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Year 2007

DNA as topological quantum computer: IX

In previous postings I, II, III, IV, V, VI, VII, VIII I have discussed various aspects of the idea that DNA could acts as a topological quantum computer using fundamental braiding operation as a universal 2-gate.

There are several grand visions about TGD Universe. One of them is as a topological quantum computer in a very general sense. This kind of visions are always oversimplifications but the extreme generality of the braiding mechanism suggest that also simpler systems than DNA might be applying tqc. The detailed model for tqc performed by DNA indeed leads to the idea that so called water memory could be realized in terms of braidings.

A. Braid strands as flux tubes of color magnetic body

The flux tubes defining braid strands carry magnetic field when the supra current is on. In TGD Universe all classical fields are expressible in terms of the four CP2 coordinates and their gradients so that em, weak, color and gravitational fields are not independent as in standard model framework. In particular, the ordinary classical em field is necessarily accompanied by a classical color field in the case of non-vacuum extremals. This predicts color and ew fields in arbitrary long scales and quantum classical correspondence forces to conclude that there exists fractal hierarchy of electro-weak and color interactions.

Since the classical color gauge field is proportional to Kähler form, its holonomy group is Abelian so that effectively U(1)× U(1)subset SU(3) gauge field is in question. The generation of color flux requires colored p"../articles/ at the ends of color flux tube so that the presence of pairs of quark and antiquark assignable to the pairs of wormhole throats at the ends of the tube is unavoidable if one accepts quantum classical correspondence.

In the case of cell, a highly idealized model for color magnetic flux tubes is as flux tubes of a dipole field. The preferred axis could be determined by the position of the centrosomes forming a T shaped structure. DNA strands would define the idealized dipole creating this field: DNA is indeed negatively charged and electronic currents along DNA could create the magnetic field. The flux tubes of this field would go through nuclear and cell membrane and return back unless they end up to another cell. This is indeed required by the proposed model of tqc.

It has been assumed that the initiation of tqc means that the supra current ceases and induces the splitting of braid strands. The magnetic flux need not however disappear completely. As a matter fact, its presence forced by the conservation of magnetic flux seems to be crucial for the conservation of braiding. Indeed, during tqc magnetic and color magnetic flux could return from lipid to DNA along another space-time sheet at a distance of order CP2 radius from it. For long time ago I proposed that this kind of structures -which I christened "wormhole magnetic fields" - might play key role in living matter. The wormhole contacts having quark and antiquark at their opposite throats and coding for A, T, C, G would define the places where the current flows to the "lower" space-time sheet to return back to DNA. Quarks would also generate the remaining magnetic field and supra current could indeed cease.

The fact that classical em fields and thus classical color fields are always present for non-vacuum extremals means that also the motion of any kind of p"../articles/ (space-time sheets), say water flow, induces a braiding of magnetic flux tubes associated with molecules in water if the temporary splitting of flux tubes is possible. Hence the prerequisites for tqc are met in extremely general situation and tqc involving DNA could have developed from a much simpler form of tqc performed by water giving perhaps rise to what is known as water memory (see this, this and this). This would also suggest that the braiding operation is induced by the a controlled flow of cellular water.

B. Water memory: general considerations

With few exceptions so called "serious" scientists remain silent about the experiments of Benveniste and others relating to water memory (see this, this and this) in order to avoid association with the very ugly word "homeopathy".

The Benveniste's discovery of water memory initiated quite dramatic sequence of events. The original experiment involved the homeopathic treatment of water by human antigene. This meant dilution of the water solution of antigene so that the concentration of antigene became extremely low. In accordance with homeopathic teachings human basophils reacted on this solution.

The discovery was published in Nature and due to the strong polemic raised by the publication of the article, it was decided to test the experimental arrangement. The experimental results were reproduced under the original conditions. Then it was discovered that experimenters knew which bottles contained the treated water. The modified experiment in which experimenters did not possess this information failed to reproduce the results and the conclusion was regarded as obvious and Benveniste lost his laboratory among other things. Obviously any model of the effect taking it as a real effect rather than an astonishingly simplistic attempt of top scientists to cheat should explain also this finding.

The model based on the notion of field body and general mechanism of long term memory allows to explain both the memory of water and why it failed under the conditions described.

  1. Also molecules have magnetic field bodies acting as intentional agents controlling the molecules. Nano-motors do not only look co-operating living creatures but are such. The field body of molecule contains besides the static magnetic and electric parts also dynamical parts characterized by frequencies and temporal patterns of fields. To be precise, one must speak both field and relative field bodies characterizing interactions of molecules. Right brain sings-left brain talks metaphor might generalize to all scales meaning that representations based on both frequencies and temporal pulse with single frequency could be utilized.

    The effects of complex bio-molecule to other bio-molecules (say antigene on basofil) in water could be characterized to some degree by the temporal patterns associated with the dynamical part of its field body and bio-molecules could recognize each other via these patterns. This would mean that symbolic level in interactions would be present already in the interactions of bio-molecules.

    If water is to mimic the field bodies of molecules using water molecule clusters, at least vibrational and rotational spectra, then water can produce fake copies of say antigenes recognized by basofils and reacting accordingly.

    Also the magnetic body of the molecule could mimic the vibrational and rotational spectra using harmonics of cyclotron frequencies. Cyclotron transitions could produce dark photons, whose ordinary counterparts resulting in de-coherence would have large energies due to the large value of hbar and could thus induce vibrational and rotational transitions. This would provide a mechanism by which molecular magnetic body could control the molecule. Note that also the antigenes possibly dropped to the larger space-time sheets could produce the effect on basofils.

  2. There is a considerable experimental support for the Benveniste's discovery that bio-molecules in water environment are represented by frequency patterns, and several laboratories are replicating the experiments of Benveniste as I learned from the lecture of Yolene Thomas in the 7:th European SSE Meeting held in Röros [4]. The scale of the frequencies involved is around 10 kHz and as such does not correspond to any natural molecular frequencies. Cyclotron frequencies associated with electrons or dark ions accompanying these macromolecules would be a natural identification if one accepts the notion of molecular magnetic body. For ions the magnetic fields involved would have a magnitude of order .03 Tesla if 10 kHz corresponds to scaled up alpha band. Also Josephson frequencies would be involved if one believes that EEG has fractally scaled up variants in molecular length scales.

  3. Suppose that the representations of bio-molecules in water memory rely on pulse patterns representing bit sequences. The simplest realization of bit would be as a laser like system with bit 1 represented by population inverted state and bit 0 by the ground state. Bits could be arranged in sequences spatially or by variation of zero point energy defining the frequency: for instance increase of frequency with time would define temporal bit sequence. Many-sheeted lasers are the natural candidates for laser like systems are in question since they rely on universal metabolic energy quanta. Memory recall would involve sending of negative energy phase conjugate photons inducing a partial transition to the ground state. The presence of metabolic energy feed would be necessary in order to preserve the memory representations.

C. Water memory in terms of molecular braidings

It is interesting to look water memory from the point of view of tqc. Suppose that the molecules and water p"../articles/ (space-time sheet of size of say cell length scale) are indeed connected by color flux tubes defining the braid strands and that splitting of the braid strands can take place so that water flow can gives rise to a braiding pattern and tqc like process.

The shaking of the bottle containing the diluted homeopathic remedy is an essential element in the buildup of water memories also in the experiments of Benveniste. Just like the vigorous flow of sol near the inner monolayer, this process would create a water flow and this flow creates a braiding pattern which could provide a representation for the presence of the molecules in question. Note that the hardware of braiding could carry information about molecules (cyclotron frequencies for ions for instance).

The model for the formation of scaled down variants of memories in hippocampus discussed above suggests that each half period of theta rhythm corresponds to tqc followed by a non-computational period during which the outcome of tqc is expressed as 4-D nerve pulse patterns involving cyclotron frequencies and Josephson frequency. Josephson currents at the second half period would generate dark Josephson radiation communicating the outcome of the calculation to the magnetic body. Entire hierarchy of EEGs with varying frequency scale would be present corresponding to the onion like structure of magnetic body. This pattern would provide an electromagnetic representation for the presence of the antigene and could be mimicked artificially [1,2,3].

This picture might apply be the case also in the case of water memory.

  1. The shaking might drop some fraction of antigene molecules to dark space-time sheets where they generate a dark color magnetic field. Because of the large value of Planck constant super-conductivity along color flux tubes running from molecular space-time sheets could still be present.

  2. TGD based model of super conductivity involves double layered structures with same p-adic length scale scale as cell membrane (see this). The universality of p-adic length scale hierarchy this kind of structures but with a much lower voltage over the bilayer could be present also in water. Interestingly, Josephson frequency ZeV/hbar would be much lower than for cell membrane so that the time scale of memory could be much longer than for cell membrane for given value of hbar meaning longer time scale of memory recall.

  3. Also in the case of homeopathic remedy the communication of the result of tqc to the magnetic body would take place via Josephson radiation. From the point of view of magnetic body Josephson radiation resulting in shaking induced tqc induced would replace the homeopathic remedy with a field pattern. The magnetic bodies of basophils could be cheated to produce allergic reaction by mimicking the signal representing the outcome of this tqc. This kind of cheating was indeed done in the later experiments of Benveniste involving very low frequency electromagnetic fields in kHz region allowing no identification in terms of molecular transitions (magnetic body and cyclotron frequencies) [1].

D. Why experimenter had to know which bottle contained the treated water?

Why experimenter had to know which bottle contained the treated water? The role of experimenter eliminates the possibility that the (magnetic bodies of) clusters of water molecules able to mimic the (magnetic bodies of) antigene molecules electromagnetically are present in the solution at geometric now and produce the effect. The earlier explanation for experimenter's role was based on the idea that memory storage requires metabolic energy and that experimenter provides it. Tqc picture suggests a variant of this model in which experimenter makes possible the recall of memories of water represented as braiding patterns and realized via tqc.

D.1 Does experimenter provide the metabolic energy needed to store the memories of water?

What could be then the explanation for the failure of the modified experiment? Each memory recall reduces the occupation of the states representing bit 1 and a continual metabolic energy feed is needed to preserve the bit sequence representations of antibodies using laser light systems as bit. This metabolic energy feed must come from some source.

By the universality of metabolic energy currencies population inverted many-sheeted lasers in living organisms define the most natural source of the metabolic energy. Living matter is however fighting for metabolic energy so that there must be some system willing to provide it. The biological bodies of experimenters are the best candidates in this respect. In this case experimenters had even excellent motivations to provide the metabolic energy. If this interpretation is correct then Benveniste's experiment would demonstrate besides water memory also psychokinesis and direct action of desires of experimenters on physics at microscopic level. Furthermore, the mere fact that we know something about some object or direct attention to it would mean a concrete interaction of our magnetic with the object.

D.2 Does experimenter make possible long term memory recall?

The alternative explanation is that experimenter makes possible long term memory recall which also requires metabolic energy.

  1. If braiding pattern represents, the water memory the situation changes since the robustness of the braiding pattern suggests that this representation is still in the geometric past (which is replaced with a new one many times). If the dark variants of molecules created in the process are still in the water, the braid representation of water memories could be available even in the geometric now but it is better to not make this assumption. The challenge is to understand how this information can be made conscious.

  2. What is certainly needed is that the system makes the tqc again. This would mean a fractal quantum jump involving unitary U process and state function reduction leading to the generation of generalized EEG pattern. Only the sums and differences of cyclotron frequency and Josephson frequency would matter so that the details of the flow inducing braiding do not matter. The shaking process might be continuing all the subjective time in the geometric past so that the problem is how to receive information about its occurrence. Experimenter might actually help in this respect since the mechanism of intentional action initiates the action in the geometric past by a negative energy signal.

  3. If the magnetic body of the water in the geometric now can entangle with the geometric past, tqc would regenerate the experience about the presence of antigene by sharing and fusion of mental images. One can however argue that water cannot have memory recall in this time scale since water is quite simple creature and levels with large enough hbar might not be present. It would seem that here the experimenter must come in rescue.

  4. The function of experimenter's knowledge about which bottle contains the homeopathic solution could be simply to generate time-like entanglement in the required long time scale by serving as a relay station. The entanglement sequence would be water now - experimenter now - water in the past with "now" and "past" understood in the geometric sense. The crucial entanglement bridge between the magnetic body of water and experimenter would be created in the manufacturing of the homeopathic remedy.

Note that these explanations do not exclude each other. It is quite possible that experimenter provides also the metabolic energy to the bit representation of water memories possibly induced by the long term memory recall.

This picture is of course just one possible model and cannot be taken literally. The model however suggest that magnetic bodies of molecules indeed define the braiding; that the generalized EEG provides a very general representation for the outcome of tqc; that liquid flow provides the manner to build tqc programs - and also that shaking and sudden pulses is the concrete manner to induce visible-dark phase transitions. All this might be very valuable information if one some day in the distant future tries to build topological quantum computers in laboratory.

