Experimental support for binocular rivalry as a quantum phenomenon

For years ago I constructed a quantum model for binocular rivalry and generalized it to a general model of volitional act as a quantum jump selecting not only between alternative motor actions but also between percepts. In this model different alternatives were represented as superpositions of neural firing patterns. The model allows to see sensory perception as an active volitional process (at some level of hierarchy of selves) and explains sensory rivalry as a quantum phenomenon.

The work of Efstratios Manaosakis

I learned from New Scientist that physicist Efstratios Manousakis has published an interesting work about binocular rivalry providing experimental support for this model.

Recall that the classical demonstration of binocular rivalry is a pattern experienced either as a vase or two opposite faces. The two percepts alternate with some frequency and it is not possible to consciously experience both patterns simultaneously. This has led Manousakis to consider the idea that binocular rivalry could provide direct evidence for the notion of quantum consciousness. The obvious idea is that either of the percepts results by a state function reduction from the superposition of both percepts.

The model predicts that the flip rate correlates with neuronal firing rate. The prediction is confirmed by using as subjects persons who have a reduced firing rate due to the use of LSD. The work of Manousakis might turn out to be an important step of progress in the development of theories of quantum consciousness and might help also main stream physicists to get rid of their atavistic fears relating quantum consciousness.

Justification for the model in TGD framework

The finding conforms with TGD view about quantum jump in which $U$ process creates a quantum superposition and state function reduction selects either of the percepts. TGD however brings in new elements.

1. In the conceptual framework of the standard quantum mechanics there is no known mechanism making possible macroscopic quantum coherence in the time scales involved. If dark matter with large $\hbar$ is involved with the formation of conscious percept there is no problem in understanding the time scales in question. Actually a hierarchy of rivalries of various kinds in various time scales is predicted corresponding to the p-adic time scale hierarchy and hierarchy of Planck constants.

2. Another ingredient which is new from the point of view of standard quantum mechanics is that the hierarchy of Planck constants implies self hierarchy actually identifiable actually as a hierarchy of quantum jumps having quantum jumps within quantum jumps ..... The fractal structure of state function reduction process means that it is possible have macroscopic quantum behavior in given time scale but dissipative self-organization in shorter time scales.

   This is actually not new: in hadron physics hadrons are described as quantum systems whereas parton dynamics in the shorter time scales is assumed to be dissipative. In the recent case this means the possibility of quantum superposition of dissipative self-organization processes involved with the formation of neuronal correlates of percepts and proceeding in time scales of order milliseconds considerably shorter than the time scale of binocular rivalry.

TGD based model for rivalry and its generalization

The TGD based quantum model for binocular rivalry relies on the idea that the formation of quantum superposition of competing percepts is somewhat analogous to quantum computing in which large number of quantum parallel computations are carried out and one computation is selected as the computation halts.

In TGD framework one does not assign a conscious experience to the mere state function reduction part of quantum jump and the question arises whether the transitions periods are experienced consciously as a kind of inability to disentangle what is there and if so what is the subjective time duration of these periods and is it very short in absence of some other periodic sensory input defining a clock. The TGD prediction would be that the mental image defined by the percept is absent but consciousness is not lost.

The formation of quantum superposition of right and left percepts has evolutionary advantages which suggest also a generalization to a model of volitional action as a selection between neural firing patterns leading to alternative motor actions.

1. The formation of superposition would be metabolically advantageous. In the classical world one should form both right and left percept simultaneously. The associated self-organization process requires a metabolic energy feed. When only single brain hemisphere forms the percept and one has quantum superposition of right and left percepts metabolic energy feed is reduced by factor 1/2. A highly synchronous neural firing distinguishes the perceived stimulus from non-perceived so that a quantum superposition of patterns of two neural firing patterns would be in question.

2. This picture leads naturally to a proposal that one function of sleep is to make possible quantum superposition of large number of neural firing patterns via quantum entanglement with external systems (perhaps other sleeping brains) so that sleep would be a process analogous to quantum computation.

3. The formation of alternative percepts would have an obvious evolutionary advantage in a situation in which several percepts are consistent with the sensory input. For instance, bipolar mood disorders seem to involve sticking of consciousness to either hemisphere. This generalizes also to cognition: of course, percepts actually consist of sensory input plus cognition.

4. This framework is behind TGD based model of volitional action applying to both motor actions and selection of sensory percepts (see this). For a brain living in jungle it would be highly advantageous to develop in a difficult situation a quantum superposition of alternative motor actions and select the proper one only at the eleventh moment.

5. Sensory rivalry is analogous to an ability to move fluently between - say - skeptic and new age views about world. There is also a parallel at the level of society and in TGD framework the rivalry of various views (religions, political parties, competing scientific theories,...) might perhaps be seen as counterpart of binocular rivalry at the level of collective consciousness. The complete dominance of only single view - be it religious or materialistic world view, market economy or communism, or super-string model or loop quantum gravity - would be something comparable to a bimodal mood disorder.

For more details see the chapter Quantum Model for Cognition of "TGD Inspired Theory of Consciousness".

Krishnamurti and revolution of consciousness as the only possible revolution

I found Jiddu Krishnamurti for about two decades ago as I tried to understand my own "Great Experience". A couple of days ago, after realization that the writing hours are over for this particular day, I went to Youtube. It occurred to me that I could try to find some material about Krishnamurti and indeed found some videos and looked them through. Krishnamurti is a representative of a long tradition in Eastern philosophy and there are also many other thinkers and gurus representing similar vision about what is good life (see for instance this) but it somehow seems that Krishnamurti has been able to catch the attention of westeners especially well.

I have experienced Krishnamurti's strange charisma through written texts translated also into Finnish. As a matter fact, I do not know of any other writer who would be so concretely present through his texts. His charisma is equally powerful in these videos. He talks slowly with eyes directed to the distance, there is no hurry to anywhere, no need to impress anyone, he just allows the thoughts to articulate themselves. What a difference to the raging Hitler or a slippery politician. In one video Krishamurti talks about attention, about what a totally devoted listening is, and manages to get his listeners to this state: audience indeed seems to become a part of Krishnamurti just as he told to happen in full attentiveness. Something about his charisma tells the fact that he has been offered the honor of being declared to be a God! Also something about Krishamurti tells his refusal from the honor.

What is so special in Krishnamurti that he refused to take the role of a religious leader or a guru or a philosopher. No doubt he was and still is one of the most influental spiritual leaders and thinkers but regards all systems, be they religions, political ideologies, or scientific or philosophical dogmas only as forms of spiritual, psychological, or intellectual violence and thus sources of suffering. His view about good life could perhaps be called loving anarchism.

For Krishnamurti philosophy is not a library of thick dusty books containing academic articles written with boring jargon with long lists of references at the end. For Krishnamurti philosophy is very simple and concrete. Not longwindy ontologizing or epistemologizing but talk about the importance of inner harmony, about why this harmony is lost, and about how this harmony could be achieved again.

Changing the external world does not help

All of use have a lot of problems and the key message of Krishnamurti is that fear is our basic problem. Do I get a job? Do I lose my job? Do I suffer burnout? Does this happen to some of my beloved ones? There are myriads of fears about social humiliations. Perhaps I am not so good as my competitor in this or that respect and I will be rejected. Or perhaps some of these small and not always so small lies that me and everyone find so difficult to avoid in our social games become to the daylight in some unhappy situation and I lose my face totally.

We try get get rid of our inner problems basically created by these fears by projecting them to the external world and by manipulating it to find the solution. We have created magnificent technology, we have developed a lot of well-fare and safety, we have developed political ideologies providing recipes for achieving a better world. We are however not happy. We are unable to live in this moment enjoying it, a skill that every five year old has.

Mechanical thinking as the source of problems

Why this failure? Why are we still afraid and suffering. And what is the source of all these fears? What creates them? The answer is simple: thinking!

This is a statement that certainly induces protests in scientist who are so proud of the thinking abilities. But what thinking - as Krishnamurti defines it - is? Thinking means for Krishnamurti a mechanical simulation, using memories of the past as rules allowing to build predictive and deterministic models or reality. Thinking is based on the belief that if A implied B in the past it does so also in the future. Mechanistic thinking - trying to force something which is inherently non-deterministic to be deterministic - is the quintessence of all violence and violence always creates unhappiness.

Comparison, a decision that this is better than that, is second basic element of thought. This comparison aspect is already built in the sensory perception and at the level of emotions leads to this endless comparison of what I have to what others have.

Division of the world into internal and external is third basic aspect of the state of consciousness that we call thinking. The external world is by definition something which we can manipulate at will at least if it is "dead". This is a practical conceptual division the reality of which we rarely challenge.

To avoid misunderstandings, it must be said that Krishnamurti distinguishes carefully between thinking and intelligence (understanding, insight, intuition, a direct perception of truth: as you wish). The latter is a creative process of discovery rather than an application of a mere mechanical deduction based on rules deduced by making the wrong assumtion that the world is deterministic machine. Somewhat like discovering a new mathematical truth not deducible from the axioms of an existing mathematical system.

Fear is created when you start to live in an imagined deterministic model reality constructed from this memory stuff and performing these endless comparisons between the real outcome and the prediction simulation and finding that they do not fit. Fear is fear of non-determinism. Fear is created when your attention is not in now, but in the deterministic simulation: either in the past which is also a simulation of the world created by our brain or in the imagined future constructed from the memories. Ironically, when you are in a real danger, you are not afraid at all: you are fully attentive and alert and you act instead of thinking mechanically!

Controlling the external world

To get rid of these fears that mechanical thinking creates, we invent the idea of controlling and manipulating the external world to make it predictable.

At the social level the only manner to achieve predictability is power but you cannot gain it without some violence since you are not the only one who discovered this magic substance. Everywhere a merciless fight for this social resource is raging behind the facade of polite manners. Someone manages to steal it more than others but knows that he can lose it any time and competitors do not hesitate at the moment of revenge. The winner meets his old friend fear again: at this time this old friend threatens to make him a hallucinating paranoid unable to trust even his closest friends.

