squared (see this). In TGD framework right handed covariantly constant neutrino generates a super-symmetry in CP₂ degrees of freedom and it might be better to regard left-handed neutrino mass as a longitudinal mass.

This list justifies my own hunch that Glashow and Cohen might have discovered something very important.

Reader interested in background can consult to the article Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams and the new chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-Matrix".

MY HOMEPAGE ADDRESS CHANGES!!

I accidentally learned that the host providing the computer storage finishes all this kind of activity. Accidentally because the finnish enterprises behind Wippie - Saunalahti and Elisa - had decided to do the whole thing in complete secrecy and had cut both phone and email connections related to wippie-space.

In trying to contact the only response was that your password is wrong. Many extremely frustrated and irritated people in web told about this. The big bosses have however calculated that the people practically losing their how life work in this manner are a minority which they need not care of.

It was extremely difficult to find what is happening and what I should do. It turned out that I cannot use the old addresses.

The old web addresses

tgd.wippiespace. com

and

tgdq.wippiespace.com

will be replaced by

tgdtheory.com.

For some time I will disappear from web just as I did for three years ago when Helsinki University decided to get rid of my web presence and also managed quite well.

I will do best to update the web addresses appearing in the books and articles in home page and also in viXra.org but this will take time. Apologies. The new web page should be at use within few days.

Any concrete ideas to help in the situation are welcome!! In particular, there should be manners to communicate search engines about the change of the address.

Number theoretical universality and quantum arithmetics, renormalization, and relation of TGD to N=4 SYM

In the previous posting I already discussed the proposal for how twistorial construction could generalize to apply to generalized Feynman diagrams in TGD framework. During last days I have made a further progress in understanding of the number theoretical aspects of the proposed construction.

In particular, I finally have simple and general justification for the hypothesis that length scales coming as powers of two and p-adic length scales associated p-adic primes near powers of 2 are very special. The explanation is extremely simple: quantum arithmetics is characterized by prime p and for p=2 all odd quantum integers are identical with ordinary integers so that only powers of two mapped to their inverses distinguish 2-adic quantum arithematics from the ordinary one.

I have also corrected some erratic statements in the view about coupling constant evolutiont and compared the approach to that of N=4 SYM developed by Nima Arkani Hamed and others. Therefore this posting contains some material overlapping with the previous posting.

Number theoretical universality

The construction of the amplitudes should be number theoretically universal meaning that amplitudes should make sense also in p-adic number fields or perhaps in adelic sense in the tensor product of p-adic numbers fields. Quantum arithmetics is characterized by p-adic prime and canonical identification mapping p-adic amplitudes to real amplitudes is expected to make number theoretical universality possible.

This is achieved if the amplitudes should be expressible in terms of quantum rationals and rational functions having quantum rationals as coefficients of powers of the arguments. This would be achieved by simply mapping ordinary rationals to quantum rationals if they appear as coefficients of polynomials appearing in rational functions.

Quantum rationals are characterized by p-adic prime p and p-adic momentum with mass squared interpreted as p-adic integer appears in the propagator. If M² mass squared is proportional to this p-adic prime p, propagator behaves as 1/P²∝ 1/p, which means that one has pole like contribution for these on mass shell longitudinal masses. p-Adic mass calculations indeed give mass squared proportional to p. The real counterpart of propagator in canonical identification is proportional to p. This would select the all CD characterized by n divisible by p as analogs of propagator poles. Note that the infrared singularity is moved and the largest p-adic prime appearing as divisor of integer characterizing the largest CD indeed serves as a physical IR cufoff.

It would seem that one must allow different p-adic primes in the generalized Feynman diagram since physical particles are in general characterized by different p-adic primes. This would require the analog of tensor product for different quantum rationals analogous to adeles. These numbers would be mapped to real (or complex) numbers by canonical identification.

How to get only finite number of diagrams in a given IR and UV resolution?

In gauge theory one obtains infinite number of diagrams. In zero energy ontology the overall important additional constraint comes from on mass shell conditions at internal lines and external lines and from the requirement that the M² momentum squared is quantized for super-conformal representation in terms of stringy mass squared spectrum.

This condition alone does not however imply that the number of diagrams is finite. If forward scattering diagram is non-vanishing also scattering without on mass shell massive conditions on final state lines is possible. One can construct diagrams representing a repeated n→ n scattering and combining these amplitudes with non-forward scattering amplitude one obtains infinite number of scattering diagrams with fixed initial and final states. Number theoretic universality however requires that the number of the contributing diagrams must be finite unless some analytic miracles happens.