E. Little personal reminiscence about flow

I cannot resist a temptation to bore the reader with something which I have already told quite too many times. The reason why I started to seriously ponder consciousness was the wonderful experience around 1985 or so, which lasted from week two two - I do not remember precisely. To tell quite honestly and knowing the reactions induced in some hard nosed "serious" scientists: my experience was that I was enlightened. The depth and beauty of this state of consciousness was absolutely stunning and it was very hard to gradually realize that I would not get this state back.

To characterize the period of my life which I would without a hesitation choose if I had to select the most important weeks of my life, the psychologist needed only two magic words - acute psychosis. The psychologist had even firmly predicted that I would soon fall in a totally autistic state! This after some routine examinations (walking along straight line and similar tests). What incredible idiots can an uncritical belief on science make of us!

This experience made with single stroke clear that in many respects the existing psychology does not differ much from the medicine at middle ages. The benevolent people believing in this trash - modern psychologists - can cause horrible damage and suffering to their patients. As I started serious building of consciousness theory and learned neuroscience and biology, I began to grasp at more general level how insane the vision of the official neuroscience and biology about consciousness was. We laugh for the world view of people of middle ages but equally well they could laugh for our modern views about what we are.

Going back to the experience. During it I saw my thoughts as extremely vivid and colorful patterns bringing in mind paintings of Dali and Bosch. What was strange was the continual and very complex flow at the background consisting of separate little dots. I can see this flow also now by closing my eyes lightly when in a calm state of mind. I have proposed many explanations for it and tried to figure out what this flow tries to tell to me. Sounds pompous and a little bit childish in this cynic world, but this is the first time that I dare hope of having understood the deeper message I know is there.


[1] J. Benveniste et al (1988). Human basophil degranulation triggered by very dilute antiserum against IgE. Nature 333:816-818.

[2] J. Benveniste et al (198?). Transatlantic transfer of digitized antigen signal by telephone link. Journal of Allergy and Clinical Immunology. 99:S175 (abs.). For recent work about digital biology and further references about the work of Benveniste and collaborators see this .

[3] L. Milgrom (2001), Thanks for the memory. An article in Guardian about the work of professor M. Ennis of Queen's University Belfast supporting the observations of Dr. J. Benveniste about water memory.

[4] E. Strand (editor) (2007), Proceedings of the 7th European SSE Meeting August 17-19, 2007, Röros, Norway. Society of Scientific Exploration.

For details see the chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: VIII

In previous postings I, II, III, IV, V, VI, VII I have discussed various aspects of the idea that DNA could acts as a topological quantum computer using fundamental braiding operation as a universal 2-gate.

In the following I will consider first the realization of the basic braiding operation: this requires some facts about phospholipids which are summarized first. Also the realization of braid color is discussed. This requires the coding of the DNA color A,T,C,G to a property of braid strand which is such that it is conserved meaning that after halting of tqc only strands with same color can reconnect. This requires long range correlation between lipid and DNA nucleotide. It seems that strand color cannot be chemical. Quark color is essential element of TGD based model of high Tc superconductivity and provides a possible solution to the problem: the four neutral quark-antiquark pairs with quark and antiquark at the ends of color flux tube defining braid strand would provide the needed four colors.

A. Some facts about phospholipids

Phospholipids - which form about 30 per cent of the lipid content of the monolayer - contain phosphate group. The dance of lipids requires metabolic energy and the hydrophilic ends of the phospholipid could provide it. They could also couple the lipids to the flow of water in the vicinity of the lipid monolayer possibly inducing the braiding. Of course, the causal arrow could be also opposite.

The hydrophilic part of the phospholipid is a nitrogen containing alcohol such as serine, inositol or ethanolamine, or an organic compound such as choline. Phospholipids are classified into 3 kinds of phosphoglycerides and sphingomyelin.

A.1 Phosphoglycerides

In cell membranes, phosphoglycerides are the more common of the two phospholipids, which suggest that they are involved with tqc. One speaks of phosphotidyl X, where X= serine, inositol, ethanolamine is the nitrogen containing alcohol and X=Ch the organic compound. The shorthand notion OS, PI, PE, PCh is used.

The structure of the phospholipid is most easily explained using the dancer metaphor. The two fatty chains define the hydrophobic feet of the dancer, glycerol and phosphate group define the body providing the energy to the dance, and serine, inositol, ethanolamine or choline define the hydrophilic head of the dancer (perhaps "deciding" the dancing pattern).

There is a lipid asymmetry in the cell membrane. PS, PE, PI in cytoplasmic monolayer (alcohols). PC (organic) and sphingomyelin in outer monolayer. Also glycolipids are found only in the outer monolayer. The asymmetry is due to the manner that the phospholipids are manufactured.

PS in the inner monolayer is negatively charged and its presence is necessary for the normal functioning of the cell membrane. It activates protein kinase C which is associated with memory function. PS slows down cognitive decline in animals models. This encourages to think that the hydrophilic polar end of at least PS is involved with tqc, perhaps to the generation of braiding via the coupling to the hydrodynamic flow of cytoplasm in the vicinity of the inner monolayer.

A. 2. Fatty acids

The fatty acid chains in phospholipids and glycolipids usually contain an even number of carbon atoms, typically between 14 and 24 making 5 possibilities altogether. The 16- and 18-carbon fatty acids are the most common. Fatty acids may be saturated or unsaturated, with the configuration of the double bonds nearly always cis. The length and the degree of unsaturation of fatty acids chains have a profound effect on membranes fluidity as unsaturated lipids create a kink, preventing the fatty acids from packing together as tightly, thus decreasing the melting point (increasing the fluidity) of the membrane. The number of unsaturaded cis bonds and their positions besides the number of Carbon atoms characterizes the lipid. Quite generally, there are 3n Carbons after each bond. The creation of unsatured bond by removing H atom from the fatty acid could be an initiating step in the basic braiding operation creating room for the dancers. The bond should be created on both neighboring lipids simultaneously.

B. How the braiding operation could be induced?

One can imagine several models for what might happen during the braiding operation in the lipid bilayer. One such view is following.

  1. The creation of unsaturated bond and involving elimination of H atom from fatty acid would lead to cis configuration and create the room needed by dancers. This operation should be performed for both lipids participating in the braiding operation. After the braiding it might be necessary to add H atom back to stabilize the situation. The energy needed to perform either or both of these operations could be provided by the phosphate group.

  2. The hydrophilic ends of lipids couple the lipids to the surrounding hydrodynamic flow in the case that the lipids are able to move. This coupling could induce the braiding. The primary control of tqc would thus be by using the hydrodynamic flow by generating localized vortices. There is considerable evidence for water memory but its mechanism remains to be poorly understood. If also water memory is realized in terms of the braid strands connecting fluid p"../articles/, DNA tqc could have evolved from water memory.

  3. Sol-gel phase transition is conjectured to be important for the quantum information processing of cell (see this). In the transition which can occur cyclically actin filaments (also at EEG frequencies) are assembled and lead to a gel phase resembling solid. Sol phase could correspond to tqc and gel to the phase following the halting of tqc. Actin filaments might be assignable with braid strands or bundles of them and shield the braiding. Also microtubules might shield bundles of braid strands.

  4. Only inner braid strands are directly connected to DNA which also supports the view that only the inner monolayer suffers a braiding operation during tqc and that the outer monolayer should be in a "freezed" state during it. There is a net negative charge associated with the inner mono-layer possibly relating to its participation to the braiding. The vigorous hydrodynamical flows known to take place below the cell membrane could induce the braiding.

C. How braid color could be realized?

The conserved braid color is not necessary for the model but would imply genetic coding of the tqc hardware so that sexual reproduction would induce an evolution of tqc hardware. Braid color would also make the coupling of foreign DNA to the tqc performed by the organism difficult and realize an immune system at the level of quantum information processing.

The conservation of braid color poses however considerable problems. The concentration of braid strands of same color to patches would guarantee the conservation but would restrict the possible braiding dramatically. A more attractive option is that the strands of same color find each other automatically by energy minimization after the halting of tqc. Electromagnetic Coulomb interaction would be the most natural candidate for the interaction in question. Braid color would define a faithful genetic code at the level of nucleotides. It would induce long range correlation between properties of DNA strand and the dynamics of cell immediately after the halting of tqc.

C.1 Chemical realization of color is not plausible

The idea that color could be a chemical property of phospholipids does not seem plausible. The lipid asymmetry of the inner and outer monolayers excludes the assignment of color to the hyrdrophilic group PS, PI, PE, PCh. Fatty acids have N=14,...,24 carbon atoms and N=16 and 18 are the most common cases so that one could consider the possibility that the 4 most common feet pairs could correspond to the resulting combinations. It is however extremely difficult to understand how long range correlation between DNA nucleotide and fatty acid pair could be created.

C.2 Could quark pairs code for braid color?

It seems that the color should be a property of the braid strand. In TGD inspired model of high Tc super-conductivity (see this) wormhole contacts having u and dc and d and uc quarks at the two wormhole throats feed electron's gauge flux to larger space-time sheet. The long range correlation between electrons of Cooper pairs is created by color confinement for an appropriate scaled up variant of chromo-dynamics which are allowed by TGD. Hence the neutral pairs of colored quarks whose members are located the ends of braid strand acting like color flux tube connecting nucleotide to the lipid could code DNA color to QCD color.

For the pairs udc with net em charge the quark and anti-quark have same sign of em charge and tend to repel each other. Hence the minimization of electro-magnetic Coulomb energy favors the neutral configurations uuc, ddc and uuc, and ddc coding for A, G (say) and their conjugates T and C. After the halting of tqc only these pairs would form with a high probability. The reconnection of the strands would mean a formation of a short color flux tube between the strands and the annihilation of quark pair to gluon. Note that single braid strand would connect DNA color and its conjugate rather than identical colors so that braid strands connecting two DNA strands (conjugate strands) should always traverse through an even (odd) number of cell membranes.

For details see the chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: VII

In previous postings I, II, III, IV, V, VI I have discussed how DNA topological quantum computation could be realized.

If Josephson current through cell membrane ceases during tqc, tqc manifests itself as the presence of only EEG rhythm characterized by an appropriate cyclotron frequency (see posting VI). Synchronous neuron firing might therefore relate to tqc. The original idea that a phase shift of EEG is induced by the voltage initiating tqc - although wrong - was however useful in that it inspired the question whether the initiation of tqc could have something to do with what is known as a place coding by phase shifts performed by hippocampal pyramidal cells (see this and this). Playing with this idea provides important insights about the construction of quantum memories and demonstrates the amazing explanatory power of the paradigm once again.

The model also makes explicit important conceptual differences between tqc a la TGD and in the ordinary sense of word: in particular those related to different view about the relation between subjective and geometric time.

  1. In TGD tqc corresponds to the unitary process U taking place following by a state function reduction and preparation. It replaces configuration space ("world of classical worlds") spinor field with a new one. Configuration space spinor field represent generalization of time evolution of Schrödinger equation so that a quantum jump occurs between entire time evolutions. Ordinary tqc corresponds to Hamiltonian time development starting at time t=0 and halting at t=T to a state function reduction.

  2. In TGD the expression of the result of tqc is essentially 4-D pattern of gene expression (spiking pattern in the recent case). In usual tqc it would be 3-D pattern emerging as the computation halts at time t. Each moment of consciousness can be seen as a process in which a kind of 4-D statue is carved by starting from a rough sketch and proceeding to shorter details and building fractally scaled down variants of the basic pattern. Our life cycle would be a particular example of this process and would be repeated again and again but of course not as an exact copy of the previous one.

1. Empirical findings

The place coding by phase shifts was discovered by O'Reefe and Recce. Y. Yamaguchi describes the vision in which memory formation by so called theta phase coding is essential for the emergence of intelligence. It is known that hippocampal pyramidal cells have "place property" being activated at specific "place field" position defined by an environment consisting of recognizable objects serving as landmarks. The temporal change of the percept is accompanied by a sequence of place unit activities. The theta cells exhibit change in firing phase distributions relative to the theta rhythm and the relative phase with respect to theta phase gradually increases as the rat traverses the place field. In a cell population the temporal sequence is transformed into a phase shift sequence of firing spikes of individual cells within each theta cycle.

Thus a temporal sequence of percepts is transformed into a phase shift sequence of individual spikes of neurons within each theta cycle along linear array of neurons effectively representing time axis. Essentially a time compressed representation of the original events is created bringing in mind temporal hologram. Each event (object or activity in perceptive field) is represented by a firing of one particular neuron at time τn measured from the beginning of the theta cycle. τn is obtained by scaling down the real time value tn of the event. Note that there is some upper bound for the total duration of memory if scaling factor is constant.