A more refined manner to control external world is to become a guru: religious leader, a founder of some ism in arts, or a scientist creating a new paradigm. The idea is ingenious: if you get the mind of the followers into your control by brainwashing them, they assimilate the survival of the belief system with their own survival. No need for a direct control anymore. System becomes self sustaining! You can even avoid death since you thoughts continue to control the brains of the followers. But again! There is this merciless competition and debunking into heretics and crackpots, and there is no guarantee that the leading figure of today is tomorrow's crackpot. Theories can turn out to be wrong and isms are replaced with new isms.

We can also try to gain a control over the external physical world instead of humans around us. This sounds much more innocent, rational, and also ethical. We have indeed developed a refined technology - all kinds of refined technologies - to make the world safer and more predictable place to live. Now we are trying to transform the entire society to an enormous computer program in order to achieve the optimal predictability. Unfortunately, the environment does not seem to like at all of this predictability: species are suffering extinction, there is a huge environmental crisis, our natural resources are coming to an end, climate is warming,...: the list is long.

Why this? Somehow the sad outcome must relate to the fact that predictability means less information, less variation, mono-culture. At more profound level the explanation is that this environent, this external world, is not actually dead matter, it lives, and our engineering activities are doing violence for it and it is suffering. Even more: in some hidden sense it seems that we ourselves are the environment: we are suffering!

Indeed, despite all this top technology and all these environmental sacrifices very many of us agree that our life has become miserable. Never have human beings been so desperately lonely as in this post-modern western society. Never have they seen so much ugliness. The life as a part of computer program is predictable but extremely boring. But again technology promises a cure. For instance, in a form of program formats (predictability!). We can replace our life with virtual life through TV programs and through "Reality TV", versions of Wonderful Nature programs with animals replaced by individuals of our species behaving as vulgar Darwinistic animals are expected to behave.

These programs actually give a school example about how fear transforms to a sadistic misuse of power. We want to participate coffee table discussions since we are afraid that otherwise we would experience the same fate as those kicked out from the survival series of reality TV. To participate we must know something about the stuff that went in TV last evening. This knowledge has become a social currency. So valuable a currency that we are ready to devote our attention evening after evening to a totally disgusting trash. Not only all these silly TV series telling about ego-centered idiots behaving like monsters towards each other but also these "reality TV" series, where people are humiliated in all manners that a totally insane psychopath can invent. Those who are responsible for this have long time ago realized that they have quite impressive power over people: they can force us to eat shit just because we want to belong to the group.

The revolution

The above represents essentially Krishnamurti's diagnosis of what is wrong with our consciousness. The crisis of our civilization is a crisis of consciousness and can be solved only by a profound revolution in the qualitative character of consciousness. What could be this transformation?

Everyone has had moments when fear and worries about tomorrow are absent: when we are laughing in a good company we are not conscious of our worries and fears. Those laughing friendly people around me have replaced my worrying ego, I am conscious but conscious about something else. It is not this dull ego who invents a brilliant joke but this happy group of people which talks and laughs through me and others. Get rid of ego who tries to force the universe to be a deterministic machine and let go; find the real me, the creator, masked by this ego; discover the forgotten connection with the world so that it ceases to be external. Accept as a fact that the universe is constantly recreating itself. This is the crux of the revolution.

The new way to see the world

This revolution of consciousness means many things and every ego interprets it in this own manner. So do I and what follows contains my own ideas and beliefs and is far not being so pure and crystallized as the vision of Krishnamurti.

Certainly the revolution means realizing that I am in a deeper spiritual sense absolutely unique: there are no copies about me. There can be no external authorities able to tell how I should live and think and feel and there are no measures allowing to tell how valuable this particular me is.

There is also the mystic experience for which many names have been given, Brahman=Atman is only one of them. Physicist would talk about hologram. The division intop external world and me disappears. This real me is or represents or experiences in some profound sense the entire Universe. Since there is nothing exterior to me to threat me, there can be no fear. And there is no need to give lectures in ethics to a person experiencing the world as part of himself and able to feel directly the pain that he produces to another human beings by acting unethically.

The fate of this biological body of mine is an endless source of worries and fears. The revolution of consciousness means getting rid of the attention directed only to this biological body and its worries masking the experience of the unity with the surrounding world which is actually always at the background. This revolution of consciousness brings also the discovery that there is no real death. After all, biological body is only one particular mental image and biological death is just directing attention to something more interesting. Could it be simpler and more obvious!

Thinking and fears relate closely to our common mis-understandings about time. Also the view about time changes. One can experience the paradoxical identification of single moment as eternity repeating itself in poetry: nothing but the temporal counterpart for Brahman = Atman. The recent linguistic reportoare of science is still far from being able to transform this paradox to an apparent one.

Also the view about relation of death to time changes. It is hard to articulate this linguistically and the following is only my own atttempt. Any physical object has boundary in space and there is nothing dramatic in this. Any physical object has also boundary in time direction and there is nothing dramatic in this either: biological death becomes only a boundary of the sensory me in time direction. Visiting the boundary does not mean that what exists behind me disappears: it is still there -also in temporal sense- and biological death becomes an exotic experience, not the end point of bodily humiliations followed by a final loss of the tortured consciousness.

One root of sorrow is the belief that what is done cannot be undone. Many scientists still work hard to believe that we live in a clockwork universe where everything is predetermined so that we are powerless victims of our fate. The discovery that the basic aspect of consciousness is to re-create the universe again and again allows to get rid of this source of sorrow: in this infinitely creative existence it is never too late and everything is possible.

How to achieve the revolution and why it is so difficult?

Krishnamurti does not promise any miracle healings. The revolution of consciousness does not mean central committers nor local committees nor politruks. It requires only hard work to become conscious of all these fears, to admit and accept them, to see them in action as a healer rather than a patient, and finally to realize that the fears have disappeared. Certainly this happens at the moment of biological death when the worries about the future of this biological body dissolve. But why should we wait so long: we can study and at the same time engineer - or rather heal - our own consciousness just like we can study and engineer as healers the world which we once regarded as external.

Simple! Isn't it! But I know from my personal experience that for a scientist it is especially difficult to achieve this goal. All this bitterness and hatred due to these continual humiliations that scientists are so cleverly doing to each other behind polite protocols: this is an extremely heavy and painful load which manages again and again to suck you you to the cycles of mechanical thought. And many scientists regard ideal thinking as something extremely valuable and regard insights and visions as something which should but unfortunately cannot be totally avoided!

The basic dogma of the materialistic science is a declaration of war against reality: the universe is deterministic: to think otherwise is non-scientific. Quite a many neuro scientist tries to see brain as a deterministic machine and develops extremely complex arguments in an attempt to understand free will and consciousness as an illusion (think carefully what this concept might mean!). Biologists in turn make their best to believe on genetic determinism.

We have also heavy religious backgrounds. Christian religion is based on supreme external authority: the Bible. And Christianity is a religion of violence. Christian God is a merciless psychopath. He teased Job like a cat teases mouse after having made a bet with Devil! He tortured and killed his own Son. And what was his motivation? What would you do if someone would come and say "You have done many bad things to me. I should punish you but I murdered my own family so that your sins are forgiven! Be happy!".

Luther took this idea to extreme. The God of Luther was Stalin: he gives you a mercy or not and it has absolutely nothing to do with what you are or what you do. Around the age of nine I experienced a kind of religious awaking (fortunately rather short-lasting!) due to the Lutherian brainwashing in school: I prayed every evening that this Mad Man would not decide to kill my family and my relatives during the night and would save also others. But not because of me! This I had learned to add so that this Psychopath would not get a reason to kill someone for my selfishness.

Our religous leaders are images of our God as sons of their father. Luther gave his blessing for a mass murder of peasants who had been inspired by Luther's own rebel against authorities. The former Pope did not allow abort for a ten year old girl raped by his father. Pope had quite different attitude towards pedophilia of homosexual priests: fortunately the situation is changing now.

We have our backgrounds and it takes long time time to get rid of them. Christianity shares a lot of good with other forms of spirituality and it has also created wonderful art but insanity is insanity and we should get rid of insanity.

Connes tensor product and perturbative expansion in terms of generalized braid diagrams

Many steps of progress have occurred in TGD lately.

1. In a given measurement resolution characterized by the inclusion of HFFs of type II₁ Connes tensor product defines an almost universal M-matrix apart from the non-uniqueness due to the facts that one has a direct sum of hyper-finite factors of type II₁ (sum over conformal weights at least) and the fact that the included algebra defining the measurement resolution can be represented in a reducible manner. The S-matrices associated with irreducible factors would be unique in a given measurement resolution and the non-uniqueness would make possible non-trivial density matrices and thermodynamics.

2. Higgs vacuum expectation is proportional to the generalized position dependent eigenvalue of the modified Dirac operator and its minima define naturally number theoretical braids as orbits for the minima of the universal Higgs potential: fusion and decay of braid strands emerge naturally. Thus the old speculation about a generalization of braid diagrams to Feynman diagram likes objects, which I already began to think to be too crazy to be true, finds a very natural realization.

In the previous posting I explained how generalized braid diagrams emerge naturally as orbits of the minima of Higgs defined as a generalized eigenvalue of the modified Dirac operator. I also explained how Connes tensor product relates to a diagrammatic expansion in terms of generalized braid diagrams. Because of the utmost importance of this result I decided to move it from the end of the earlier posting here. Sorry for any inconvenience.

The association of generalized braid diagrams to incoming and outgoing 3-D partonic legs and possibly also vertices of the generalized Feynman diagrams forces to ask whether the generalized braid diagrams could give rise to a counterpart of perturbation theoretical formalism via the functional integral over configuration space degrees of freedom.