The finite number of diagrams could be achieved if one gives for the vision about CDs within CDs a more concrete metric meaning. In spirit of Uncertainty Principle, the size scale of the CD defined by the temporal distance between its tips could correspond to the inverse of the momentum scale defined as its inverse. A further condition would be that the sub-CDs and their Lorentz boosts are indeed within the CD and do not overlap. Obviously the number of diagrams representing repeated n-n scattering forward scattering is finite if these assumptions are made. This would also suggest a scale hierarchy in powers of 2 for CDs: the reason is that given CD with scale T=nT(CP₂) can contain two non-overlapping sub-CDs with the same rest frame only if sub-CD has size scale smaller than nT_CP2/2. This applies also to the Lorentz boosts of the sub-CDs.

Amplitudes would be constructed by labeling the CDs by integer n defining its size scale. p-Adicity suggests that the factorization of n to primes must be important and if n=p condition holds true, a new resonant like contribution appears corresponding to p-adic diagrams involving propagator.

Should one allow all M² momenta in the loops in all scales or should one restrict the M² momenta to have a particular mass squared scale determined somehow by the size of CD involved? If this kind of constraint is posed it must be posed in mathematically elegant manner and it is not clear how to to this.

Is this kind of constraint really necessary? Quantum arithmetics for the length scale characterized by p-adic prime p would make M² mass squared values divisible by p to almost poles of the propagators, and this might be enough to effectively select the particular p and corresponding momentum scale and CD scale. Consider only the Mersenne prime M₁₂₇=2¹²⁷-1 as a concrete example.

How to realize the number theoretic universality?

One should be able to realized the p-adicity in some elegant manner. One must certainly allow different p-adic primes in the same diagram and here adelic structure seems unavoidable as tensor product of amplitudes in different p-adic number fields or rather - their quantum arithmetic counterparts characterized by a preferred prime p and mapped to reals by the substitution p→ 1/p. What does this demand?

1. One must be able to glue amplitudes in different p-adic number fields together so that the lines in some case must have dual interpretation as lines of two p-adic number fields. It also seems that one must be able to assign p-adic prime and quantum arithmetics characterized by a given prime p to to a given propagator line. This prime is probably not arbitrarily and it will be found that it should not be larger than the largest prime dividing n characterizing the CD considered.

2. Should one assign p-adic prime to a given vertex?
   1. Suppose first that bare 3-vertices reduce to algebraic numbers containing no rational factors. This would guarantee that they are same in both real and p-adic sense. Propagators would be however quantum rationals and depend on p and have almost pole when the integer valued mass squared is proportional to p.

   2. The radiative corrections to the vertex would involve propagators and this suggests that they bring in the dependence on p giving rise to p-adic coupling constant evolution for the real counterparts of the amplitudes obtained by canonical identification.
      1. Should also vertices obey p-adic quantum arithmetics for some p? What about a vertex in which particles characterized by different p-adic primes enter? Which prime defines the vertex or should the vertex somehow be multi-p p-adic? It seems that vertex cannot contain any prime as such although it could depend on incoming p-adic primes in algebraic or transcendental manner.

      2. Could the radiative corrections sum up to algebraic number depending on the incoming p-adic primes? Or are the corrections transcendental as ordinary perturbation theory suggests and involve powers of π and logarithm of mass squared and basically logarithms of some primes requiring infinite-dimensional transcendental extension of p-adic numbers? If radiative corrections depend only on the logarithms of these primes p-adic coupling constant evolution would be obtained. The requirement that radiative vertex corrections vanish does not look physically plausible.

   3. Only CDs corresponding to integers m< n would be possible as sub-CDs. A geometrically attractive possibility is that CD characterized by integer n allows only propagator lines which correspond to prime factors of integers not larger than the largest prime dividing n in their quantum arithmetics. Bare vertices in turn could contain only primes larger than the maximal prime dividing n. This would simplify the situation considerably- This could give rise to coupling constant evolution even in the case that the radiative corrections are vanishing since the rational factors possibly present in vertices would drop away as n would increase.