This scaling down - story telling - seems to be a fundamental aspect of memory. Our memories can even abstract the entire life history to a handful of important events represented as a story lasting only few seconds. This scaling down is thought to be important not only for the representation of the contextual information but also for the memory storage in the hippocampus. Yamaguchi and collaborators have also found that the gradual phase shift occurs at half theta cycle whereas firings at the the other half cycle show no correlation. One should also find an interpretation for this.

2. TGD based interpretation of findings

How this picture relates to TGD based 4-D view about memory in which primary memories are stored in the brain of the geometric past?

  1. The simplest option is the initiation of tqc like process in the beginning of each theta cycle of period T and having geometric duration T/2. The transition T→ T/2 conforms nicely with the fundamental hierarchy of time scales comings as powers defining the hierarchy of measurement resolutions and associated with inclusions of HFFs. That firing is random at second half of cycle could simply mean that no tqc is performed and that the second half is used to code the actual events of "geometric now".

  2. In accordance with the vision about the hierarchy of Planck constants defining a hierarchy of time scales of long term memories and of planned action, the scaled down variants of memories would be obtained by down-wards scaling of Planck constant for the dark space-time sheet representing the original memory. In principle a scaling by any factor 1/n (actually by any rational) is possible and would imply the scaling down of the geometric time span of tqc and of light-like braids. One would have tqcs inside tqcs and braids within braids (flux quanta within flux quanta). The coding of the memories to braidings would be an automatic process as almost so also the formation of their zoomed down variants.

  3. A mapping of the time evolution defining memory to a linear array of neurons would take place. This can be understood if the scaled down variant (scaled down value of hbar) of the space-time sheet representing original memory is parallel to the linear neuron array and contains at scaled down time value tn a stimulus forcing nth neuron to fire. The 4-D character of the expression of the outcome of tqc allows to achieve this automatically without complex program structure.

To sum up, it seems that the scaling of Planck constant of time like braids provides a further fundamental mechanism not present in standard tqc allowing to build fractally scaled down variants of not only memories but tqc:s in general. The ability to simulate in shorter time scale is a certainly very important prerequisite of intelligent and planned behavior. This ability has also a space-like counterpart: it will be found that the scaling of Planck constant associated with space-like braids connecting bio-molecules might play a fundamental role in DNA replication, control of transcription by proteins, and translation of mRNA to proteins. A further suggestive conclusion is that the period T associated with a given EEG rhythm defines a sequence of tqc:s having geometric span T/2 each: the rest of the period would be used to perceive the environment of the geometric now. The fractal hierarchy of EEGs would mean that there are tqcs within tqcs in a very wide range of time scales.

For details see the new chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: VI

In previous postings I have discussed how DNA topological quantum computation could be realized (see this, this, this , this, and this). A more detailed model for braid strands leads to the understanding of how high Tc super conductivity assigned with cell membrane (see this) could relate to tqc.

1. Are space-like braids A-braids or B-braids or hybrids of these?

If space-like braid strands are identified as idealized structures obtained from 3-D tube like structures by replacing them with 1-D strands, one can regard the braiding as a purely geometrical knotting of braid strands.

The simplest realization of the braid strand would be as a hollow cylindrical surface connecting conjugate DNA nucleotide to cell membrane and going through 5- and/or 6- cycles associated with the sugar backbone of conjugate DNA nucleotides. The free electron pairs associated with the aromatic cycles would carry the current creating the magnetic field needed.

There are two extreme options. For B-option magnetic field is parallel to the strand and vector potential rotates around it. For A-option vector potential is parallel to the strand and magnetic field rotates around it. The general case corresponds to the hybrid of these options and involves helical magnetic field, vector potential, and current.

  1. For B-option current flowing around the cylindrical tube in the transversal direction would generate the magnetic field. The splitting of the flux tube would require that magnetic flux vanishes requiring that the current should go to zero in the process. This would make possible selection of a part of DNA strand participating to tqc.

  2. For A-option the magnetic field lines of the braid would rotate around the cylinder. This kind of field is created by a current in the direction of cylinder. In the beginning of tqc the strand would split and the current of electron pairs would stop flowing and the magnetic field would disappear. Also now the initiation of computation would require stopping of the current and should be made selectively at DNA.

The general conclusion would be that the control of the tqc should rely on currents of electron pairs (perhaps Cooper pairs) associated with the braid strands.

Supra currents would have quantized values and are therefore very attractive candidates. The (supra) currents could also bind lipids to pairs so that they would define single dynamical unit in 2-D hydrodynamical flow. One can also think that Cooper pairs with electrons assignable to different members of lipid pair bind it to single dynamical unit.

2. Do supra currents generate the magnetic fields?

Energetic considerations favor the possibility that supra currents create the magnetic fields associated with the braid strands. Supra current would be created by a voltage pulse Δ V, which gives rise to a constant supra current after it has ceased. Supra current would be destroyed by a voltage pulse of opposite sign. Therefore voltage pulses could define an elegant fundamental control mechanism allowing to select the parts of genome participating to tqc. This kind of voltage pulse could be collectively initiated at cell membrane or at DNA. Note that constant voltage gives rise to an oscillating supra current.

Josephson current through the cell membrane would be also responsible for dark Josephson radiation determining that part of EEG which corresponds to the correlate of neuronal activity (see this). Note that TGD predicts a fractal hierarchy of EEGs and that ordinary EEG is only one level in this hierarchy. The pulse initiating or stopping tqc would correspond in EEG to a phase shift by a constant amount

Δ Φ= ZeΔ VT/hbar ,

where T is the duration of pulse and Δ V its magnitude.

The contribution of Josephson current to EEG responsible for beta and theta bands interpreted as satellites of alpha band should be absent during tqc and only EEG rhythm would be present. The periods dominated by EEG rhythm should be observed as EEG correlates for problem solving situations (say mouse in a maze) presumably involving tqc. The dominance of slow EEG rhythms during sleep and meditation would have interpretation in terms of tqc.

3. Topological considerations

The existence of supra current for A- or B-braid requires that the flow allows a complex phase exp(iΨ) such that supra current is proportional to grad Ψ. This requires integrability in the sense that one can assign to the flow lines of A or B (combination of them in the case of A-B braid) a coordinate variable Ψ varying along the flow lines. In the case of a general vector field X this requires grad Ψ= Φ X giving rot X= -grad Φ/Φ as an integrability condition. This condition defines what is known as Beltrami flow (see this).

The perturbation of the flux tube, which spoils integrability in a region covering the entire cross section of flux tube means either the loss of super-conductivity or the disappearance of the net supra current. In the case of the A-braid, the topological mechanism causing this is the increase the dimension of the CP2 projection of the flux tube so that it becomes 3-D (see this), where I have also considered the possibility that 3-D character of CP2 projection is what transforms the living matter to a spin glass type phase in which very complex self-organization patterns emerge. This would conform with the idea that in tqc takes place in this phase.

For details see the new chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: V

In previous postings I have discussed how DNA topological quantum computation could be realized (see this, this , this, and this). It is useful to try to imagine how gene expression might relate to the halting of tqc. There are of course myriads of alternatives for detailed realizations, and one can only play with thoughts to build a reasonable guess about what might happen.

1. Qubits for transcription factors and other regulators

Genetics is consistent with the hypothesis that genes correspond to those tqc moduli whose outputs determine whether genes are expressed or not. The naive first guess would be that the value of single qubit determines whether the gene is expressed or not. Next guess replaces " is " with " can be".

Indeed, gene expression involves promoters, enhancers and silencers (see this). Promoters are portions of the genome near genes and recognized by proteins known as transcription factors. Transcription factors bind to the promoter and recruit RNA polymerase, an enzyme that synthesizes RNA. In prokaryotes RNA polymerase itself acts as the transcription factor. For eukaryotes situation is more complex: at least seven transcription factors are involved with the recruitment of the RNA polymerase II catalyzing the transcription of the messenger RNA. There are also transcription factors for transcription factors and transcription factor for the transcription factor itself.

The implication is that several qubits must have value "Yes" for the actual expression to occur since several transcription factors are involved with the expression of the gene in general. In the simplest situation this would mean that the computation halts to a measurement of single qubit for subset of genes including at least those coding for transcription factors and other regulators of gene expression.

2. Intron-exon qubit

Genes would have very many final states since each nucleotide is expected to correspond to at least single qubit. Without further measurements that state of nucleotides would remain highly entangled for each gene. Also these other qubits are expected to become increasingly important during evolution.

For instance, eukaryotic gene expression involves a transcription of RNA and splicing out of pieces of RNA which are not translated to amino-acids (introns). Also the notion of gene is known to become increasingly dynamical during the evolution of eukaryotes so that the expressive power of genome increases. A single qubit associated with each codon telling whether it is spliced out or not would allow maximal flexibility. Tqc would define what genes are and the expressive power of genes would be due to the evolution of tqc programs: very much like in the case of ordinary computers. Stopping sign codon and starting codon would automatically tell where the gene begins and ends if the corresponding qubit is "Yes". In this picture the old fashioned static genes of prokaryotes without splicings would correspond to tqc programs for which the portions of genome with a given value of splicing qubit are connected.

3. What about braids between DNA, RNA, tRNA and aminoacids

This simplified picture might have created the impression that aminoacids are quantum outsiders obeying classical bio-chemistry. For instance, transcription factors would in this picture end up to the promoter by a random process and "Print" would only increase the density of the transcription factor. If DNA is able to perform tqc, it would however seem very strange if it would be happy with this rather dull realization of other central functions of the genetic apparatus.

One can indeed consider besides dark braids connecting DNA and its conjugate - crucial for the success of replication - also braids connecting DNA to mRNA and other forms of RNA, mRNA to tRNA, and tRNA to aminoacids. These braids would provide the topological realization of the genetic code and would increase dramatically the precision and effectiveness of the transcription and translation if these processes correspond to quantum transitions at the level of dark matter leading more or less deterministically to the desired outcome at the level of visible matter be it formation of DNA doublet strand, of DNA-mRNA association, of mRNA-tRNA association or tRNA-aminoacid association.

For instance, a temporary reduction of the value of Planck constant for these braids would contract these to such a small size that these associations would result with a high probability. The increase of Planck constant for braids could in turn induce the transfer of mRNA from the nucleus, the opening of DNA double strand during transcription and mitosis.

Also DNA-aminoacid braids might be possible in some special cases. The braiding between regions of DNA at which proteins bind could be a completely general phenomenon. In particular, the promoter region of gene could be connected by braids to the transcription factors of the gene and the halting of tqc computation to printing command could induce the reduction of Planck constant for these braids inducing the binding of the transcription factor binds to the promoter region. In a similar manner, the region of DNA at which RNA polymerase binds could be connected by braid strands to the RNA polymerase.

For details see the new chapter DNA as Topological Quantum Computer of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

One element field, quantum measurement theory and its q-variant, and the Galois fields associated with infinite primes

Kea mentioned John Baez's This Week's Finds 259, where John talked about one-element field - a notion inspired by the q=exp(i2π/n)→1 limit for quantum groups. This limit suggests that the notion of one-element field for which 0=1 - a kind of mathematical phantom for which multiplication and sum should be identical operations - could make sense. Physicist might not be attracted by this kind of identification.

In the following I want to articulate some comments from the point of view of quantum measurement theory and its generalization to q-measurement theory which I proposed for some years ago (see this).

I also consider and alternative interpretation in terms of Galois fields assignable to infinite primes which form an infinite hierarchy. These Galois fields have infinite number of elements but the map to the real world effectively reduces the number of elements to 2: 0 and 1 remain different.

1. q→ 1 limit as transition from quantum physics to effectively classical physics?

The q→limit of quantum groups at q-integers become ordinary integers and n-D vector spaces reduce to n-element sets. For quantum logic the reduction would mean that 2N-D spinor space becomes 2N-element set. N qubits are replaced with N bits. This brings in mind what happens in the transition from wave mechanism to classical mechanics. This might relate in interesting manner to quantum measurement theory.

Strictly speaking, q→1 limit corresponds to the limit q=exp(i2π/n), n→∞ since only roots of unity are considered. This also correspond to Jones inclusions at the limit when the discrete group Zn or or its extension-both subgroups of SO(3)- to contain reflection has infinite elements. Therefore this limit where field with one element appears might have concrete physical meaning. Does the system at this limit behave very classically?

In TGD framework this limit can correspond to either infinite or vanishing Planck constant depending on whether one consider orbifolds or coverings. For the vanishing Planck constant one should have classicality: at least naively! In perturbative gauge theory higher order corrections come as powers of g2/4πhbar so that also these corrections vanish and one has same predictions as given by classical field theory.