The question is how the functional integral over configuration space degrees of freedom relates to the generalized braid diagrams. The basic conjecture motivated also number theoretically is that radiative corrections in this sense sum up to zero for critical values of Kähler coupling strength and Kähler function codes radiative corrections to classical physics via the dependence of the scale of M⁴ metric on Planck constant. Cancellation occurs only for critical values of Kähler coupling strength α_K: for general values of α_K cancellation would require separate vanishing of each term in the sum and does not occur.

The natural guess is that finite measurement resolution in the sense of Connes tensor product can be described as a cutoff to the number of generalized braid diagrams. Suppose that the cutoff due to the finite measurement resolution can be described in terms of inclusions and M-matrix can be expressed as a Connes tensor product. Suppose that the improvement of the measurement resolution means the introduction of zero energy states and corresponding light-like 3-surfaces in shorter time scales bringing in increasingly complex 3-topologies.

This would mean following.

1. One would not have perturbation theory around a given maximum of Kähler function but as a sum over increasingly complex maxima of Kähler function. Radiative corrections in the sense of perturbative functional integral around a given maximum would vanish (so that the expansion in terms of braid topologies would not make sense around single maximum). Radiative corrections would not vanish in the sense of a sum over 3-topologies obtained by adding radiative corrections as zero energy states in shorter time scale.

2. Connes tensor product with a given measurement resolution would correspond to a restriction on the number of maxima of Kähler function labelled by the braid diagrams. For zero energy states in a given time scale the maxima of Kähler function could be assigned to braids of minimal complexity with braid vertices interpreted in terms of an addition of radiative corrections. Hence a connection with QFT type Feyman diagram expansion would be obtained and the Connes tensor product would have a practical computational realization.

3. The cutoff in the number of topologies (maxima of Kähler function contributing in a given resolution defining Connes tensor product) would be always finite in accordance with the algebraic universality.

4. The time scale resolution defined by the temporal distance between the tips of the causal diamond defined by the future and past light-cones applies to the addition of zero energy sub-states and one obtains a direct connection with p-adic length scale evolution of coupling constants since the time scales in question naturally come as negative powers of two. More precisely, p-adic p-adic primes near power of two are very natural since the coupling constant evolution comes in powers of two of fundamental 2-adic length scale.

There are still some questions. Radiative corrections around given 3-topology vanish. Could radiative corrections sum up to zero in an ideal measurement resolution also in 2-D sense so that the initial and final partonic 2-surfaces associated with a partonic 3-surface of minimal complexity would determine the outcome completely? Could the 3-surface of minimal complexity correspond to a trivial diagram so that free theory would result in accordance with asymptotic freedom as measurement resolution becomes ideal?

The answer to these questions seems to be 'No'. In the p-adic sense the ideal limit would correspond to the limit p→ 0 and since only p→ 2 is possible in the discrete length scale evolution defined by primes, the limit is not a free theory. This conforms with the view that CP₂ length scale defines the ultimate UV cutoff.

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

Number theoretic braids and global view about anti-commutations of induced spinor fields

The anti-commutations of induced spinor fields are reasonably well understood locally. The basic objects are 3-dimensional light-like 3-surfaces. These surfaces can be however seen as random light-like orbits of partonic 2-surfaces taking which would thus seem to take the role of fundamental dynamical objects. Conformal invariance in turn seems to make the 2-D partons 1-D objects and number theoretical braids in turn discretizes strings. And it also seems that the strands of number theoretic braid can in turn be discretized by considering the minima of Higgs potential in 3-D sense.

Somehow these apparently contradictory views should be unifiable in a more global view about the situation allowing to understand the reduction of effective dimension of the system as one goes to short scales. The notions of measurement resolution and number theoretic braid indeed provide the needed insights in this respect.

1. Anti-commutations of the induced spinor fields and number theoretical braids

The understanding of the number theoretic braids in terms of Higgs minima and maxima allows to gain a global view about anti-commutations. The coordinate patches inside which Higgs modulus is monotonically increasing function define a division of partonic 2-surfaces X²_t= X³_l\intersection δ M⁴_+/-,t to 2-D patches as a function of time coordinate of X³_l as light-cone boundary is shifted in preferred time direction defined by the quantum critical sub-manifold M²× CP₂. This induces similar division of the light-like 3-surfaces X³_l to 3-D patches and there is a close analogy with the dynamics of ordinary 2-D landscape.

In both 2-D and 3-D case one can ask what happens at the common boundaries of the patches. Do the induced spinor fields associated with different patches anti-commute so that they would represent independent dynamical degrees of freedom? This seems to be a natural assumption both in 2-D and 3-D case and correspond to the idea that the basic objects are 2- resp. 3-dimensional in the resolution considered but this in a discretized sense due to finite measurement resolution, which is coded by the patch structure of X³_l. A dimensional hierarchy results with the effective dimension of the basic objects increasing as the resolution scale increases when one proceeds from braids to the level of X³_l.

If the induced spinor fields associated with different patches anti-commute, patches indeed define independent fermionic degrees of freedom at braid points and one has effective 2-dimensionality in discrete sense. In this picture the fundamental stringy curves for X²_t correspond to the boundaries of 2-D patches and anti-commutation relations for the induced spinor fields can be formulated at these curves. Formally the conformal time evolution scaled down the boundaries of these patches. If anti-commutativity holds true at the boundaries of patches for spinor fields of neighboring patches, the patches would indeed represent independent degrees of freedom at stringy level.

The cutoff in transversal degrees of freedom for the induced spinor fields means cutoff n≤ n_max for the conformal weight assignable to the holomorphic dependence of the induced spinor field on the complex coordinate. The dropping of higher conformal weights should imply the loss of the anti-commutativity of the induced spinor fields and its conjugate except at the points of the number theoretical braid. Thus the number theoretic braid should code for the value of n_max: the naive expectation is that for a given stringy curve the number of braid points equals to n_max.

2. The decomposition into 3-D patches and QFT description of particle reactions at the level of number theoretic braids

What is the physical meaning of the decomposition of 3-D light-like surface to patches? It would be very desirable to keep the picture in which number theoretic braid connects the incoming positive/negative energy state to the partonic 2-surfaces defining reaction vertices. This is not obvious if X³_l decomposes into causally independent patches. One can however argue that although each patch can define its own fermion state it has a vanishing net quantum numbers in zero energy ontology, and can be interpreted as an intermediate virtual state for the evolution of incoming/outgoing partonic state.

Another problem - actually only apparent problem -has been whether it is possible to have a generalization of the braid dynamics able to describe particle reactions in terms of the fusion and decay of braid strands. For some strange reason I had not realized that number theoretic braids naturally allow fusion and decay. Indeed, cusp catastrophe is a canonical representation for the fusion process: cusp region contains two minima (plus maximum between them) and the complement of cusp region single minimum. The crucial control parameter of cusp catastrophe corresponds to the time parameter of X³_l. More concretely, two valleys with a mountain between them fuse to form a single valley as the two real roots of a polynomial become complex conjugate roots. The continuation of light-like surface to slicing of X⁴ to light-like 3-surfaces would give the full cusp catastrophe.

In the catastrophe theoretic setting the time parameter of X³_l appears as a control variable on which the roots of the polynomial equation defining minimum of Higgs depend: the dependence would be given by a rational function with rational coefficients.

This picture means that particle reactions occur at several levels which brings in mind a kind of universal mimicry inspired by Universe as a Universal Computer hypothesis. Particle reactions in QFT sense correspond to the reactions for the number theoretic braids inside partons. This level seems to be the simplest one to describe mathematically. At parton level particle reactions correspond to generalized Feynman diagrams obtained by gluing partonic 3-surfaces along their ends at vertices. Particle reactions are realized also at the level of 4-D space-time surfaces. One might hope that this multiple realization could code the dynamics already at the simple level of single partonic 3-surface.

3. About 3-D minima of Higgs potential

The dominating contribution to the modulus of the Higgs field comes from δ M⁴_+/- distance to the axis R₊ defining quantization axis. Hence in scales much larger than CP₂ size the geometric picture is quite simple. The orbit for the 2-D minimum of Higgs corresponds to a particle moving in the vicinity of R₊ and minimal distances from R₊ would certainly give a contribution to the Dirac determinant. Of course also the motion in CP₂ degrees of freedom can generate local minima and if this motion is very complex, one expects large number of minima with almost same modulus of eigenvalues coding a lot of information about X³_l.

It would seem that only the most essential information about surface is coded: the knowledge of minima and maxima of height function indeed provides the most important general coordinate invariant information about landscape. In the rational category where X³_l can be characterized by a finite set of rational numbers, this might be enough to deduce the representation of the surface.

What if the situation is stationary in the sense that the minimum value of Higgs remains constant for some time interval? Formally the Dirac determinant would become a continuous product having an infinite value. This can be avoided by assuming that the contribution of a continuous range with fixed value of Higgs minimum is given by the contribution of its initial point: this is natural if one thinks the situation information theoretically. Physical intuition suggests that the minima remain constant for the maxima of Kähler function so that the initial partonic 2-surface would determine the entire contribution to the Dirac determinant.

Apologies: I have moved the earlier text about the connection between Connes tensor product and Feynman diagrammatics based on generalized braid diagrams to the next posting!

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

Oddly behaving kilogram

The definition of kilogram is not the topics number one in coffee table discussions and definitely not so because it could lead to heated debates. The fact however is that even the behavior of standard kilogram can open up fascinating questions about the structure of space-time.

The 118-year old International Prototype Kilogram is an alloy with 90 per cent Platinum and 10 per cent Iridium by weight (gravitational mass). It is held in an environmentally monitored vault in the basement of the BIPM’s House of Breteuil in Sèvres on the outskirts of Paris. It has forty copies located around the world which are compared with Sevres copy with a period of 40 years.

The problem is that the Sevres kilogram seems to behave in a manner totally in-appropriate taking into account its high age if the behaviour of its equal age copies around the world is taken as the norm (see Wikipedia article and the more popular article here). The unavoidable conclusion from the comparisons is that the weight of Sevres kilogram has been reduced by about 50 μg during 118 years which makes about

dlog(m)/dt= -4.2×10^-10/year

for Sevres copy or relative increase of same amout for its copies.