   4. Integers n=2^k give rise to an objection. They would allow only 2-adic propagators and vertices containing no powers of 2. For p=2 the quantum arithmetics reduces to ordinary arithmetics and ordinary rationals correspond to p=2 apart from the fact that powers of 2 mapped to their inverses in the canonical identification. This is not a problem and might relate to the fact that primes near powers of 2 are physically preferred. Indeed, the CDs with n=2^k would be in a unique position number theoretically. This would conform with the original - and as such wrong - hypothesis that only these time scales are possible for CDs. The preferred role of powers of two supports also p-adic length scale hypothesis.

These observations give rather strong clues concerning the construction of the amplitudes. Consider a CD with time scale characterized by integer n.

1. For given CD all sub-CDs with m<n are allowed and all p-adicities corresponding to the primes appearing as prime factors of given m are possible. m=2^k are in a preferred position since p=2 quantum rationals not containing 2 reduce to ordinary rationals.

2. The geometric condition that sub-CDs and their boosts remain inside CD and do not overlap together with momentum conservation and on-mass-shell conditions on internal lines implies that only a finite number of generalized Feynman diagrams are possible for given CD. This is essential for number theoretical universality. To each sub-CD one must assign its moduli spaces including its not-too-large boosts. Also the planes M² associated with sub-CDs should be regarded as independent and one should integrate over their moduli.

3. The construction of amplitudes with a given resolution would be a process involving a finite number of steps. The notion of renormalization group evolution suggests a generalization as a change of the amplitude induced by adding CDs with size smaller than smallest CDs and their boosts in a given resolution.

4. It is not clear whether increase of the upper length scale interpreted as IR cutoff makes sense in the similar manner although physical intuition would encourage this expectation.

How to understand renormalization flow in twistor context?

In twistor context the notion of mass renormalization is not straightforward since everything is massless. In TGD framework p-adic mass scale hypothesis suggests a solution to the problem.

1. At the fundamental level all elementary particles are massless and only their composites forming physical particles are massive.
2. M² mass squared is given by p-adic mass calculations and should correspond to the mass squared of the physical particle. There are contributions from magnetic flux tubes and in the case of baryons this contribution dominates.
3. p-Adic physics discretizes coupling constant flow. Once the p-adic length scale of the particle is fixed its M² momentum squared is fixed and massless takes care of the rest.

Consider now how renormalization flow would emerge in this picture. At the level of generalized Feynman diagrams the change of the IR (UV) resolution scale means that the maximal size of the CDs involve increases (the minimal size of the sides decreases).

Concerning the question what CD scales should be allowed, the situation is not completely clear.

1. The most general assumption allows integer multiples of CP₂ scale and would guarantee that the products of hermitian matrices and powers of S-matrix commuting with them define Kac-Moody type algebra assignable to M-matrices. If one uses in renormalization group evolution equation CDs corresponding to integer multiples of CP₂ length scale, the equation would become a difference equation for integer valued variable.

2. p-Adicity would suggest that the scales of CDs come as prime multiples of CP₂ scale. The proposed realization of p-adicity indeed puts CDs characterized by p-adic primes p in a special position since they correspond to the emergence of a vertex corresponding to p-adic prime p which depends on p in the sense that the radiative corrections to 3-vertex can give it a dependence on log(p). This requires infinite-D transcendental extension of p-adic numbers.

   As far as coupling constant evolution in strict sense is considered, a natural looking choice is evolution of vertices as a function of p-adic primes of the particles arriving to the vertex since radiative corresponds are expected to depend on their logarithms.

3. p-Adic length scale hypothesis would allow only p-adic length scales near powers of two. There are excellent reasons to expect that these scales are selected by a kind of evolutionary process favoring those scales for CDs for which particles are maximally stable. The fact that quantum arithmetics for p=2 reduces to ordinary arithmetics when quantum integers do not contain 2 raises with size scales coming as powers of 2 in a special position and also supports p-adic length scale hypothesis.

Renormalization group equations are based on studying what happens in an infinitesimal reduction of UV resolution scale would mean. Now the change cannot be infinitesimal but must correspond to a change in the scale of CD by one unit defined by CP₂ size scale.

1. The decrease of UV cutoff means addition of new details represented as bare 3-vertices represented by truncated triangle having size below the earlier length scale resolution. The addition can be done inside the original CD and inside any sub-CD would be in question taking care that the details remain inside CD. The hope is that this addition of details allows a recursive definition. Typically addition would involve attaching two sub-CDs to propagator line or two propagator lines and connecting them with propagator. The vertex in question would correspond to a p-adic prime dividing the integer characterizing the sub-CDs. Also the increase of the shortest length scale makes sense and means just the deletion of the corresponding sub-CDs. Note that also the positions of sub-CDs inside CD manner since the number of allowed boosts depends on the position. This would mean an additional complication.