2. Q-measurement theory and q→ 1 limit.

Q-measurement theory differs from quantum measurement theory in that the coordinates of the state space, say spinor space, are non-commuting. Consider in the sequel q-spinors for simplicity.

Since the components of quantum spinor do not commute, one cannot perform state function reduction. One can however measure the modulus squared of both spinor components which indeed commute as operators and have interpretation as probabilities for spin up or down. They have a universal spectrum of eigen values. The interpretation would be in terms of fuzzy probabilities and finite measurement resolution but may be in different sense as in case of HFF:s. Probability would become the observable instead of spin for q not equal to 1.

At q→ 1 limit quantum measurement becomes possible in the standard sense of the word and one obtains spin down or up. This in turn means that the projective ray representing quantum states is replaced with one of n possible projective rays defining the points of n-element set. For HFF:s of type II1 it would be N-rays which would become points, N the included algebra. One might also say that state function reduction is forced by this mapping to single object at q→ 1 limit.

On might say that the set of orthogonal coordinate axis replaces the state space in quantum measurement. We do this replacement of space with coordinate axis all the time when at blackboard. Quantum consciousness theorist inside me adds that this means a creation of symbolic representations and that the function of quantum classical correspondences is to build symbolic representations for quantum reality at space-time level.

q→ 1 limit should have space-time correlates by quantum classical correspondence. A TGD inspired geometro-topological interpretation for the projection postulate might be that quantum measurement at q→1 limit corresponds to a leakage of 3-surface to a dark sector of imbedding space with q→ 1 (Planck constant near to 0 or ∞ depending on whether one has n→∞ covering or division of M4 or CP2 by a subgroup of SU(2) becoming infinite cyclic - very roughly!) and Hilbert space is indeed effectively replaced with n rays. For q not equal to 1 one would have only probabilities for different outcomes since things would be fuzzy.

In this picture classical physics and classical logic would be the physical counterpart for the shadow world of mathematics and would result only as an asymptotic notion.

3. Could 1-element fields actually correspond to Galois fields associated with infinite primes?

Finite field Gp corresponds to integers modulo p and product and sum are taken only modulo p. An alternative representation is in terms of phases exp(ik2π/p), k=0,...,p-1 with sum and product performed in the exponent. The question is whether could one define these fields also for infinite primes (see this) by identifying the elements of this field as phases exp(ik2π/Π) with k taken to be finite integer and Π an infinite prime (recall that they form infinite hierarchy). Formally this makes sense. 1-element field would be replaced with infinite hierarchy of Galois fields with infinite number of elements!

The probabilities defined by components of quantum spinor make sense only as real numbers and one can indeed map them to real numbers by interpreting q as an ordinary complex number. This would give same results as q→ 1 limit and one would have effectively 1-element field but actually a Galois field with infinite number of elements.

If one allows k to be also infinite integer but not larger than than Π in real sense, the phases exp(k2π/Π) would be well defined as real numbers and could differ from 1. All real numbers in the range [-1,1] would be obtained as values of cos(k2π/Π) so that this limit would effectively give real numbers.

This relates also interestingly to the question whether the notion of p-adic field makes sense for infinite primes. The p-adic norm of any infinite-p p-adic number would be power of π either infinite, zero, or 1. Excluding infinite normed numbers one would have effectively only p-adic integers in the range 1,...Π-1 and thus only the Galois field GΠ representable also as quantum phases.

I conclude with a nice string of text from John'z page:

What's a mathematical phantom? According to Wraith, it's an object that doesn't exist within a given mathematical framework, but nonetheless "obtrudes its effects so convincingly that one is forced to concede a broader notion of existence".

and unashamedely propose that perhaps Galois fields associated with infinite primes might provide this broader notion of existence! In equally unashamed tone I ask whether there exists also hierarchy of conscious entities at q=1 levels in real sense and whether we might identify ourselves as this kind of entities? Note that if cognition corresponds to p-adic space-time sheets, our cognitive bodies have literally infinite geometric size in real sense.

For details see the chapter Was von Neumann Right After All?.

DNA as topological quantum computer: IV

In order to have a more concrete view about realization of topological quantum computation (see the previous posting and links from it), one must understand how quantum computation can be reduced to a construction of braidings from fundamental unitary operations. The article Braiding Operators are Universal Quantum Gates by Kaufmann and Lomonaco contains a very lucid summary of how braids can be used in topological quantum computation.

  1. The identification of the braiding operator R - a unitary solution of Yang-Baxter equation - as a universal 2-gate is discussed. In the following I sum up only those points which are most relevant for the recent discussion.

  2. One can assign to braids both knots and links and the assignment is not unique without additional conditions. The so called braid closure assigns a unique knot to a given braid by connecting nth incoming strand to nth outgoing strand without generating additional knotting. All braids related by so called Markov moves yield the same knot. The Markov trace (q-trace actually) of the unitary braiding S-matrix U is a knot invariant characterizing the braid closure.

  3. Braid closure can be mimicked by a topological quantum computation for the original n-braid plus trivial n-braid and this leads to a quantum computation of the modulus of the Markov trace of U. The probability for the diagonal transition for one particular element of Bell basis (whose states are maximally entangled) gives the modulus squared of the trace. The closure can be mimicked quantum computationally.

1. Universality of tqc

Quantum computer is universal if all unitary transformations of nth tensor power of a finite-dimensional state space V can be realized. Universality is achieved by using only two kinds of gates. The gates of first type are single particle gates realizing arbitrary unitary transformation of U(2) in case of qubits. Only single 2-particle gate is necessary and universality is guaranteed if the corresponding unitary transformation is entangling for some state pair. The standard choice for the 2-gate is CNOT acting on bit pair (t,c). The value of the control bit c remains of course unchanged and the value of the target bit changes for c=1 and remains unchanged for c=0.

2. The fundamental braiding operation as a universal 2-gate

The realization of CNOT or gate equivalent to it is the key problem in topological quantum computation. For instance, the slow de-coherence of photons makes quantum optics a promising approach but the realization of CNOT requires strongly nonlinear optics. The interaction of control and target photon should be such that for second polarization of the control photon target photon changes its direction but keeps it for the second polarization direction.

For braids CNOT can be be expressed in terms of the fundamental braiding operation en representing the exchange of the strands n and n+1 of the braid represented as a unitary matrix R acting on Vn\otimes Vn+1.

The basic condition on R is Yang-Baxter equation expressing the defining condition enen+1en= en+1enen+1 for braid group generators. The solutions of Yang-Baxter equation for spinors are well-known and CNOT can be expressed in the general case as a transformation of form A1\otimes A2 R A3\otimes A4 in which single particle operators Ai act on incoming and outgoing lines. 3-braid is the simplest possible braid able to perform interesting tqc, which suggests that genetic codons are associated with 3-braids.

The dance of lipids on chessboard defined by the lipid layer would reduce R to an exchange of neighboring lipids. For instance, the matrix R= DS, D =diag(1,1,1,-1) and S=e11+e23+e32+e44 the swap matrix permuting the neighboring spins satisfies Yang-Baxter equation and is entangling.

3. What the replacement of linear braid with planar braid could mean?

Standard braids are essentially linear objects in plane. The possibility to perform the basic braiding operation for the nearest neighbors in two different directions must affect the situation somehow.

  1. Classically it would seem that the tensor product defined by a linear array must be replaced by a tensor product defined by the lattice defined by lipids. Braid strands would be labelled by two indices and the relations for braid group would be affected in an obvious manner.

  2. The fact that DNA is a linear structure would suggests that the situation is actually effectively one-dimensional, and that the points of the lipid layer inherit the linear ordering of nucleotides of DNA strand. One can however ask whether the genuine 2-dimensionality could provide a mathematical realization for possible long range correlations between distant nucleotides n and n+N for some N. p-Adic effective topology for DNA might become manifest via this kind of correlations and would predict that N is power of some prime p which might depend on organism's evolutionary level.

  3. Quantum conformal invariance would suggest effective one-dimensionality in the sense that only the observables associated with a suitably chosen linear braid commute. One might also speak about topological quantum computation in a direction transversal to the braid strands giving a slicing of the cell membrane to parallel braid strands. This might mean an additional computational power.

  4. Partonic picture would suggest a generalization of the linear braid to a structure consisting of curves defining the decomposition of membrane surface regions such that conformal invariance applies separately in each region: this would mean breaking of conformal invariance and 2-dimensionality in discrete sense. Each region would define a one parameter set of topological quantum computations. These regions might corresponds to genes. If each lipid defines its own conformal patch one would have a planar braid.

4. Single particle gates

The realization of single particle gates as U(2) transformations leads naturally to the extension of the braid group by assigning to the strands sequences of group elements satisfying the group multiplication rules. The group elements associated with a nth strand commute with the generators of braid group which do not act on nth strand. G would be naturally subgroup of the covering group of rotation group acting in spin degrees of spin 1/2 object. Since U(1) transformations generate only an overall phase to the state, one the presence of this factor might not be necessary. A possible candidate for U(1) factor is as a rotation induced by a time-like parallel translation defined by the electromagnetic scalar potential Φ=At.

The natural realization for single particle gate s subset SU(2) would be as SU(2) rotation induced by a magnetic pulse. This transformation is fixed by the rotation axis and rotation angle around this axes. This kind of transformation would result by applying to the strand a magnetic pulse with magnetic field in the direction of rotation axes. The duration of the pulse determines the rotation angle. Pulse could be created by bringing a magnetic flux tube to the system, letting it act for the required time, and moving it away. U(1) phase factor could result from the electromagnetic gauge potential as a non-integrable phase factor exp(ie∫ Atdt/hbar) coming from the presence of scale potential Φ=At in the Hamiltonian.

What could then be the simplest realization of the U(2) transformation in the case of cell membrane?

  1. There should be a dark spin 1/2 particle associated with each lipid, electron or proton most plausibly. A more complex realization would use J=2 Cooper pairs of electrons.

  2. One should a apply the magnetic pulse on the braid strands ending at the lipid layer. The model for the communication of sensory data to the magnetic body requires that magnetic flux tubes go through the cell membrane. This would suggest that the direction of the magnetic flux tube is temporarily altered and that the flux tube then covers part of the lipid for the required period of time.

    The realization of the single particle gates requires electromagnetic interactions. That single particle gates are not purely topological transformations could bring in the problems caused by a de-coherence due to electromagnetic perturbations. The large values of Planck constant playing a key role in the TGD based model of living matter could save the situation. The large value of hbar would be also required by the anyonic character of the system necessary to obtain R-matrix defining a universal 2-gate.

For details see the new chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: III

I have discussed various ideas about topological quantum computation in two previous postings. In DNA as a topological quantum computer I discussed general ideas, and made a general suggestion about how DNA might act as a topological quantum computer. In Some ideas about topological quantum computation in TGD Universe I continued with futher general ideas about braiding and its relation to tqc.

Braids code for topological quantum computation. One can imagine many possible identifications of braids but this is not essential for what I am going to say below.

  1. What is highly non-trivial that the motion of the ends of strands defines both time-like and space-like braidings with latter defining in a well-defined sense a written version of the tqc program, kind of log file. The manipulation of braids is a central element of tqc and if DNA really performs tqc, the biological unit modifying braidings should be easy to identify. An obvious signature is the 2-dimensional character of this unit and the alert and informed reader might be able to guess the rest.

  2. One can also wonder exactly what part of DNA performs tqc and alert and informed reader might have answer also to this question. In the following I propose an improved view about tqc performed by DNA inspired by these guesses.

1. Sharing of labor

The braid strands must begin from DNA double strands. Precisely which part of DNA does perform tqc? Genes? Introns? Or could it be conjugate DNA which performs tqc? The function of conjugate DNA has indeed remain mystery and sharing of labor suggests itself.

Conjugate DNA would do tqc and DNA would "print" the outcome of tqc in terms of RNA yielding aminoacids in case of exons. RNA could result in the case of introns. The experience about computers and the general vision provided by TGD suggests that introns could express the outcome of tqc also electromagnetically in terms of standardized field patterns. Also speech would be a form of gene expression. The quantum states braid would entangle with characteristic gene expressions. This hypothesis will be taken as starting point in the following considerations.

2. Cell membranes as modifiers of braidings defining tqc programs?

What part of the cell or nucleus is specialized to perform braiding operations? The first guess was that nucleotides of the intronic part of DNA are permuted without any change in the sequence: the argument was that if introns do not express themselves chemically this activity does not perturb tqc. At the second thinking this does not look a good idea at all. First of all, introns are transcribed but then spliced out from the transcript. Secondly, they are now known to express themselves by producing RNA having some function as I had myself explained earlier (and forgotten it!). Something much more elegant is required. Two days ago I started to reconsider the problem and ended up with a nice little argument allowing to understand why cell membrane is necessary and why it is liquid crystal.