Specialists have not been able to identify any convincing explanation for the strange phenomenon. For instance, there is condensation of matter from the air in the vault which increases the weight and there is periodic cleaning procedure which however should not cause the effect.

1. Could the non-conservation of gravitational energy explain the mystery?

The natural question is whether there could be some new physics mechanism involved. If the copies were much younger than the Sevres copy, one could consider the possibility that gravitational mass of all copies is gradually reduced. This is not the case. One can still however look what this could mean.

In TGD Equvalence Principle is not a basic law of nature and in the generic case gravitational energy is non-conserved whereas inertial energy is conserved (I will not go to the delicacies of zero energy ontology here). This occurs even in the case of stationary metrics such as Reissner-Nordström exterior metric and the metrics associated with stationary spherically symmetric star models imbedded as vacuum extremals (for details see this).

The basic reason is that Schwartschild time t relates by a shift to Minkowski time m⁰:

m⁰= t+h(r)

such that the shift depends on the distance r to the origin. The Minkowski shape of the 3-volume containing the gravitational energy changes with M⁴ time but this does not explain the effect. The key observation is that the vacuum extremal of Kähler action is not an extremal of the curvature scalar (these correspond to asymptotic situations). What looks first really paradoxical is that one obtains a constant value of energy inside a fixed constant volume but a non-vanishing flow of energy to the volume. The explanation is that the system simply destroys the gravitational energy flowing inside it! The increase of gravitational binding energy compensating for the feed of gravitational energy gives a more familiar looking articulation for the non-conservation.

Amusingly, the predicted rate for the destruction of the inflowing gravitational energy is of same order of magnitude as in the case of kilogram. Note also that the relative rate is of order 1/a, a the value of cosmic time of about 10¹⁰ years. The spherically symmetric star model also predicts a rate of same order.

This approach of course does not allow to understand the behavior of the kilogram since it predicts no change of gravitational mass inside volume and does not even apply in the recent situation since all kilograms are in same age. The co-incidence however suggests that the non-conservation of gravitational energy might be part of the mystery. The point is that if the inflow satisfies Equivalence Principle then the inertial mass of the system would slowly increase whereas gravitational mass would remain constant: this would hold true only in steady state.

2. Is the change of inertial mass in question?

It would seem that the reduction in weight should correspond to a reduction of the inertial mass in Sevres or its increase of its copies. What would distinguish between Sevres kilogram and its cousins? The only thing one can imagine is that the cousins are brought to Sevres periodically. The transfer process could increase the kilogram or stop its decrease.

Could it be that the inertial mass of every kilogram increases gradually until a steady state is achieved? When the system is transferred to another place the saturation situation is changed to a situation in which genuine transfer of inertial and gravitational mass begins again and leads to a more massive steady state. The very process of transferring the comparison masses to Sevres would cause their increase.

In TGD Universe the increase of the inertial (and gravitational) mass is due to the flow of matter from larger space-time sheets to the system. The additional mass would not enter in via the surface of the kilogram but like a Trojan horse from the interior and it would be thus impossible to control using present day technology. The flow would continue until a flow equilibrium would be reached with as much mass leaving the kilogram as entering it.

3. A connection with gravitation after all?

Why the in-flow of the inertial energy should be of same order of magnitude as that for the gravitational energy predicted by simple star models? Why Equivalence Principle should hold for the in-flow alhough it would not hold for the body itself? A possible explanation is in terms of the increasing gravitational binding energy which in a steady situation leaves gravitational energy constant although inertial energy could still increase.

This would however require rather large value of gravitational binding energy since one has

Δ E_gr=ΔM_I/M .

The Newtonian estimate for E/M is of order GM/R, where R ≈ 1 m the size of the system. This is of order 10^-26 and too small by 16 orders of magnitude.

TGD predicts that gravitational constant is proportional to p-adic length scale squared

G propto L_p².

Ordinary gravitation can be assigned to the Mersenne prime M₁₂₇ associated with electron and thus to p-adic length scale of L(127)≈ 2.5×10^-14 meters. The open question has been whether the gravities corresponding to other p-adic length scales are realized or not.

This question together with the discrepancy encourages to ask whether the value of the p-adic prime could be larger inside massive bodies (analogous to black holes in many respects in TGD framework) and make gravitation strong? In the recent case the p-adic length scale should correspond to a length scale of order 10⁸L(127). L(181)≈ 3.2× 10^-4 m (size of a large neuron by the way) would be a good candidate for the p-adic scale in question and considerably smaller than the size scale of order .1 meter defining the size of the kilogram.

This discrepancy brings in mind the strange finding of Tajmar and collaborators suggesting that rotating super-conductors generate a gravimagnetic field with a field strength by a factor of order 10²⁰ larger than predicted by General Relativity. I have considered a model of the finding based on dark matter (see this). An alternative model could rely on the assumption that Newton's constant can in some situations correspond to p larger than M₁₂₇. In this case the p-adic length scale needed would be around L(193)≈ 2 cm.

The action of zero energy algebra on positive and negative energy parts of zero energy states

I have summarized in the previous posting how the Connes tensor product (or actually its slight modification) guaranteing that the zero energy sub-algebras N of the algebra M creating positive/negative energy states in a time scale shorter that that assignable to positive/negative energy states act like complex numbers on positive and negative energy parts of zero energy states. This is necessary in order that one can speak about the replacement of the complex rays of state space with N-rays.

This leads to an M-matrix which by simple argument is unique for Jones inclusions with M:N<4. In the more general case one obtains an M-matrix with non-trivial density matrix, whose dimension correspond to the number of irreducibles in the representation of N in M induced by inclusion and to which different summands of M-matrix correspond to. The presence of type I factors complicates of course this picture but does not bring anything mathematically new. This picture is extremely attractive physically but should be made more precise. In the following an attempt in this direction is made.

1. How to define the inclusion of N physically?

The overall picture looks beautiful but it is not clear how one could define the inclusion N subset M precisely. One must distinguish between two cases corresponding to the unitary U-matrix representing unitary process associated with the quantum jump and defined between zero energy states and M-matrix defining the time-like entanglement between positive and negative energy states.

1. In the case of U-matrix both N and M corresponds to zero energy states. The time scale of the zero energy state created by N should be shorter than that for the state defined naturally as the temporal distance t_+- between the tips of the light-cones M⁴_+/- associated with the state and defining diamond like structure.

2. In the case of M-matrix one has zero energy subalgebra of algebra creating positive or negative energy states in time scale t_+-. In this case the time scale for zero energy states is smaller than t_+-/2. The defining conditions for the Connes tensor product are analogous to crossing symmetry but with the restriction that the crossed operators create zero energy states.

Quantum classical correspondence requires a precise formulation for the action of N at space-time level and this is a valuable guideline in attempts to understand what is involved. Consider now the definition of the action of N in the case of M-matrix.

1. In standard QFT picture the action of the element of N multiplies the positive or negative energy parts of the state with an operator creating a zero energy state.

2. At the space-time level one can assign positive/negative energy states to the incoming/outcoing 3-D lines of generalized Feynman diagrams (recall that in vertices the 3-D light-like surfaces meet along their ends). At the parton level the addition of a zero energy state would be simply addition of a collection of light-like partonic 3-surfaces describing a zero energy state in a time scale shorter than that associated with incoming/outgoing positive/negative energy space-time sheet. The points of the discretized number theoretic braid would naturally contain the insertions of the second quantized induced spinor field in the description of M-matrix element in terms of N-point function.

3. At first look this operation looks completely trivial but this is not the case. The point is that the 3-D lines of zero energy diagram and those of the original positive/negative energy diagram must be assigned to single connected 4-D space-time surface. Note that even the minima of the generalized eigenvalue &lamdba; (to which Higgs vacuum expectation is proportional) are not same as for the original positive energy state and free zero energy state since the minimization is affected by the constraint that the resulting space-time sheet is connected.

4. What happens if one allows several disconnected space-time sheets in the initial state? Could/should one assign the zero energy state to a particular incoming space-time sheet? If so, what space-time sheet of the final state should one attach the *-conjugate of this zero energy state? Or should one allow a non-unique assignment and interpret the result in terms of different phases? If one generalizes the connectedness condition to the connectedness of the entire space-time surface characterizing zero energy state one would bet rid of the question but can still wonder how unique the assignment of the 4-D space-time surface to a given collection of light-like 3-surfaces is.

2. How to define Hermitian conjugation physically?

Second problem relates to the realization of Hermitian conjugation N → N^* at the space-time level. Intuitively it seems clear that the conjugation must involve M⁴ time reflection with respect to some origin of M⁴ time mapping partonic 3-surfaces to their time mirror images and performing T-operation for induced spinor fields acting at the points of discretized number theoretic braids.

Suppose that incoming and outgoing states correspond to light-cones M⁴₊ and M⁴_- with tips at points m⁰=0 and m⁰=t_+-. This does not require that the preferred sub-manifolds M² and S²_II are same for positive/negative energy states and inserted zero energy states. In this case the point (m⁰=t_+-/2,m^k=0) would be the natural reflection point and the operation mapping the action of N to the action of N^* would be unique.

Can one allow several light cones in the initial and final states or should one restrict M-matrix to single diamond like structure defined by the two light-cones? The most reasonable option seems to be an assignment of a diamond shape pair of light-cones to each zero energy component of the state. The temporal distance t_+- between the tips of the light-cones would assign a precise time scale assigned to the zero state. The zero energy states inserted to a state characterized by a time scale t_+- would correspond to time scales t<t_+-/2 so that a hierarchy in powers of 2 would emerge naturally. Note that the choice of quantization axes (manifolds M² and S²_II) could be different at different levels of hierarchy.

This picture would apply naturally also in the case of U-matrix and make the cutoff hierarchy discrete in accordance with p-adic length scale hypothesis bringing in also quantization of the time scales t_+-. In the case of U-matrix N would contain besides the zero energy algebra of M-matrix also the subalgebra for which the positive and negative energy parts reside at different sides of the center of the diamond.