2. The increase of IR cutoff length means that the size of the largest CD increases. The physical interpretation would be in terms of the time scale in which one observes the process. If this time scale is too long, the process is not visible. For instances, the study of strong interactions between quarks requires short enough scale for CD. At long scales one only observes hadrons and in even longer scales atomic nuclei and atoms.

3. One could also allow the UV scale to depend on the particle. This scale should correspond to the p-adic mass scales assignable to the stable particle. In hadron physics this kind of renormalization is standard operation.

Comparison with N=4 SYM

The ultimate hope is to formulate all these ideas using precise formulas. This goal is still far away but one can make trials. Let us first compare the above proposal to the formalism in N=4 SYM.

1. In the construction of twistorial amplitudes the 4-D loop integrals are interpreted as residue integrals in complexified momentum space and reduces to residues around the poles. This is analogous to using "on mass shell states" defined by this poles. In TGD framework the situation is different since one explicitly assigns massless on-mass-shell fermions to braid strands and allows the sign of the energy to be both positive and negative.

2. Twistor formalism and description of momentum and helicity in terms of the twistor (λ,μ) certainly makes sense for any spin. The well-known complications relate to the necessity to use complex twistors for M⁴ signature: this would correspond to complexified space-time or momentum space. Also region momenta and associated momentum twistors are the TGD counterparts so that the basic building bricks for defining the analogs of twistorial amplitudes exist.

An important special feature is that the gauge potential is replaced with its N=4 super version.

1. This has some non-generic implications. In particular gluon helicity -1 is obtained from 1 ground state by "adding" four spartners with helicity +1/2 each. This interpretation of the two helicities of a massless particle is not possible in N<4 theories nor in TGD and the question is whether this is something deep or not remains open.

2. In TGD framework it is natural to interpret all fermion modes associated with partonic 2-surface (and corresponding light-like 3-surfaces) as generators of super-symmetry and fermions are fundamental objects instead of helicity +1 gauge bosons. Right-handed neutrino has special role since it has no electroweak or color interactions and generates SUSY for which breaking is smallest.

3. The N=2 SUSY generated by right-handed neutrino and antineutrino is broken since the propagator for states containing three fermion braid strands at the same wormhole throat behaves like 1/p³: this is already an anyon-like state. The least broken SUSY is N=1 SUSY with spartners of fermions being spin zero states. The proposal is that one could construct scattering amplitudes by using a generalize chiral super-field associated with N equal to the number of spinor modes acting on ground state that has vanishing helicity. For N=4 it has helicity +1. This would suggest that the analogs of twistorial amplitudes exist and could even have very similar formulas in terms of twistor variables.

4. The all-loop integrand for scattering amplitudes in planar N=4 SYM relies of BCFW formula allowing to sew two n-particle three amplitudes together using single analog of propagator line christened as BCFW bridge. Denote by Yn,k,l n-particle amplitudes with k positive helicity gluons and l loops. One can glue Y_nL,kL,lL and Y_nR,kR,lR by using BCFW bridge and add "entangled " removal of two external lines of Y(n+2,k+1,l-1) amplitude with n= n_L+n_R-2,k=k_L+k_R,l=l_L+l_R to get Y_n,k,l amplitude recursively by starting from just two amplitudes defining the 3-vertices. The procedure involves only residue integral over the Gl(k,n) for a quantity which is Yangian invariant. The question is whether one could apply this procedure by replacing N=4 SUSY with SUSY in TGD sense and generalizing the fundamental three particle vertices appropriately by requiring that they are Yangian invariants?

5. One can also make good guesses for the BCFW bridge and entangled removal. By looking the structure of the amplitudes obtained by the procedure from 3-amplitudes, one learns that one obtains tree diagrams for which some external lines are connected to give loop. The simplest situation would be that BCFW bridge corresponds to M² fermion propagator for a given braid strand and entangled removal corresponds to a short cut of two external lines to internal loop line. One would have just ordinary Feynman graphs but vertices connected with Yangian invariants (not that there is sum over loop corrections). It should be easy to kill this conjecture.

Reader interested in background can consult to the article Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams and the new chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-Matrix".