The manipulation of braid strands transversal to DNA must take place at 2-D surface. The ends of the space-like braid are dancers whose dancing pattern defines the time-like braid, the running of classical tqc program. Space-like braid represents memory storage and tqc program is automatically written to memory during the tqc. The inner membrane of the nuclear envelope and cell membrane with entire endoplasmic reticulum included are good candidates for dancing halls. The 2-surfaces containing the ends of the hydrophobic ends of lipids could be the parquets and lipids the dancers. This picture seems to make sense.

  1. Consider first the anatomy of membranes. Cell membrane and membranes of nuclear envelope consist of 2 lipid layers whose hydrophobic ends point towards interior. There is no water here nor any direct perturbations from the environment or interior milieu of cell. Nuclear envelope consists of two membranes having between them an empty volume of thickness 20-40 nm. The inner membrane consists of two lipid layers like ordinary cell membrane and outer membrane is connected continuously to endoplasmic reticulum which forms a part of highly folded cell membrane. Many biologists believe that cell nucleus is a prokaryote, which began to live in symbiosis with prokaryote defining the cell membrane.

  2. What makes dancing possible is that the phospholipid layers of the cell membrane are liquid crystals: the lipids can move freely in the horizontal direction but not vertically. "Phospho" could relates closely to the metabolic energy needs of dancers. If these lipids are self-organized around braid strands, their dancing patterns along the membrane surface would be an ideal manner to modify braidings since the lipids would have standard positions in a lattice. This would be like dancing on a chessboard. As a matter fact, living matter is full of self-organizing liquid crystals and one can wonder whether the deeper purpose of their life be running and simultaneous documentation of tqc programs?

  3. Ordinary computers have operating system: a collection of standard programs -the system - and similar situation should prevail now. The "printing" of outputs of tqc would correspond to this kind of standard program. This tqc program should not receive any input from the environment of the nucleus and should therefore correspond to braid strands connecting conjugate strand with strand. Braid strands would go only through the inner nuclear membrane and return back and would not be affected much since the volume between inner and outer nuclear membranes is empty. This assumption looks ad hoc but it will be found that the requirement that these programs are inherited as such in the cell replication necessitates this kind of structure.

  4. The braid strands starting from the conjugate DNA could traverse several time through the highly folded endoplasmic reticulum but without leaving cell interior and return back to nucleus and modify tqc by intracellular input. Braid strands could also traverse the cell membrane and thus receive information about the exterior of cell. Both of these tqc programs could be present also in monocellulars (prokaryotes) but the braid strands would always return back to the nucleus. In multicellulars (eukaryotes) braid strands could continue to another cell and give rise to "social" tqc programs performed by the multicellular organisms. Note that the topological character of braiding does not require isolation of braiding from environment. It might be however advantageous to have some kind of sensory receptors amplifying sensory input to standardized re-braiding patterns. Various receptors in cell membrane would serve this purpose.

  5. If braid strands carry 4-color (A,T,C,G) then also lipid strands should carry this kind of 4-color. The lipids whose hydrophobic ends can be joined to form longer strand should have same color. This color need not be chemical in TGD Universe.

  6. Braid strands can end up at the parquet defined by ends of the inner phospholipid layer: their distance of inner and outer parquet is few nanometers. They could also extend further.

    1. If one is interested in connecting cell nucleus to the membrane of another cells, the simpler option is formation of hole defined by a protein attached to cell membrane. In this case only the environment of second cell affects the braiding assignable to the first cell nucleus.
    2. The bi-layered structure of the cell membrane could be essential for the build-up of more complex tqc programs since the strands arriving at two nearby hydrophobic 2-surfaces could combine to form longer strands. The formation of longer strands could mean the fusion of the two nearby hydrophobic two-surfaces in the region considered. This would allow to connect cell nucleus and cell membrane to a larger tqc unit and cells to multicellular tqc units so that the modification of tqc programs by feeding the information from the exteriors of cells - essential for the survival of multicellulars - would become possible. It would be essential that only braid strands of same color are connected in this process and splitting of strands and their reconnection defines a manner to change braidings.

4. Quantitative test for the proposal

There is a simple quantitative test for the proposal. A hierarchy of tqc programs is predicted, which means that the number of lipids in the nuclear inner membrane should be larger or at least of same order of magnitude that the number of nucleotides. For definiteness take the radius of the lipid molecule to be about 5 Angstroms (probably somewhat too large) and the radius of the nuclear membrane about 2.5 μm.

For our own species the total length of DNA strand is about one meter and there are 30 nucleotides per 10 nm. This gives 6.3×107 nucleotides: the number of intronic nucleotides is only by few per cent smaller. The total number of lipids in the nuclear inner membrane is roughly 108. The number of lipids is roughly twice the number nucleotides. The number of lipids in the membrane of a large neuron of radius of order 10-4 meters is about 1011. The fact that the cell membrane is highly convoluted increases the number of lipids available. Folding would make possible to combine several modules in sequence by the proposed connections between hydrophobic surfaces.

5. Cell replication and tqc

One can look what happens in the cell replication in the hope of developing more concrete ideas about tqc in multicellular system. This process must mean a replication of the braid's strand system and a model for this process gives concrete ideas about how multicellular system performs tqc.

  1. During mitosis chromosomes are replicated. During this process the strands connecting chromosomes become visible: the pattern brings in mind flux tubes of magnetic field. For prokaryotes the replication of chromosomes is followed by the fission of the cell membrane. Also plant nuclei separated by cellulose walls suffer fission after the replication of chromosomes. For animals nuclear membranes break down before the replication suggesting that nuclear tqc programs are reset and newly formed nuclei start tqc from a clean table. For eukaryotes cell division is controlled by centrosomes. The presence of centrosomes is not necessary for the survival of the cell or its replication but is necessary for the survival of multicellular. This conforms with the proposed picture.

  2. If the conjugate strands are specialized in tqc, the formation of new double strands does not involve braids in an essential manner. The formation of conjugate strand should lead to also to a generation of braid strands unless they already exist as strands connecting DNA and its conjugate and are responsible for "printing". These strands need not be short. The braiding associated with printing would be hardware program which could be genetically determined or at least inherited as such so that the strands should be restricted inside the inner cell membrane or at most traverse the inner nuclear membrane and turn back in the volume between inner membrane and endoplasmic reticulum.

    The return would be most naturally from the opposite side of nuclear membrane which suggest a breaking of rotational symmetry to axial symmetry. The presence of centriole implies this kind of symmetry breaking: in neurons this breaking becomes especially obvious. The outgoing braid strands would be analogous to axon and returning braid strands to dendrites. Inner nuclear membrane would decompose the braiding to three parts: one for strand, second for conjugate strand, and a part in the empty space inside nuclear envelope.

  3. The formation of new DNA strands requires recognition relying on "strand color" telling which nucleotide can condense at it. The process would conserve the braidings connecting DNA to the external world. The braidings associated with the daughter nuclei would be generated from the braiding between DNA and its conjugate. As printing software they should be identical so that the braiding connecting DNA double strands should be a product of a braiding and its inverse. This would however mean that the braiding is trivial. The division of the braid to three parts hinders the transformation to a trivial braid if the braids combine to form longer braids only during the "printing" activity.

  4. The new conjugate strands are formed from the old strands associated with printing. In the case of plants the nuclear envelope does not disintegrate and splits only after the replication of chromosomes. This would suggest that plant cells separated by cell walls perform only intracellular tqc. Hermits do not need social skills. In the case of animals nuclear envelope disintegrates. This is as it must be since the process splits the braids connecting strand and conjugate strands so that they can connect to the cell membrane. The printing braids are inherited as such which conforms with the interpretation as a fixed software.

  5. The braids connecting mother and daughter cells to extranuclear world would be different and tqc braidings would give to the cell a memory about its life-cycle. The age ordering of cells would have the architecture of a tree defined by the sequence of cell replications and the life history of the organism. The 4-D body would contain kind of log file about tqc performed during life time: kind of fundamental body memory.

  6. Quite generally, the evolution of tqc programs means giving up the dogma of genetic determinism. The evolution of tqc programs during life cycle and the fact that half of them is inherited means kind of quantum Lamarckism. This inherited wisdom at DNA level might partly explain why we differ so dramatically from our cousins.

6. Sexual reproduction and tqc

  1. Sexual reproduction of eukaryotes relies on haploid cells differing from diploid cells in that chromatids do not possess sister chromatids. Whereas mitosis produces from single diploid cell two diploid cells, meiosis gives rise to 4 haploid cells. The first stage is very much like mitosis. DNA and chromosomes duplicate but cell remains a diploid in the sense that there is only single centrosome: in mitosis also centrosome duplicates. After this the cell membrane divides into two. At the next step the chromosomes in daughter cells split into two sister chromosomes each going into its own cell. The outcome is four haploid cells.

  2. The presence of only single chromatid in haploids means that germ cells would perform only one half of the "social" tqc performed by soma cells who must spend their life cycle as a member of cell community. In some cells the tqc would be performed by chromatids of both father and mother making perhaps possible kind of stereo view about world and a model for couple - the simplest possible social structure.

  3. This brings in mind the sensory rivalry between left and right brain: could it be that the two tqc:s give competing computational views about world and how to act in it? We would have inside us our parents and their experiences as a pair of chromatids representing chemical chimeras of chromatid pairs possessed by the parents: as a hardware - one might say. Our parents would have the same mixture in software via sharing and fusion of chromatid mental images or via quantum computational rivalry. What is in software becomes hardware in the next generation.

  4. The ability of sexual reproduction to generate something new relates to meiosis. During meiosis genetic recombination occurs via chromosomal crossover which in string model picture would mean splitting of chromatids and the recombination of pieces in a new manner (A1+B1)+(A2+B2) → (A1+B2)+(A2+B1) takes place in crossover and (A1+B1+C1)+(A2+B2+C2) → (A1+B2+C1)+(A2+B1+C2) in double crossover. New hardware for tqc would result by combining pieces of existing hardware. What this means in terms of braids should be clarified.

  5. Fertilization is in well-define sense the inverse of meiosis. In fertilization the chromatids of spermatozoa and ova combine to form the chromatids of diploid cell. The recombination of genetic programs during meiosis becomes visible in the resulting tqc programs.

7. What is the role of centrosomes and basal bodies?

Centrosomes and basal bodies form the main part of Microtubule Organizing Center. They are somewhat mysterious objects and at first do not seem to fit to the proposed picture in an obvious manner.

  1. Centrosomes consist two centrioles forming a T shaped antenna like structure in the center of cell. Also basal bodies consist of two centrioles but are associated with the cell membrane. Centrioles and basal bodies have cylindrical geometry consisting of nine triplets of microtubules along the wall of cylinder. Centrosome is associated with nuclear membrane during mitosis.

  2. The function of basal bodies which have evolved from centrosomes seems to be the motor control (both cilia and flagella) and sensory perception (cilia). Cell uses flagella and cilia to move and perceive. Flagella and cilia are cylindrical structures associated with the basal bodies. The core of both structures is axoneme having 9×2+2 microtubular structure. So called primary cilia do not posses the central doublet and the possible interpretation is that the inner doublet is involved with the motor control of cilia. Microtubules of the pairs are partially fused together.

  3. Centrosomes are involved with the control of mitosis. Mitosis can take place also without them but the organism consisting of this kind of cells does not survive. Hence the presence of centrosomes might control the proper formation of tqc programs. The polymerization of microtubules is nucleated at microtubule self-organizing center which can be centriole or basal body. One can say that microtubules which are highly dynamical structures whose length is changing all the time have their second end anchored to the self-organizing center. Since this function is essential during mitosis it is natural that centrosome controls it.

  4. The key to the understanding of the role of centrosomes and basal bodies comes from a paradox. DNA and corresponding tqc programs cannot be active during mitosis. What does then control mitosis?

    1. Perhaps centrosome and corresponding tqc program represents the analog of the minimum seed program in computer allowing to generate an operating system like Windows 2000 (the files from CD containing operating system must be read!). The braid strands going through the microtubuli of centrosome might define the corresponding tqc program. The isolation from environment by the microtubular surface might be essential for keeping the braidings defining these programs strictly unchanged.
    2. The RNA defining the genome of centrosome (yes: centrosome has its own genome defined by RNA rather than DNA!) would define the hardware for this tqc. The basal bodies could be interpreted as a minimal sensory-motor system needed during mitosis.
    3. As a matter fact, centrosome and basal bodies could be seen as very important remnants of RNA era believed by many biologists to have preceded DNA era. This assumption is also made in TGD inspired model of prebiotic evolution.
    4. Also other cellular organelles possessing own DNA and own tqc could remain partly functional during mitosis. In particular, mitochondria are necessary for satisfying energy needs during the period when DNA is unable to control the situation so that they must have some minimum amount of own genome.