3. How to generalize the notion of observable?

The almost-uniqueness of M-matrix seems too good to be true and in this kind of situation it is best to try to find an argument killing the hypothesis. The first test is whether the ordinary quantum measurement theory with Hermitian operators identified as observables generalizes.

The basic implication is that M should commute with Hermitian operators of N assuming that they exist in some sense. All Hermitian elements of N could be regarded not only as observables but also as conserved charges defining symmetries of M which would be thus maximal. The geometric counterpart for this would be the fact that configuration space is a union of symmetric spaces having maximal isometry group. Super-conformal symmetries of M-matrix would be in question.

The task is to define what Hermiticity means in this kind of situation. The super-positions N+N^* and products N^*N defined in an appropriate sense should Hermitian operators. One can define what the products MN^* and NM mean. There are also two Hermitian conjugations involved: M conjugation and N conjugation.

1. Consider first Hermitian conjugation in M. The operators of N creating zero energy states on the positive energy side and N^* acting on the negative energy side are not Hermitian in the hermitian conjugation of M. If one defines MN^*== N^*M and NM== MN, the operators N+N^* and N^*N indeed commute with M by the basic condition. One could label the states created by M by eigenvalues of a maximally commuting sub-algebra of N. Clearly, the operators acting on positive and negative energy state spaces should be interpreted in terms of a polarization N=N₊+N_- such that N_+/- acts on positive/negative energy states.

2. In the Hermitian conjugation of N which does not move the operator from positive energy state to negative energy state there certainly exist Hermitian operators and they correspond to zero energy states invariant under exchange of the incoming and outgoing states but in time scale t_+-/2. These operators are not Hermitian in M. The commutativity of M with these operators follows also from the basic conditions.

It thus seems that the conjecture survives the first test.

4. Fractal hierarchy of state function reductions

In accordance with fractality, the conditions for the Connes tensor product at a given time scale imply the conditions at shorter time scales. On the other hand, in shorter time scales the inclusion would be deeper and would give rise to a larger reducibility of the representation of N in M. Formally, as N approaches to a trivial algebra, one would have a square root of density matrix and trivial S-matrix in accordance with the idea about asymptotic freedom.

M-matrix would give rise to a matrix of probabilities via the expression P(P₊→ P_-) = Tr[P₊M⁺P_-M], where P₊ and P_- are projectors to positive and negative energy energy N-rays. The projectors give rise to the averaging over the initial and final states inside N ray. The reduction could continue step by step to shorter length scales so that one would obtain a sequence of inclusions. If the U-process of next quantum jump can return the M-matrix associated with M or some larger HFF, U process would be kind of reversal for state function reduction.

Analytic thinking proceeding from vision to details; human life cycle proceeding from dreams and wild actions to the age when most decisions relate to the routine daily activities; the progress of science from macroscopic to microscopic scales in the spirit of strong reductionism; even biological decay processes: all these have an intriguing resemblance to the fractal state function reduction process proceeding to to shorter and shorter time scales. Since this means increasing thermality of M-matrix, U process as a reversal of state function reduction might break the second law of thermodynamics.

The conservative option would be that only the transformation of intentions to action by U process giving rise to new zero energy states can bring in something new and is responsible for evolution. The non-conservative option is that the biological death is the U-process of the next quantum jump leading to a new life cycle. Breathing would become a universal metaphor for what happens in quantum Universe. The 4-D body would be lived again and again.

For more details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

Yes, it rotates: but in what direction?

New Scientist contains a link to a really fascinating optical illusion representing dancing (very!) beautiful woman. The dancer rotates and the direction of rotation is told to be determined by the dominating brain hemisphere which need not be always the same. Right-brainers should see the dancer to rotate clockwise and left-brainers anticlockwise. Most people see it rotate anticlockwise.

I found that my dancer rotates clockwise. I am a right-brainer! This should mean that the following right brain functions work very well for me if I am an ideal right-brainer.

RIGHT BRAIN FUNCTIONS: uses feeling, "big picture" oriented, imagination rules, symbols and images, present and future, philosophy & religion, can "get it" (i.e. meaning), believes, appreciates, spatial perception, knows object function, fantasy based, presents possibilities, impetuous, risk taking.

I cannot deny of recognizing something familiar in this list, and probably this blog serves as a justification for this feeling. I suddenly feel very good and I realize that weather outside is also beautiful! Before going for a walk I must tell that if I had been an ideal left-brainer, I would have been happy because the following brain functions would work especially well for me.

LEFT BRAIN FUNCTIONS: uses logic, detail oriented, facts rule, words and language, present and past, math and science, can comprehend, knowing, acknowledges, order/pattern perception, knows object name, reality based, forms strategies, practical, safe.

Well, many physicist colleagues agree that I am a pathetic crackpot who is not able to comprehend even the simplest survival rules of the academic environment, and I fully agree that I am not able to form strategies: painful and humiliating experiences from the game of chess taught this to me when I was young. My friends agree that my behavior is certainly not reality based, and that I am hopelessly unpractical, and safety has not been the priority number one in my life. Language is however the manner which I use in my efforts to express myself and I occasionally try to use also logic. I make also attempts to not forget the details so that there seems to be at least a strong motivation to use these very useful left brain functions.

It would be even better if one could change the brain hemisphere dominance: if not for any other purpose then for the noble ability to feel compassion towards all those left-brainers around me worrying about practicalities of everyday life. This should be possible. Meditation practitioners tell that by blocking the other nostril and breathing through nose it is possible change the brain hemisphere dominance. I tried the right nostril but it did not work: perhaps my strong motivation to remain in the big picture mood explains partially the outcome of the experiment.

Having realized how beautiful and holistic this right-brainy world is I went for a walk to get a better view of it. It occurred to me that perhaps the brain hemisphere dominance would change if I try to put myself into the shoes of a skeptic. What if the whole thing is a big joke of bad skeptics? What if all of us see the dancer to rotate clockwise and become very happy as they discover their creativity and the ability to see the big picture? Then they go to their blog and tell everyone about their right-brainy mind and bad skeptics have big bad fun. Or could it be that it is just the creative minority which sees the dancer to rotate counterclockwise? How cruel these skeptics can be!

I came back to the computer and looked. Just in the beginning the dancer made an attempt to rotate counterclockwise but started immediately to spin clockwise. It stopped for a moment now and then but this probably relates to the delays in internet connections. Then I decided to make a new trial by blocking the right nostril and the dancer indeed made two counter-clockwise half turns but nothing more! Perhaps I must still develop my ability to feel compassion towards the suffering part of human kind!

In my attempts to become a skeptic I stumbled with the question about how global this left-right dominance really is. Is it really a property of the entire brain hemisphere as the holistic me wants to believe? Could it be that different regions of brain can be in right/left modes independently? Probably this has been studied.

How uniquely Connes tensor product defines the M-matrix?

I told already earlier about a highly unique identification of M-matrix (product of square root of density matrix and unitary S-matrix) defining time-like entanglement coefficients between positive and negative energy parts of zero energy states and characterizing quantum dynamics in zero energy ontology.

The identification is based on the notion of measurement resolution represented as Jones inclusion. The zero energy states created by included algebra N creating zero energy states for which the time interval between positive and negative energy parts of the state is shorter than that for M are below measurement resolution. Hence complex rays are replaced with N rays and M-matrix must "*-commute" with N: in other words MN=N^*M in element-wise manner. One obtains a hierarchy of M-matrixes chacterized by inclusions, which are not necessary Jones inclusions.

I conjectured that this leads to a highly unique M-matrix. This is indeed the case. The defining condition for the variant of the Connes tensor product proposed here has the following equivalent forms

MN= N^*M ,

N=M^-1N^*M ,

N^*=MNM^-1 .

If M₁ and M₂ are two M-matrices satisfying the conditions then the matrix M₁₂=M₁M₂^-1 satisfies the following equivalent conditions

N=M₁₂NM₁₂^-1 ,

[N,M₁₂]=0 .

Jones inclusions with M:N≤4 are irreducible which means that the operators commuting with N consist of complex multiples of identity. Hence one must have M₁₂=1 so that M-matrix is unique in this case. For M:N>4 the complex dimension of commutator algebra of N is 2 so that M-matrix depends should depend on single complex parameter. The dimension of the commutator algebra associated with the inclusion gives the number of parameters appearing in the M-matrix in the general case.

When the commutator has complex dimension d >1 , the representation of N in M is reducible: the matrix analogy is the representation of elements of N as direct sums of d representation matrices. M-matrix is a direct sum of form M= a₁M₁+a₂M₂+..., where M_i are unique. The condition ∑_i;|a_i|²=1 is satisfied and*-commutativity holds in each summand separately.

Questions: Could M_i define unique universal unitary S-matrices in their own blocks? Could the direct sum define a counterpart of a statistical ensemble? Could irreducible inclusions correspond to pure states and reducible inclusions to mixed states? Could different values of energy in thermodynamics and of the scaling generator L₀ in p-adic thermodynamics define direct summands of the inclusion? The values of conserved quantum numbers for the positive energy part of the state indeed naturally define this kind of direct direct summands.

For more details see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

Bird's eye of view

Mammmooth bones a source of continual frustration when you have fifteen books about everything between Planck length and primordial cosmology. Chapters tend to split into new chapters as they grow over one hundred page length, and sooner or later you realize that some abstract or introduction has practically nothing to do with the content of the chapter. There is no other way out that start again boring bureucratic activity of summing up what it is that you did in this chapter. Otherwise you will loose completely your respectability in critical eyes of possible colleague who might happen to read your writings. This bureacracy however has good side effects: you get some bird's eye of view about the big picture and manage to eliminate at least the worst conflicting statements. In the following rewritten abstract to a book trying to give overall view about TGD. I have the feeling that it catches something about what I feel just now to be the essential elements of TGD. This feeling of course reflects also the fact I am just a human with a rather limited span of attention but in any case.

Abstract of the chapter Overall View about Quantum TGD of "Topological Geometrydynamics: an Overview".

This chapter provides a summary about quantum TGD. The discussions are based on the general vision that quantum states of the Universe correspond to the modes of classical spinor fields in the "world of the classical worlds" identified as the infinite-dimensional configuration space of 3-surfaces of H=M⁴×CP₂ (more or less-equivalently, the corresponding 4-surfaces defining generalized Bohr orbits). The following topics are discussed on basis this vision.

1. Geometric ideas

TGD relies heavily on geometric ideas, which have gradually generalized during the years.

1. The basic vision is that it is possible to reduce quantum theory to configuration space geometry and spinor structure. The geometrization of loop spaces inspires the idea that the mere existence of Riemann connection fixes configuration space Kähler geometry uniquely. Accordingly, configuration space can be regarded as a union of infinite-dimensional symmetric spaces labelled by zero modes labelling classical non-quantum fluctuating degrees of freedom. The huge symmetries of the configuration space geometry deriving from the light-likeness of 3-surfaces and from the special conformal properties of the boundary of 4-D light-cone would guarantee the maximal isometry group necessary for the symmetric space property. Quantum criticality is the fundamental hypothesis allowing to fix the Kähler function and thus dynamics of TGD uniquely. Quantum criticality leads to surprisingly strong predictions about the evolution of coupling constants.
2. Configuration space spinors correspond to Fock states and anti-commutation relations for fermionic oscillator operators correspond to anti-commutation relations for the gamma matrices of the configuration space. Configuration space spinors define a von Neumann algebra known as hyper-finite factor of type II₁ (HFFs). This has led to a profound understanding of quantum TGD. The outcome of this approach is that the exponents of Kähler function and Chern-Simons action are not fundamental objects but reduce to the Dirac determinant associated with the modified Dirac operator assigned to the light-like 3-surfaces.
3. p-Adic mass calculations relying on p-adic length scale hypothesis led to an understanding of elementary particle masses using only super-conformal symmetries and p-adic thermodynamics. The need to fuse real physics and various p-adic physics to single coherent whole led to a generalization of the notion of number obtained by gluing together reals and p-adics together along common rationals and algebraics. The interpretation of p-adic space-time sheets is as correlates for cognition and intentionality. p-Adic and real space-time sheets intersect along common rationals and algebraics and the subset of these points defines what I call number theoretic braid in terms of which both configuration space geometry and S-matrix elements should be expressible. Thus one would obtain number theoretical discretization which involves no adhoc elements and is inherent to the physics of TGD.
4. The work with HFFs combined with experimental input led to the notion of hierarchy of Planck constants interpreted in terms of dark matter. The hierarchy is realized via a generalization of the notion of imbedding space obtained by gluing infinite number of its variants along common lower-dimensional quantum critical sub-manifolds. This leads to the identification of number theoretical braids as points of partonic 2-surface which correspond to the minima of generalized eigenvalue of Dirac operator, a scalar field to which Higgs vacuum expectation is proportional to. Higgs vacuum expectation has thus a purely geometric interpretation. This leads to an explicit formula for the Dirac determinant. What is remarkable is that the construction gives also the 4-D space-time sheets associated with the light-like orbits of partonic 2-surfaces: they should correspond to preferred extremals of Kähler action. Thus hierarchy of Planck constants is now an essential part of construction of quantum TGD and of mathematical realization of the notion of quantum criticality.
5. HFFs lead also to an idea about how entire TGD emerges from classical number fields, actually their complexifications. In particular, CP₂ could be interpreted as a structure related to octonions. This would mean that TGD could be seen also as a generalized number theory.

2. Ideas related to the construction of S-matrix

The construction of S-matrix involves several ideas that have emerged during last years.

1. Zero energy ontology motivated originally by TGD inspired cosmology means that physical states have vanishing net quantum numbers and are decomposable to positive and negative energy parts separated by a temporal distance characterizing the system as space-time sheet of finite size in time direction. The particle physics interpretation is as initial and final states of a particle reaction. S-matrix and density matrix are unified to the notion of M-matrix expressible as a product of square root of density matrix and of unitary S-matrix. Thermodynamics becomes therefore a part of quantum theory. One must distinguish M-matrix from U-matrix defined between zero energy states and analogous to S-matrix and characterizing the unitary process associated with quantum jump. U-matrix is most naturally related to the description of intentional action since in a well-defined sense it has elements between physical systems corresponding to different number fields.
2. The notion of measurement resolution represented in terms of inclusions of HFFs is an essential element of the picture. Measurement resolution corresponds to the action of the included sub-algebra creating zero energy states in time scales shorter than the cutoff scale. This algebra effectively replaces complex numbers as coefficient fields and the condition that its action commutes with the M-matrix implies that M-matrix corresponds to Connes tensor product. Together with super-conformal symmetries this fixes possible M-matrices to a very high degree.
3. Light-likeness of the basic fundamental objects implies that TGD is almost topological QFT so that the formulation in terms of category theoretical notions is expected to work. M-matrices form in a natural manner a functor from the category of cobordisms to the category of pairs of Hilbert spaces and this gives additional strong constraints on the theory.

3. Some general predictions of quantum TGD

TGD is consistent with the symmetries of the standard model by construction although there are definite deviations from the symmetries of standard model. TGD however predicts also a lot of new physics. Below just some examples of the predictions of TGD.

1. Fractal hierarchies meaning the existence of scaled variants of standard model physics is the basic prediction of quantum TGD. p-Adic length scale hypothesis predicts the possibility that elementary particles can have scaled variants with mass scales related by power of square root 2. Dark matter hierarchy predicts the existence of infinite number of scaled variants with same mass spectrum with quantum scales like Compton length scaling like hbar.
2. TGD predicts that standard model fermions and gauge bosons differ topologically in a profound manner. Fermions correspond to light-like wormhole throats associated with topologically condensed CP₂ type extremals whereas gauge bosons correspond to fermion-antifermion states associated with the throats of wormhole contacts connecting two space-time sheets with opposite time orientation. The implication is that Higgs vacuum expectation value cannot contribute to fermion mass: this conforms with the results of p-adic mass calculations. TGD predicts also so called super-canonical quanta and these give dominating contribution to most hadron masses. These degrees of freedom correspond to those of hadronic string and should not reduce to QCD.
3. The most fascinating applications of zero energy ontology are to quantum biology and TGD inspired theory of consciousness. Basic new element are negative energy photons making possible communications to the direction of geometric past. Here also dark matter hierarchy is involved in an essential manner.
4. In cosmology the mere imbeddability required for Robertson-Walker cosmology implies that critical and over-critical cosmologies are almost unique and characterized by their finite duration. The cosmological expansion is accelerating for them and there is no need to assume cosmological constant. Macroscopic quantum coherence of dark matter in astrophysical scales is a dramatic prediction and allows also to assign periods of accelerating expansion to quantum phase transition changing the value of gravitational Planck constant. The dark matter parts of astrophysical systems are predicted to be quantum systems.
5. The notion of generalized imbedding space suggests that the physics of TGD Universe is universal in the sense that it is possible to engineer a system able to mimic the physics of any consistent gauge theory. Kind of analog of Turing machine would be in question.

For more details see the chapter Overall View about Quantum TGD of "Topological Geometrydynamics: an Overview".

Congratulations to myself

While enjoying a few hours of leisure time yesterday, I realized that the further generalization of the notion of imbedding space (see for instance this) inspired by the hierarchy of Planck constants has led to a really spectacular progress in the understanding of TGD. Therefore I felt that it is good to try to declare for the History what have been communicated through Her humble correspondent to the humanity;-). Lists are the manner that She uses to make things clear to me and to bore the readers and this posting will not be an exception.

1. I have now a rather clear view about the eigenmodes of the modified Dirac operator. The generalized eigenvalue (I used plural earlier) which is a scalar field has an interpretation as Higgs vacuum expectation value. Why singular must be used should have been understood long time ago from the orthogonality requirement for various modes and lack of correlation between longitudinal (light-like) and transversal degrees of freedom occuring in the modified Dirac operator: this is what makes possible eigenvalues which are functions. The scale of the vacuum expectation is proportional to log(p) corresponding to the hierarchy of p-adic length scales.

2. The construction leads to an identification of number theoretical braids as a set of points for which Higgs potential has minimum value: the first and quite not correct identification was as quantum critical points in the intersection of all sectors of the generalized imbedding space forming a book like structure. This book is rather thick having infinite number of pages and the value of Planck constant gives a partial oage numbering: the back of the book is the 4-D quantum critical manifold M²× S²₂ of H. You have actually library of identical copies of this book obtained by applying Poincare and color isometries of H. The points in the intersection with critical sub-manifold, the back of the book, however correspond to zeros of Higgs which are unstable by their quantum criticality.

   By the way, it seems that physical thinking is the image of what happens in physics: first the simplest extremum, then the stable extremum. What is really beautiful is the realization of the breaking of the exact quantum criticality (corresponds to vacuum extremals) in terms of Higgs mechanism. Higgs as God particle: I begin to take this phrase more seriously!

3. Higgs potential can be taken to be the negative of the modulus squared of Higgs or any monotonically increasing function of it: the potential as such does not matter since it is only an auxiliary quantity and diffeo-invariance is involved so that only the extrema matter. This is new and sounds of course a little bit strange. The point is however that Higgs modulus is essentially the distance of point of the partonic 2-surface from the quantum critical 2-sphere of CP₂ which of course has minima. Partonic 2-surface induces the dynamics and in case of classical gauge fields and metric. One further example of the power of induction mechanism.

   Higgs potential has this utterly simple expression only in the local complex coordinate and Higgs actually is the universal local complex coordinate satisfying the number theoretical needs. At the critical line surrounding zero, the modulus of this coordinate begins to decrease and in order to avoid assignment of several points of partonic 2-surface to single coordinate value one must introduce coordinate patches. Earth's surface is very good metaphor for Higgs landscape with mountain tops is zeros and valley bottoms as minima defining the number theoretic braid: the modulus of Higgs corresponds to the height coordinate as a local coordinate. In global coordinates, naturally the complex coordinate of S², the minima become explicit and one can say that the minima of Higgs potential emerge through the geometry of space-time sheet.