  5. Neurons do not possess centrosome which explains why they cannot replicate. The centrioles are replaced with long microtubules associated with axons and dendrites. The system consisting of microtubules corresponds to a sensory-motor system controlled by the tqc programs having as a hardware the RNA of centrosomes and basal bodies. Also this system would have a multicellular part.

  6. Intermediate filaments, actin filaments, and microtubules are the basic building elements of the eukaryotic cytoskeleton. Microtubules, which are hollow cylinders with outer radius of 24 nm, are especially attractive candidates for structures carrying bundles of braid strands inside them. The microtubular outer-surfaces could be involved with signalling besides other well-established functions. It would seem that microtubules cannot be assigned with tqc associated with nuclear DNA but with RNA of centrosomes and could contain corresponding braid strand bundles. It is easy to make a rough estimate for the number of strands and this would give an estimate for the amount of RNA associated with centrosomes. Also intermediate filaments and actin filaments might relate to cellular organelles having their own DNA.

For details see the new chapter DNA as Topological Quantum Computer.

DNA as topological quantum computer: II

I have been trying to develop general ideas about topological computation in terms of braidings. There are many kinds of braidings. Number theoretic braids are defined by the orbits of minima of vacuum expectation of Higgs at lightlike partonic 3-surfaces (and also at space-like 3-surfaces). There are braidings defined by Kähler gauge potential (possibly equivalent with number theoretic ones) and by Kähler magnetic field. Magnetic flux tubes and partonic 2-surfaces interpreted as strands of define braidings whose strands are not infinitely thin. A very concrete and very complex time-like braiding is defined by the motions of people at the surface of globe: perhaps this sometimes purposeless-looking fuss has a deeper purpose: maybe those at the higher levels of dark matter hierarchy are using us to carry out complex topological quantum computations;-)!

1. General vision about quantum computation

The hierarchy of Planck constants would give excellent hopes of quantum computation in TGD Universe. The general vision about quantum computation (tqc would result as special case) would look like follows.

  1. Time-like entanglement between positive and negative energy parts of zero energy states would define the analogs of qc-programs. Space-like quantum entanglement between ends of strands whose motion defines time-like braids would provide a representation of q-information.

  2. Both time- and space-like quantum entanglement would correspond to Connes tensor product expressing the finiteness of the measurement resolution between the states defined at ends of space-like braids whose orbits define time like braiding. The characterization of the measurement resolution would thus define both possible q-data and tq-programs as representations for "laws of physics".

  3. I have discussed here a possible vision of how DNA could act as topological quantum computer. The braiding between DNA strands with each nucleotide defining one strand transversal to DNA realized in terms of magnetic flux tubes is my bet for the representation of space-like braiding in living matter. The conjectured hierarchy of genomes giving rise to quantum coherent gene expressions in various scales would correspond to computational hierarchy.

2. About the relation between space-like and time-like number theoretic braidings

The relationship between space- and time-like braidings is interesting and there might be some connections also to 4-D topological gauge theories suggested by geometric Langlands program discussed in the previous posting.

  1. The braidings along light-like surfaces modify space-like braiding if the moving ends of the space-like braids at partonic 3-surfaces define time-like braids. From tqc point of view the interpretation would be that tqc program is written to memory represented as the modification of space-like braiding in 1-1 correspondence with the time-like braiding.

  2. The orbits of space-like braids define codimension two sub-manifolds of 4-D space-time surface and can become knotted. Presumably time-like braiding gives rise to a non-trivial "2-braid". Could it be that also "2-braiding" based on this knotting be of importance? Do 2-connections of n-category theorists emerge somehow as auxiliary tools? Could 2-knotting bring additional structure into the topological QFT defined by 1-braidings and Chern-Simons action?

  3. The strands of dynamically evolving braids could in principle go through each other so that time evolution can transform braid to a new one also in this manner. This is especially clear from standard representation of knots by their planar projections. The points where intersection occurs correspond to self-intersection points of 2-surface as a sub-manifold of space-time surface. Topological QFT:s are also used to classify intersection numbers of 2-dimensional surfaces understood as homological equivalence classes. Now these intersection point would be associated with "braid cobordism".

3. Quantum computation as quantum superposition of classical computations?

It is often said that quantum computation is quantum super-position of classical computations. In standard path integral picture this does not make sense since between initial and final states represented by classical fields one has quantum superposition over all classical field configurations representing classical computations in very abstract sense. The metaphor is as good as the perturbation theory around the minimum of the classical action is as an approximation.

In TGD framework the classical space-time surface is a preferred extremal of Kähler action so that apart from effects caused by the failure of complete determism, the metaphor makes sense precisely. Besides this there is of course the computation associated with the spin like degrees of freedom in which one has entanglement and which one cannot describe in this manner.

For tqc a particular classical computation would reduce to the time evolution of braids and would be coded by 2-knot. Classical computation would be coded to the manipulation of the braid. Note that the branching of strands of generalized number theoretical braids has interpretation as classical communication.

4. The identification of topological quantum states

Quantum states of tqc should correspond to topologically robust degrees of freedom separating neatly from non-topological ones.

  1. The generalization of the imbedding space inspired by the hierarchy of Planck constants suggests an identification of this kind of states as elements of the group algebra of discrete subgroup of SO(3) associated with the group defining covering of M4 or CP2 or both in large hbar sector. One would have wave functions in the discrete space defined by the homotopy group of the covering transforming according to the representations of the group. This is by definition something robust and separated from non-topological degrees of freedom (standard model quantum numbers). There would be also a direct connection with anyons.

  2. An especially interesting group is dodecahedral group corresponding to the minimal quantum phase q=exp(2π/5) (Golden Mean) allowing a universal topological quantum computation: this group corresponds to Dynkin diagram for E8 by the ALE correspondence.

5. Some questions

A conjecture inspired by the inclusions of HFFs is that these states can be also regarded as representations of various gauge groups which TGD dynamics is conjectured to be able to mimic so that one might have connection with non-Abelian Chern-Simons theories where topological S-matrix is constructed in terms of path integral over connections: these connections would be only an auxiliary tool in TGD framework.

  1. Do these additional degrees of freedom give only rise to topological variants of gauge- and conformal field theories? Note that if the earlier conjecture that entire dynamics of these theories could be mimicked, it would be best to perform tqc at quantum criticality where either M4 or CP2 dynamical degrees of freedom or both disappear.

  2. Could it be advantageous to perform tqc near quantum criticality? For instance, could one construct magnetic braidings in the visible sector near q-criticality using existing technology and then induce phase transition changing Planck constant by varying some parameter, say temperature.

For details see the chapter DNA as Topological Quantum Computer.

DNA as a topological quantum computer: I

For years ago I developed a model of topological quantum computation combining TGD based view about space-time with basic ideas about topological quantum computation and ended up with the proposal that DNA might act as a topological quantum computer.

The first guess (see this) was that parallel DNA or RNA strands could form braids. The problem is that the number of braid strands is limited and the computations are restricted within single cell nucleus. The need to establish the hardware for each computation separately can be also seen as a restriction.

One can imagine also other manners in which DNA or RNA could act as a topological quantum computer and it good to try to state clearly what one wants.

  1. Natural requirements are that the topological quantum computer programs can be naturally combined to larger programs and evolution means this kind of process; that the programs have a natural modular structure inherited from the previous stages of evolution; and that the computation is not restricted inside single nucleus.

  2. DNA and/or RNA defines the hardware of topological computation and at least for more advanced topological quantum computers this hardware should be static so that only programs would be dynamical. This leaves only DNA in consideration and the entangled initial and quantum states at the ends of braids quantum states would be assignable to static DNA structures.

  3. The program would be determined by different braidings connecting the states of DNA in time direction or in spatial direction. Since the genomes are identical in different nuclei, the strands could connect different nuclei or conjugate strands of double DNA strand.

1. The recent progress in quantum TGD and TGD inspired quantum biology

After the advent of the first model for topological quantum computation in TGD Universe (see this), the mathematical and physical understanding of TGD has developed dramatically and the earlier quite speculative picture can be replaced with a framework which leads to a rather unique view about topological quantum computations by DNA.

1.1 Universe as a topological quantum computer

One can say that the recent formulation of quantum TGD states that the entire Universe behaves like a topological quantum computer. This notion of topological quantum computer differs however from the standard one in many respects.

  1. The emergence of hierarchy of Planck constants realized as a generalization of the notion of imbedding space is now a basic piece of TGD allowing an elegant formulation of quantum TGD (see this and this). The phases of matter with large Planck constant are interpreted as dark matter. Large values of Planck constant make possible topological quantum computations in arbitrary long time scales so that the most fundamental objection against quantum computation can be circumvented.

  2. Zero energy ontology forces to unify S-matrix and density matrix to M-matrix - the product of the square root of density matrix and S-matrix- defined as time-like (or rather light-like) entanglement coefficients between positive and negative energy parts of zero energy state (see this and this). Connes tensor product emerging naturally from the notion of finite measurement resolution described in terms of inclusions of hyperfinite factors of type II1 defines highly uniquely the M-matrix. M-matrix would be natural candidate for defining topological quantum computation in light-like direction. Connes tensor product makes sense also in space-like direction and would define quantum storage of functions represented as entanglement coefficients.

  3. The notion of number theoretic braid (see this and this) is now well-understood and has become a basic element of the formulation of quantum TGD based on the requirement of number theoretical universality. As a matter fact, the notion of braid is generalized in the sense that braid strands can fuse and decay. The physical interpretation is as motion of minima of the generalization eigenvalue of the modified Dirac operator which is function of transversal coordinates of light-like partonic 3-surface and has interpretation as vacuum expectation of Higgs field. Fusion of braid strands corresponds to fusion of minima.

    For generalized Feynman diagrams partonic light-like 3-surfaces meet at 2-dimensional vertices defined by partonic 2-surfaces (see this). This implies that braids replicate at vertices: the interpretation is as a copying of classical information. Quantum information is not copied faithfully. The exchange of partonic 2-surfaces in turn corresponds to quantum communications. Hence quantum communication and quantum copying emerge naturally as additional elements. Space-like Connes tensor product in turn defines quantum memory storage.

  4. Computation time is a fundamental restriction in both ordinary and quantum computation. Zero energy ontology makes possible communications in both directions of geometric time, which suggests the possibility of geometric time loops in topological quantum computations. Could this mean that computation time ceases to be a restriction and ordinary computations lasting for infinite amount of geometric time could be performed in a finite time interval of observer's time? This is perhaps too much to hope. The subjective time taken by the computation would be infinite if each step in the iteration corresponds to single quantum jump. If this is the case and if each quantum jump of observer corresponds to a finite increment of geometric time perceived by the observer, time loops would not allow miracles.

1.2 The notion of magnetic body and the generalization of the notion of genome

The evolution of ideas related to quantum biology provides also new valuable insights. In particular, the notion of magnetic body leads to a model of living system in which dark matter at magnetic flux quanta of the field body of biological system uses biological body as a motor instrument and sensory receptor (see this). Quantum control would be naturally via the genome and sensory input would be from cell membrane containing all kinds of receptors. This would suggest that magnetic flux sheets traverse through DNA strands and cell membranes.

The quantization of magnetic flux with unit defined by Planck constant having arbitrarily large values leads naturally to the notions of super-genome and hyper-genome (see this). Super-genome would consists of DNA strands of separate nuclei belonging to single magnetic flux sheet and these sequences of genomes would be like lines of text at the page of book. Super-genomes in turn can combine to form text lines at the pages of a bigger book, I have used the term hyper-genome. This hierarchy of genomes would give rise to a collective gene expression at the level of organs, individuals of a species, and at the collective level consisting of populations containing several species. Even biosphere could express itself coherently via all the genomes of the bio-sphere. The model of topological quantum computation performed by DNA should be consistent with this general picture.

2. Model for DNA based topological quantum computation

The most promising model of DNA as topological quantum computer relies on the hierarchy of genomes. The flux sheets or collections of parallel flux tubes assignable to a magnetic body would traverse the DNA strands of several nuclei so that strands would be analogous to lines of text on the page of a book.

DNA strands would define the intersections of magnetic or number theoretic braids with plane and braiding would be associated with with the magnetic field lines or flux tubes transversal to DNA. The M-matrix defining topological quantum computation would act on quantum states assignable to nucleotides.

2.1 The interpretation of nucleotides

The interpretation of the A,T,C,G degree of freedom is not obvious and one can consider several options.

1) The quantum numbers entangled by braids having nothing to do with (A,T,C,G) assignable to nucleotides and the braiding does not affect nucleotides.