4. Dirac determinant is identified as the product of the complex values of Higgs at the points of braids associated with the partonic 2-surface: in QFT it would be ill-defined product over all points for eigenvalues of Dirac operator. For years I have been conjecturing that Dirac determinant reduces to the product of exponents of Kähler action and Chern-Simons action. Determinant should reduce to unity for vacuum extremals of Kahler action. The pleasant news is that the construction of Higgs vacuum expectation is consistent with this condition: this is a highly, highly non-trivial result. Everything is of course finite and discrete which is rather ironic taking into account that the dynamics of arena is the infinite-dimensional world of classical worlds.

5. A concrete connection with the non-commutative geometry approach suggests itself: the non-commutative geometry of quantum critical geodesic sphere of CP₂ would be natural manner to describe the anticommutativity of induced spinor field and its conjugate at only the points of number theoretic braid rather than along line (string) as implied by the holomorphy of the spinor mode basis. Non-commutativity is equivalent with the reduction of basis to a finite sub-basis implied by the discretization of partonic 2-surface to collection of number theoretic braids defined by Higgs minima.

   The underlying philosophy is the description of the finite quantum measurement resolution in terms of Jones inclusions and leading naturally to non-commutative Hilbert space, Connes tensor product, etc... What is also new is that braiding can be seen as being induced by the motion of Higgs minima (here also the projections on the geodesic 2-sphere could define the braiding).

6. Higgs expectation has a purely geometric interpretation. The very construction of the Higgs expectation allows to assign a 4-D space-time sheet with a light-like 3-surface. The additional dimension comes via the lines connecting partonic 2-surface to the quantum critical geodesic sphere which are mathematically orbits of Kähler charged particle with a dynamically generated charge. The emergence of the additional dimension naturally is a real victory. That this space-time sheet is indeed an extremal of Kähler action remains to be demonstrated. This would not be a big surprise.

7. The zeta function defined by the values of Higgs at the points of braids associated with the partonic 2-surface codes for geometric data about it so that the earlier speculation is now become a firm fact. The zeros of zeta are excellent candidates for what I call super-canonical conformal weights and many of them should reside at the critical line of zeta since the conformal weights for the function basis depending on the radial coordinate of lightcone boundary have naturally real part equal to 1/2. A very elegant and economic reduction of most of basic physics of quantum TGD to the properties of the modified Dirac operator takes place. This means also a concrete realization of the number theoretic ideas.

8. One fascinating and emotionally stressing conjecture (I believe it just now but I cannot predict what I think after one minute) is the self referentiality of the zeta function stating that the eigenvalues λ (complex values of Higgs at minima) defining the zeta by replacing natural number n in Riemann zeta are proportional to the inverses of zeta at the points defining the number theoretical braid and continue to a map from geodesic sphere to geodesic sphere by replacing λ with general complex point of sphere. The weak form of this conjecture is satisfied by the zeros s= -2m, m=1,2,... of Riemann Zeta in the sense that zeta(s) indeed vanishes at points -2λ, &lambda= m. Now one would have stronger form holding true not only at origin but also at the points associated with the number theoretical braids and perhaps at entire sphere.

9. Also many other things remain to be done to prove the long list of purely mathematical conjectures characterizing preferred extremals of Kähler action analogous to Bohr orbits. These conjectures are obvious once TGD view about physics is expected but mathematically very non-trivial. This is one of the frustrating things in this business of re-creating universe on computer screen: you simply know that if I your philosophy is this and this then that and that which is mathematically something highly non-trivial must be true. But you have no idea about how to prove it. I wonder whether She has experienced similar frustrations and why she does not want to tell me the proof;-).

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".

Geometrization of Higgs mechanism in TGD framework

The improved understanding of the generalization of the imbedding space concept forced by the hierarchy of Planck constants led to a considerable progress in TGD. For instance, I understand now how fractional quantum Hall effect emerges in TGD framework. I have also a rather satisfactory understanding of the notion of number theoretic braid: in particular the question how the cutoff implying that the number of strands is finite, emerges from inherent geometry of the partonic 2-surface. Also a beautiful geometric interpretation of the generalized eigenstates and eigenvalues of the modified Dirac operator and understanding of super-canonical conforma weights emerges.

It became already earlier clear that the generalized eigenvalue of Dirac operator which are scalar fields correspond to Higgs expectation value physically. The problem was to deduce what this expectation value is and I have now very beautiful geometric construction of Higgs expectation value as a coder of rather simple but fundamental geometric information about partonic surface. This leads also to an expression for the zeta function associated with number theoretic braid and understanding of what geometric information it codes about partonic 2-surface. Also the finiteness of the theory becomes manifest since the number of generalized eigenvalues is finite. In the following I describe the arguments related to the geometrization of Higgs expectation. I attach the text which can be also found from the chapter Construction of Quantum Theory Symmetries of "Towards S-matrix".

Geometrization of Higgs mechanism in TGD framework

The identification of the generalized eigenvalues of the modified Dirac operator as Higgs field suggests the possibility of understanding the spectrum of D purely geometrically by combining physical and geometric constraints.

The standard zeta function associated with the eigenvalues of the modified Dirac action is the best candidate concerning the interpretation of super-canonical conformal weights as zeros of ζ. This ζ should have very concrete geometric and physical interpretation related to the quantum criticality. This would be the case if these eigenvalues, eigenvalue actually, have geometric based on geometrization of Higgs field.

Before continuing it its convenient to introduce some notations. Denote the complex coordinate of a point of X² by w, its H=M⁴× CP₂ coordinates by h=(m,s), and the H coordinates of its R₊× S²_II projection by h_c=(r₊,s_II).

1. Interpretation of eigenvalues of D as Higgs field

The eigenvalues of the modified Dirac operator have a natural interpretation as Higgs field which vanishes for unstable extrema of Higgs potential. These unstable extrema correspond naturally to quantum critical points resulting as intersection of M⁴ resp. CP₂ projection of the partonic 2-surface X² with S²_r resp. S²_II.

Quantum criticality suggests that the counterpart of Higgs potential could be identified as the modulus square of Higgs

V(H(s))= -H(s)² .

which indeed has the points s with V(H(s))=0 as extrema which would be unstable in accordance with quantum criticality. The fact that for ordinary Higgs mechanism minima of V are the important ones raises the question whether number theoretic braids might more naturally correspond to the minima of V rather than intersection points with S². This turns out to be the case. It will also turn out that the detailed form of Higgs potential does not matter: the only thing that matters is that V is monotonically decreasing function of the distance from the critical manifold.

2. Purely geometric interpretation of Higgs

Geometric interpretation of Higgs field suggests that critical points with vanishing Higgs correspond to the maximally quantum critical manifold R₊× S²_II. The value of H should be determined once h(w) and R₊× S²_II projection h_c(w) are known. H should increase with the distance between these points.

The question is whether one can assign to a given point pair (h(w),h_c(w)) naturally a value of H. The first guess is that the value of H is determined by the shortest geodesic line connecting the points (product of geodesics of δM⁴ and CP₂). The value should be in general complex and invariant under the isometries of δH affecting h and h_c(w). The minimal geodesic distance d(h,h_c) between the two points would define the first candidate for the modulus of H.

This guess turns need not be quite correct. An alternative guess is that M⁴ projection is indeed geodesic but that M⁴ projection extremizes itse length subject to the constraint that the absolute value of the phase defined by one-dimensional Kähler action ∫ A_μdx^μ is minimized: this point will be discussed below. If this inclusion is allowed then internal consistency requires also the extremization of ∫ A_μdx^μ so that geodesic lines are not allowed in CP₂.

The value should be in general complex and invariant under the isometries of δ H affecting h and h_c. The minimal distance d(h,h_c) between the two points constrained by extremal property of phase would define the first candidate for the modulus of H.

The phase factor should relate close to the Kähler structure of CP₂ and one possibility would be the non-integrable phase factor U(s,s_II) defined as the integral of the induced Kähler gauge potential along the geodesic line in question. Hence the first guess for the Higgs would be as

H(w)= d(h,h_c(w))× U(s,s_II) ,

d(h,h_c(w))=∫_h^h_cds ,

U(s,s_II) = exp[i∫_s^s_IIA_kds^k] .

This gives rise to a holomorphic function is X² the local complex coordinate of X² is identified as w= d(h,h_c)U(s,s_II) so that one would have H(w)=w locally. This view about H would be purely geometric.

One can ask whether one should include to the phase factor also the phase obtained using the Kähler gauge potential associated with S²_r having expression (A_θ,A_φ)=(k,cos(θ)) with k even integer from the requirement that the non-integral phase factor at equator has the same value irrespective of whether it is calculated with respect to North or South pole. For k=0 the contribution would be vanishing. The value of k might correlate directly with the value of quantum phase. The objection against inclusion of this term is that Kähler action defining Kähler function should contain also M⁴ part if this term is included.

In each coordinate patch Higgs potential would be simply the quadratic function V= -ww*. Negative sign is required by quantum criticality. Potential could indeed have minima as minimal distance of X²_CP2 point from S²_II. Earth's surface with zeros as tops of mountains and bottoms of valleys as minima would be a rather precise visualization of the situation for given value of r₊. Mountains would have a shape of inverted rotationally symmetry parabola in each local coordinate system.

3. Consistency with the vacuum degeneracy of Kähler action and explicit construction of preferred extremals

An important constraint comes from the condition that the vacuum degeneracy of Käahler action should be understood from the properties of the Dirac determinant. In the case of vacuum extremals Dirac determinant should have unit modulus.