2) The nucleotides (A,T,C,G) correspond to four different colors (a,t,c,g) for braid strands with conjugate nucleotides defining conjugate colors. The subgroup of allowed braidings would preserve the color patterns. The minimal assumption is that braid strands connect only identical nucleotides. A stronger - probably unrealistic - assumption is that braiding permutes nucleotides physically.

3) The entangled quantum numbers are in 1-1 correspondence with states A, T, C, G of nucleotide. In zero energy ontology this would be possible without breaking of fundamental conservation laws. One can even consider the possibility that A,T,C,G are these quantum numbers. Topological quantum computation in time direction would thus make it possible to replace the DNA strands with new ones and provide a purely quantal mechanism of genetic evolution. Only introns could be involved with topological quantum computations in this sense since they would not induce mutations visible at the level of amino-acids. The intronic portions of genome would not be evolutionary invariants: whether or not this is the case should be easily testable.

4) The combination of options 2) and 3) might make sense for topological quantum computations in time like direction. One would have superposition of topological quantum computations associated with various color patterns and the halting of the computation would mean in general the occurrence of a mutation.

The option 2) with braid strands connecting only identical nucleotides is rather attractive since it explains several facts about genome (as do also options 3) and 4)).

  1. The model requires that the genomes in different nuclei must be identical: otherwise it is not possible to realize braidings as symmetry transformations mapping portions of DNA to itself (as noticed, this map need not occur at the chemical level). An interesting question is whether also the permutations of nucleotides of different codons are allowed or whether only codons are permuted so that they would define fundamental sub-programs.

  2. One can understand why the minimum number of nucleotides in a codon is three. The point is that braid group is non-commutative only when the number of strands is larger than 2. The braidings acting as symmetries would correspond to a subgroup of ordinary braidings leaving the color pattern of braid invariant. Obviously the group is generated by some minimal number of combinations of ordinary braid generators. For instance, for two braid strands with different colors the generator is e12 rather than e1 (two exchange operations/full 2π twist). For codons one would have four different subgroups of full braid group corresponding to codons of type XXX, XYY, XXY, and XYZ. Each gene would be characterized by its own subgroup of braid group and thus by an M-matrix defining topological quantum computation.

  3. One could understand the "junk DNA" character of introns. Introns are the most natural candidates for the portions of genome participating topological quantum computations The transcription process would disturb topological quantum computation so that introns should be chemically passive. Since the portion of "junk DNA" increases with the evolutionary level of the species evolution would indeed correspond to an increase the amount of topological quantum computations performed.

2.2 Two realizations of topological quantum computation

One can imagine two basic realizations of topological quantum computation like processes- or to be more precise - entanglement by braiding. In TGD framework this entanglement could be interpreted in terms of Connes tensor product.

1. Space-like entanglement The first realization would rely space-like braids. Braid strands would connect identical lines of text at the page of book defined by sequences of genomes of different nuclei. Inside nucleus the strands would connect DNA and its conjugate. The braiding operation would take place between lines.

In this case it would be perhaps more appropriate to speak about quantum memory storage of a function realized as entanglement. These functions could represent various rules about the behavior of and survival in the physical world. For this option A,T,C,G cannot correspond to entangled quantum numbers and the interpretation as braid colors is natural. Braiding cannot correspond to a physical braiding of nucleotides so that (A,T,C,G) could correspond to braid color (strands would connect only identical nucleotides).

Strands would not connect strand and its conjugate like hydrogen bonds do but would be like long flux lines of dipole field starting from nucleotide and ending to its conjugate so that braiding would emerge naturally. Color magnetic flux tube structures of almost atom size appear in the TGD based model of nucleus and have light quarks and anti-quarks at their ends (see this). This could be the case also now since quarks and anti-quarks appear also in the model of high Tc superconductivity which should be present also in living matter (see this).

2. Light-like entanglement

Second realization would rely on light-like braids at the boundaries of light-like 3-surfaces connecting 2-surfaces assignable to single genome at different moments of time. Braiding would be dynamical and dance metaphor would apply. The light-like surface could intersect genomes only at initial and final moments and strands would connect only identical nucleotides. Light-likeness in the induced metric of course allows the partonic 3-surface to look static at the level of imbedding space. The fundamental number theoretic braids defined by the minima of the Higgs like field associated with the modified Dirac operator would be very natural in this case.

Genes would define only the hardware unless they code for the magnetic body of DNA too, which looks implausible. The presence of quantum memory and quantum programs would mean a breakdown of genetic determinism since the braidings representing memories and programs would develop quantum jump by quantum jump and distinguish between individuals with the same genome. Also the personal development of individual would take place at this level. It would be these programs (that is magnetic bodies) which would differentiate between us and our cousins with almost identical genome.

3. Biological evolution as an evolution of topological quantum computation

This framework allows to understand biological evolution as an evolution of topological quantum computation like processes in which already existing programs become building blocks of more complex programs.

  1. The transition from RNA era to DNA era (for TGD inspired model for pre-biotic evolution (see this) involving also the emergence of cell membrane bounded structures would mean the emergence of the topological quantum computation using a static hardware.

  2. For mono-cellulars double DNA strands define space-like topological quantum computations involving only single step if the braids connect the nucleotides of the two DNA strands: obviously a reason why for double DNA strands.

  3. For multicellular organisms more complex space-like topological quantum computations would emerge and could code rules about environment and multicellular survival in it. At this step also introns specialized to topological quantum computation would emerge.

  4. A further evolution as a generation of super-genomes in turn forming hyper-genomes and even higher structures would have a coso that topological quantum computations would become increasingly complex and program module structure would emerge very naturally.

For details see the new chapter Topological Quantum Computation in TGD Universe.

Fractional Quantum Hall effect in TGD framework

The generalization of the imbedding space discussed in previous posting allows to understand fractional quantum Hall effect (see this and this).

The formula for the quantized Hall conductance is given by

σ= ν× e2/h,ν=m/n.

Series of fractions in ν=1/3, 2/5 3/7, 4/9, 5/11, 6/13, 7/15..., 2/3, 3/5, 4/7 5/9, 6/11, 7/13..., 5/3, 8/5, 11/7, 14/9... 4/3 7/5, 10/7, 13/9... , 1/5, 2/9, 3/13..., 2/7 3/11..., 1/7.. with odd denominator have bee observed as are also ν=1/2 and ν=5/2 state with even denominator.

The model of Laughlin [Laughlin] cannot explain all aspects of FQHE. The best existing model proposed originally by Jain [Jain] is based on composite fermions resulting as bound states of electron and even number of magnetic flux quanta. Electrons remain integer charged but due to the effective magnetic field electrons appear to have fractional charges. Composite fermion picture predicts all the observed fractions and also their relative intensities and the order in which they appear as the quality of sample improves.

I have considered earlier a possible TGD based model of FQHE not involving hierarchy of Planck constants. The generalization of the notion of imbedding space suggests the interpretation of these states in terms of fractionized charge and electron number.

  1. The easiest manner to understand the observed fractions is by assuming that both M4 an CP2 correspond to covering spaces so that both spin and electric charge and fermion number are quantized. With this assumption the expression for the Planck constant becomes hbar/hbar0 =nb/na and charge and spin units are equal to 1/nb and 1/na respectively. This gives ν =nna/nb2. The values n=2,3,5,7,.. are observed. Planck constant can have arbitrarily large values. There are general arguments stating that also spin is fractionized in FQHE and for na=knb required by the observed values of ν charge fractionization occurs in units of k/nb and forces also spin fractionization. For factor space option in M4 degrees of freedom one would have ν= n/nanb2.

  2. The appearance of nb=2 would suggest that also Z2 appears as the homotopy group of the covering space: filling fraction 1/2 corresponds in the composite fermion model and also experimentally to the limit of zero magnetic fiel [Jain]. Also ν=5/2 has been observed.

  3. A possible problematic aspect of the TGD based model is the experimental absence of even values of nb except nb=2. A possible explanation is that by some symmetry condition possibly related to fermionic statistics kn/nb must reduce to a rational with an odd denominator for nb>2. In other words, one has k propto 2r, where 2r the largest power of 2 divisor of nb smaller than nb.

  4. Large values of nb emerge as B increases. This can be understood from flux quantization. One has eBS= nhbar= n(nb/na)hbar0. The interpretation is that each of the nb sheets contributes n/na units to the flux. As nb increases also the flux increases for a fixed value of na and area S: note that magnetic field strength remains more or less constant so that kind of saturation effect for magnetic field strength would be in question. For na=knb one obtains eBS/hbar0= n/k so that a fractionization of magnetic flux results and each sheet contributes 1/knb units to the flux. ν=1/2 correspond to k=1,nb=2 and to a non-vanishing magnetic flux unlike in the case of composite fermion model.

  5. The understanding of the thermal stability is not trivial. The original FQHE was observed in 80 mK temperature corresponding roughly to a thermal energy of T≈ 10-5 eV. For graphene the effect is observed at room temperature. Cyclotron energy for electron is (from fe= 6× 105 Hz at B=.2 Gauss) of order thermal energy at room temperature in a magnetic field varying in the range 1-10 Tesla. This raises the question why the original FQHE requires so low a temperature? The magnetic energy of a flux tube of length L is by flux quantization roughly e2B2S≈ Ec(e)meL(hbar0=c=1) and exceeds cyclotron energy roughly by factor L/Le, Le electron Compton length so that thermal stability of magnetic flux quanta is not the explanation.

    A possible explanation is that since FQHE involves several values of Planck constant, it is quantum critical phenomenon and is characterized by a critical temperature. The differences of the energies associated with the phase with ordinary Planck constant and phases with different Planck constant would characterize the transition temperature. Saturation of magnetic field strength would be energetically favored.


[Laughlin] R. B. Laughlin (1983), Phys. Rev. Lett. 50, 1395.
[Jain] J. K. Jain (1989), Phys. Rev. Lett. 63, 199.

For more details see the chapter Does TGD Predict the Spectrum of Planck Constants? .

A further generalization of the notion of imbedding space

The hypothesis that Planck constant is quantized having in principle all possible rational values but with some preferred values implying algebraically simple quantum phases has been one of the main ideas of TGD during last years. The mathematical realization of this idea leads to a profound generalization of the notion of imbedding space obtained by gluing together infinite number of copies of imbedding space along common 4-dimensional intersection. The hope was that this generalization could explain charge fractionization but this does not seem to be the case. This problem led to a futher generalization of the imbedding space and this is what I want to discussed below.

1. Original view about generalized imbedding space

The original generalization of imbedding space was basically following. Take imbedding space H=M4×CP2. Choose submanifold M2×S2, where S2 is homologically non-trivial geodesic sub-manifold of CP2. The motivation is that for a given choice of Cartan algebra of Poincare algebra (translations in time direction and spin quantization axis plus rotations in plane orthogonal to this plane plus color hypercharge and isospin) this sub-manifold remains invariant under the transformations leaving the quantization axes invariant.

Form spaces M4= M4\M2 and CP2 = CP2\S2 and their Cartesian product. Both spaces have a hole of co-dimension 2 so that the first homotopy group is Z. From these spaces one can construct an infinite hierarchy of factor spaces M4/Ga and CP2/Gb where Ga is discrete group of SU(2) leaving quantization axis invariant. In case of Minkowski factor this means that the group in question acts essentially as a combination reflection and to rotations around quantization axies of angular momentum. The generalized imbedding space is obtained by gluing all these spaces together along M2×S2.

The hypothesis is that Planck constant is given by the ratio hbar= na/nb, where ni is the order of maximal cyclic subgroups of Gi. The hypothesis states also that the covariant metric of the Minkowski factor is scaled by the factor (na/nb)2. One must take care of this in the gluing procedure. One can assign to the field bodies describing both self interactions and interactions between physical systems definite sector of generalized imbedding space characterized partially by the Planck constant. The phase transitions changing Planck constant correspond to tunnelling between different sectors of the imbedding space.

2. Fractionization of quantum numbers is not possible if only factor spaces are allowed

The original idea was that the modification of the imbedding space inspired by the hierarchy of Planck constants could explain naturally phenomena like quantum Hall effect involving fractionization of quantum numbers like spin and charge. This does not however seem to be the case. Ga× Gb implies just the opposite if these quantum numbers are assigned with the symmetries of the imbedding space. For instance, quantization unit for orbital angular momentum becomes na where Zna is the maximal cyclic subgroup of Ga.

One can however imagine obtaining fractionization at the level of imbedding space for space-time sheets, which are analogous to multi-sheeted Riemann surfaces (say Riemann surfaces associated with z1/n since the rotation by 2π understood as a homotopy of M4 lifted to the space-time sheet is a non-closed curve. Continuity requirement indeed allows fractionization of the orbital quantum numbers and color in this kind of situation. Lifting up this idea to the level of imbedding space leads to the generalization of the notion of imbedding space.