Suppose that the space-time sheet associated with the vacuum parton X² is indeed vacuum extremal. This requires that also X³_l is a vacuum extremal: in this case Dirac determinant must be real although it need not be equal to unity. The CP₂ projection of the vacuum extremal belongs to some Lagrangian sub-manifold Y² of CP₂. For this kind of vacuum partons the ratio of the product of minimal H distances to corresponding M⁴_+/- distances must be equal to unity, in other words minima of Higgs potential must belong to the intersection X²\cap S²_II or to the intersection X²\cap R₊ so that distance reduces to M⁴ or CP₂ distance and Dirac determinant to a phase factor. Also this phase factor should be trivial.

It seems however difficult to understand how to obtain non-trivial phase in the generic case for all points if the phase is evaluated along geodesic line in CP₂ degrees of freedom. There is however no deep reason to do this and the way out of difficulty could be based on the requirement that the phase defined by the Kähler gauge potential is evaluated along a curve either minimizing the absolute value of the phase modulo 2π.

One must add the condition that curve is not shorter than the geodesic line between points. For a given curve length s₀ the action must contain as a Lagrange multiplier the curve length so that the action using curve length s as a coordinate reads as

S= ∫ A_sds +λ(∫ ds-s₀).

This gives for the extremum the equation of motion for a charged particle with Kähler charge Q_K= 1/λ:

D²s^k/ds² + (1/λ)× J^k_lds^l/ds=0 ,

D²m^k/ds²=0 .

The magnitude of the phase must be further minimized as a function of curve length s.

If the extremum curve in CP₂ consists of two parts, first belonging to X²_II and second to Y², the condition is satisfied. Hence, if X²_CP2× Y² is not empty, the phases are trivial. In the generic case 2-D sub-manifolds of CP₂ have intersection consisting of discrete points (note again the fundamental role of 4-dimensionality of CP₂). Since S²_II itself is a Lagrangian sub-manifold, it has especially high probably to have intersection points with S²_II. If this is not the case one can argue that X³_l cannot be vacuum extremal anymore.

The construction gives also a concrete idea about how the 4-D space-time sheet X⁴(X³_l) becomes assigned with X³_l. The point is that the construction extends X² to 3-D surface by connecting points of X² to points of S²_II using the proposed curves. This process can be carried out in each intersection of X³_l and M⁴₊ shifted to the direction of future. The natural conjecture is that the resulting space-time sheet defines the 4-D preferred extremum of Käahler action.

4. About the definition of the Dirac determinant and number theoretic braids

The definition of Dirac determinant should be independent of the choice of complex coordinate for X² and local complex coordinate implied by the definition of Higgs is a unique choice for this coordinate.

The physical intuition based on Higgs mechanism suggests strongly that the Dirac determinant should be defined simply as products of the eigenvalues of D, that is those of Higgs field, associated with the number theoretic braid. If only single kind of braid is allowed then the minima of Higgs field define the points of the braid very naturally. The points in R₊× S²_II cannot contribute to the Dirac determinant since Higgs vanishes at the critical manifold. Note that at S²_II criticality Higgs values become real and the exponent of Kähler action should become equal to one. This is guaranteed if Dirac determinant is normalized by dividing it with the product of δM⁴_+/-distances of the extrema from R₊. The value of the determinant would equal to one also at the limit R₊× S²_II.

One would define the Dirac determinant as the product of the values of Higgs field over all minima of local Higgs potential

det(D)= [∏_k H(w_k)]/[∏_k H₀(w_k)]= ∏_k[w_k/w⁰_k].

Here w⁰_k are M⁴ distances of extrema from R₊. Equivalently: one can identify the values of Higgs field as dimensionless numbers w_k/w⁰_k. The modulus of Higgs field would be the ratio of H and M⁴_+/- distances from the critical sub-manifold. The modulus of the Dirac determinant would be the product of the ratios of H and M⁴ depths of the valleys.

This definition would be general coordinate invariant and independent of the topology of X². It would also introduce a unique conformal structure in X² which should be consistent with that defined by the induced metric. Since the construction used relies on the induced metric this looks natural. The number of eigen modes of D would be automatically finite and eigenvalues would have a purely geometric interpretation as ratios of distances on one hand and as masses on the other hand. The inverse of CP₂ length defines the natural unit of mass. The determinant is invariant under the scalings of H metric as are also Kähler action and Chern-Simons action. This excludes the possibility that Dirac determinant could also give rise to the exponent of the area of X².

Number theoretical constraints require that the numbers w_k are algebraic numbers and this poses some conditions on the allowed partonic 2-surfaces unless one drops from consideration the points which do not belong to the algebraic extension used.

5. Physical identification of zeta function

The proposed picture supports the identification of the eigenvalues of D in terms of a Higgs fields having purely geometric meaning. The identification of Higgs as the inverse of ζ function is not favored. It also seems that number theoretic braids must be identified as minima of Higgs potential in X². Furthermore, the braiding operation could be defined for all intersections of X³_l defined by shifts M⁴_+/- as orbits of minima of Higgs potential. Second option is braiding by Kähler magnetic flux lines.

The question is then how to understand super-canonical conformal weights for which the identification as zeros of a zeta function of some kind is highly suggestive. The natural answer would be that the eigenvalues of D defines this zeta function as

ζ(s)= ∑_k [H(w_k)/H(w⁰_k)]^-s .

The number of eigenvalues contributing to this function would be finite and H(w_k)/H(w⁰_k should be rational or algebraic at most. ζ function would have a precise meaning consistent with the usual assignment of zeta function to Dirac determinant.

The ζ function would directly code the basic geometric properties of X² since the moduli of the eigenvalues characterize the depths of the valleys of the landscape defined by X² and the associated non-integrable phase factors. The degeneracies of eigenvalues would in turn code for the number of points with same distance from a given zero intersection point.

The zeros of this ζ function would in turn define natural candidates for super-canonical conformal weights and their number would thus be finite in accordance with the idea about inherent cutoff also in configuration space degrees of freedom. Note that super-canonical conformal weights would be functionals of X². The scaling of λ by a constant depending on p-adic prime factors out from the zeta so that zeros are not affected: this is in accordance with the renormalization group invariance of both super-canonical conformal weights and Dirac determinant.

The zeta function should exist also in p-adic sense. This requires that the numbers λ:s at the points s of S²_II which corresponds to the number theoretic braid are algebraic numbers. The freedom to scale λ could help to achieve this.

6. The relationship between λ and Higgs field

The generalized eigenvalue λ(w) is only proportional to the vacuum expectation value of Higgs, not equal to it. Indeed, Higgs and gauge bosons as elementary particles correspond to wormhole contacts carrying fermion and antifermion at the two wormhole throats and must be distinguished from the space-time correlate of its vacuum expectation as something proportional to λ. In the fermionic case the vacuum expectation value of Higgs does not seem to be even possible since fermions do not correspond to wormhole contacts between two space-time sheets but possess only single wormhole throat (p-adic mass calculations are consistent with this). Gauge bosons can have Higgs expectation proportional to λ. The proportionality must be of form <H> propto λ/p^n/2 if gauge boson mass squared is of order 1/pⁿ. The p-dependent scaling factor of λ is expected to be proportional to log(p) from p-adic coupling constant evolution.

7. Possible objections related to the interpretation of Dirac determinant

Suppose that that Dirac determinant is defined as a product of determinants associated with various points z_k of number theoretical braids and that these determinants are defined as products of corresponding eigenvalues.

Since Dirac determinant is not real and is not invariant under isometries of CP₂ and of δ M⁴_+/-, it cannot give only the exponent of Kähler function which is real and SU(3)× SO(3,1) invariant. The natural guess is that Dirac determinant gives also the Chern-Simons exponential.

The objection is that Chern-Simons action depends not only on X² but its light-like orbit X³_l.

1. The first manner to circumvent this objection is to restrict the consideration to maxima of Kähler function which select preferred light-like 3-surfaces X³_l. The basic conjecture forced by the number theoretic universality and allowed by TGD based view about coupling constant evolution indeed is that perturbation theory at the level of configuration space can be restricted to the maxima of Kähler function and even more: the radiative corrections given by this perturbative series vanish being already coded by Kähler function having interpretation as analog of effective action.

2. There is also an alternative way out of the difficulty: define the Dirac determinant and zeta function using the minima of the modulus of the generalized Higgs as a function of coordinates of X³_l so that continuous strands of braids are replaced by a discrete set of points in the generic case.

The fact that general Poincare transformations fail to be symmetries of Dirac determinant is not in conflict with Poincare invariance of Kähler action since preferred extremals of Kähler action are in question and must contain the fixed partonic 2-surfaces at δ M⁴_+/- so that these symmetries are broken by boundary conditions which does not require that the variational principle selecting the preferred extremals breaks these symmetries.

One can exclude the possibility that the exponent of the stringy action defined by the area of X² emerges also from the Dirac determinant. The point is that Dirac determinant is invariant under the scalings of H metric whereas the area action is not.

The condition that the number of eigenvalues is finite is most naturally satisfied if generalized ζ coding information about the properties of partonic 2-surface and expressible as a rational function for which the inverse has a finite number of branches is in question.

8. How unique the construction of Higgs field really is?

Is the construction of space-time correlate of Higgs as λ really unique? The replacement of H with its power H^r, r>0, leaves the minima of H invariant as points of X² so that number theoretic braid is not affected. As a matter fact, the group of monotonically increasing maps real-analytic maps applied to H leaves number theoretic braids invariant. Polynomials with positive rational coefficients suggest themselves.

The map H→ H^r scales Kähler function to its r-multiple, which could be interpreted in terms of 1/r-scaling of the Kähler coupling strength. Also super-canonical conformal weights identified as zeros of ζ are scaled as h→ h/r and Chern-Simons charge k is replaced with k/r so that at least r=1/n might be allowed.

One can therefore ask whether the powers of H could define a hierarchy of quantum phases labelled by different values of k and α_K. The interpretation as separate phases would conform with the idea that D in some sense has entire spectrum of generalized eigenvalues. Note however that this would imply fractional powers for H.