3. Both covering spaces and factor spaces are possible

The observation above stimulates the question whether it might be possible in some sense to replace H or its factors by their multiple coverings.

  1. This is certainly not possible for M4, CP2, or H since their fundamental groups are trivial. On the other hand, the fixing of quantization axes implies a selection of the sub-space H4= M2× S2subset M4× CP2, where S2 is a geodesic sphere of CP2. M4=M4\M2 and CP2=CP2\S2 have fundamental group Z since the codimension of the excluded sub-manifold is equal to two and homotopically the situation is like that for a punctured plane. The exclusion of these sub-manifolds defined by the choice of quantization axes could naturally give rise to the desired situation.

  2. H4 represents a straight cosmic string. Quantum field theory phase corresponds to Jones inclusions with Jones index M:N<4. Stringy phase would by previous arguments correspond to M:N=4. Also these Jones inclusions are labelled by finite subgroups of SO(3) and thus by Zn identified as a maximal Abelian subgroup.

    One can argue that cosmic strings are not allowed in QFT phase. This would encourage the replacement M4×CP2 implying that surfaces in M4×S2 and M2×CP2 are not allowed. In particular, cosmic strings and CP2 type extremals with M4 projection in M2 and thus light-like geodesic without zitterwebegung essential for massivation are forbidden. This brings in mind instability of Higgs=0 phase.

  3. The covering spaces in question would correspond to the Cartesian products M4na× CP2nb of the covering spaces of M4 and CP2 by Zna and Znb with fundamental group is Zna× Znb. One can also consider extension by replacing M2 and S2 with its orbit under Ga (say tedrahedral, octahedral, or icosahedral group). The resulting space will be denoted by M4×Ga resp. CP2×Gb. Product sign does not signify for Caretsian product here.

  4. One expects the discrete subgroups of SU(2) emerge naturally in this framework if one allows the action of these groups on the singular sub-manifolds M2 or S2. This would replace the singular manifold with a set of its rotated copies in the case that the subgroups have genuinely 3-dimensional action (the subgroups which corresponds to exceptional groups in the ADE correspondence). For instance, in the case of M2 the quantization axes for angular momentum would be replaced by the set of quantization axes going through the vertices of tedrahedron, octahedron, or icosahedron. This would bring non-commutative homotopy groups into the picture in a natural manner.

    Also the orbifolds M4/Ga× CP2/Gb can be allowed as also the spaces M4/Ga× (CP2×Gb) and (M4×GaCP2/Gb. Hence the previous framework would generalize considerably by the allowance of both coset spaces and covering spaces.

4. Do factor spaces and coverings correspond to the two kinds of Jones inclusions?

What could be the interpretation of these two kinds of spaces?

  1. Jones inclusions appear in two varieties corresponding to M:N<4 and M:N=4 and one can assign a hierarchy of subgroups of SU(2) with both of them. In particular, their maximal Abelian subgroups Zn label these inclusions. The interpretation of Zn as invariance group is natural for M: N< 4 and it naturally corresponds to the coset spaces. For M:N=4 the interpretation of Zn has remained open. Obviously the interpretation of Zn as the homology group defining covering would be natural.

  2. M:N=4 should correspond to the allowance of cosmic strings and other analogous objects. Does the introduction of the covering spaces bring in cosmic strings in some controlled manner? Formally the subgroup of SU(2) defining the inclusion is SU(2) would mean that states are SU(2) singlets which is something non-physical. For covering spaces one would however obtain the degrees of freedom associated with the discrete fiber and the degrees of freedom in question would not disappear completely and would be characterized by the discrete subgroup of SU(2).

    For anyons the non-trivial homotopy of plane brings in non-trivial connection with a flat curvature and the non-trivial dynamics of topological QFTs. Also now one might expect similar non-trivial contribution to appear in the spinor connection of M2×Ga and CP2×Gb. In conformal field theory models non-trivial monodromy would correspond to the presence of punctures in plane.

  3. For factor spaces the unit for quantum numbers like orbital angular momentum is multiplied by na resp. nb and for coverings it is divided by this number. These two kind of spaces are in a well defined sense obtained by multiplying and dividing the factors of H by Ga resp. Gb and multiplication and division are expected to relate to Jones inclusions with M:N< 4 and M:N=4, which both are labelled by a subset of discrete subgroups of SU(2).

  4. The discrete subgroups of SU(2) with fixed quantization axes possess a well defined multiplication with product defined as the group generated by forming all possible products of group elements as elements of SU(2). This product is commutative and all elements are idempotent and thus analogous to projectors. Trivial group G1, two-element group G2 consisting of reflection and identity, the cyclic groups Zp, p prime, and tedrahedral, octahedral, and icosahedral groups are the generators of this algebra.

    By commutativity one can regard this algebra as an 11-dimensional module having natural numbers as coefficients ("rig"). The trivial group G1, two-element group G2 generated by reflection, and tedrahedral, octahedral, and icosahedral groups define 5 generating elements for this algebra. The products of groups other than trivial group define 10 units for this algebra so that there are 11 units altogether. The groups Zp generate a structure analogous to natural numbers acting as analog of coefficients of this structure. Clearly, one has effectively 11-dimensional commutative algebra in 1-1 correspondence with the 11-dimensional "half-lattice" N11 (N denotes natural numbers). Leaving away reflections, one obtains N7. The projector representation suggests a connection with Jones inclusions. An interesting question concerns the possible Jones inclusions assignable to the subgroups containing infinitely manner elements. Reader has of course already asked whether dimensions 11, 7 and their difference 4 might relate somehow to the mathematical structures of M-theory with 7 compactified dimensions.

  5. How do the Planck constants associated with factors and coverings relate? One might argue that Planck constant defines a homomorphism respecting the multiplication and division (when possible) by Gi. If so, then Planck constant in units of hbar0 would be equal to na/nb for H/Ga× Gb option and nb/na for H×(Ga× Gb) with obvious formulas for hybrid cases. This option would put M4 and CP2 in a very symmetric role and allow much more flexibility in the identification of symmetries associated with large Planck constant phases.
For more details see the chapter Does TGD Predict the Spectrum of Planck Constants?.

A little crazy speculation about knots and infinite primes<

Kea told about some mathematical results related to knots.

  1. Knots are very algebraic objects. Product of knots is defined in terms of connected sum. Connected sum quite generally defines a commutative and associative product (or sum, as you wish), and one can decompose any knot into prime knots.

  2. Knots can be mapped to Jones polynomials J(K) (for instance -there are many other polynomials and there are very general mathematical results about them which go over my head) and the product of knots is mapped to a product of corresponding polynomials. The polynomials assignable to prime knots should be prime in a well-defined sense, and one can indeed define the notion of primeness for polynomials J(K): prime polynomial does not factor to a product of polynomials of lower degree in the extension of rationals considered.

This raises the idea that one could define the notion of zeta function for knots. It would be simply the product of factors 1/(1-J(K)-s) where K runs over prime knots. The new (to me) but very natural element in the definition would be that ordinary prime is replaced with a polynomial prime.

1. Do knots correspond to the hierarchy of infinite primes?

I have been pondering the problem how to define the counterpart of zeta for infinite primes. The idea of replacing primes with prime polynomials would resolve the problem since infinite primes can be mapped to polynomials. For some reason this idea however did not occur to me.

The correspondence of both knots and infinite primes with polynomials inspires the question whether d=1-dimensional prime knots might be in correspondence (not necessarily 1-1) with infinite primes. Rational or Gaussian rational infinite primes would be naturally selected: these are also selected by physical considerations as representatives of physical states although quaternionic and octonionic variants of infinite primes can be considered.

If so, knots could correspond to the subset of states of a super-symmetric arithmetic quantum field theory with bosonic single particle states and fermionic states labelled by quaternionic primes.

  1. The free Fock states of this QFT are mapped to first order polynomials and irreducible polynomials of higher degree have interpretation as bound states so that the non-decomposability to a product in a given extension of rationals would correspond physically to the non-decomposability into many-particle state. What is fascinating that apparently free arithmetic QFT allows huge number of bound states.

  2. Infinite primes form an infinite hierarchy which corresponds to an infinite hierarchy of second quantizations for infinite primes meaning that n-particle states of the previous level define single particle states of the next level. At space-time level this hierarchy corresponds to a hierarchy defined by space-time sheets of the topological condensate: space-time sheet containing a galaxy can behave like an elementary particle at the next level of hierarchy.

  3. Could this hierarchy have some counterpart for knots?In one realization as polynomials, the polynomials corresponding to infinite prime hierarchy have increasing number of variables. Hence the first thing that comes into my uneducated mind is as the hierarchy defined by the increasing dimension d of knot. All knots of dimension d would in some sense serve as building bricks for prime knots of dimension d+1. A canonical construction recipe for knots of higher dimensions should exist.

  4. One could also wonder whether the replacement of spherical topologies for d-dimensional knot and d+2-dimensional imbedding space with more general topologies could correspond to algebraic extensions at various levels of the hierarchy bringing into the game more general infinite primes. The units of these extensions would correspond to knots which involve in an essential manner the global topology (say knotted non-contractible circles in 3-torus). Since the knots defining the product would in general have topology different from spherical topology the product of knots should be replaced with its category theoretical generalization making higher-dimensional knots a groupoid in which spherical knots would act diagonally leaving the topology of knot invariant. The assignment of d-knots with the notion of n-category, n-groupoid, etc.. by putting d=n is a highly suggestive idea. This is indeed natural since are an outcome of a repeated abstraction process: statements about statements about ...

  5. The lowest d=1, D=3 level would be the fundamental one and the rest would be somewhat boring repeated second quantization;-). This is why dimension D=3 (number theoretic braids at light-like 3-surfaces!) would be fundamental for physics.

2. Further speculations

Some further comments about the proposed structure of all structures are in order.

  1. The possibility that algebraic extensions of infinite primes could allow to describe the refinements related to the varying topologies of knot and imbedding space would mean a deep connection between number theory, manifold topology, sub-manifold topology, and n-category theory.

  2. n-structures would have very direct correspondence with the physics of TGD Universe if one assumes that repeated second quantization makes sense and corresponds to the hierarchical structure of many-sheeted space-time where even galaxy corresponds to elementary fermion or boson at some level of hierarchy. This however requires that the unions of light-like 3-surfaces and of their sub-manifolds at different levels of topological condensate should be able to represent higher-dimensional manifolds physically albeit not in the standard geometric sense since imbedding space dimension is just 8. This might be possible.

    1. As far as physics is considered, the disjoint union of submanifolds of dimensions d1 and d2 behaves like a d1+d2-dimensional Cartesian product of the corresponding manifolds. This is of course used in standard manner in wave mechanics (the configuration space of n-particle system is identified as E3n/Sn with division coming from statistics).

    2. If the surfaces have intersection points, one has a union of Cartesian product with punctures (intersection points) and of lower-dimensional manifold corresponding to the intersection points.

    3. Note also that by posing symmetries on classical fields one can effectively obtain from a given n-manifold manifolds (and orbifolds) with quotient topologies.

    The megalomanic conjecture is that this kind of physical representation of d-knots and their imbedding spaces is possible using many-sheeted space-time. Perhaps even the entire magnificient mathematics of n-manifolds and their sub-manifolds might have a physical representation in terms of sub-manifolds of 8-D M4×CP2 with dimension not higher than space-time dimension d=4. Could crazy TOE builder dream of anything more ouf of edge;-)!

3. The idea survives the most obvious killer test

All this looks nice and the question is how to give a death blow to all this reckless speculation. Torus knots are an excellent candidate for permorming this unpleasant task but the hypothesis survives!

  1. Torus knots are labelled by a pair integers (m,n), which are relatively prime. These are prime knots. Torus knots for which one has m/n= r/s are isotopic so that any torus knot is isotopic with a knot for which m and n have no common prime power factors.

  2. The simplest infinite primes correspond to free Fock states of the supersymmetric arithmetic QFT and are labelled by pairs (m,n) of integers such that m and n do not have any common prime factors. Thus torus knots would correspond to free Fock states! Note that the prime power pkp appearing in m corresponds to kp-boson state with boson "momentum" pk and the corresponding power in n corresponds to fermion state plus kp-1 bosons.

  3. A further property of torus knots is that (m,n) and (n,m) are isotopic: this would correspond at the level of infinite primes to the symmetry mX +n→nX+m, X product of all finite primes. Thus infinite primes are in 2→ correspondence with torus knots and the hypothesis survives also this murder attempt.

4. How to realize the representation of the braid hierarchy in many-sheeted space-time?

One can consider a concrete construction of higher-dimensional knots and braids in terms of the many-sheeted space-time concept.

  1. The basic observation is that ordinary knots can be constructed as closed braids so that everything reduces to the construction of braids. In particular, any torus knot labelled by (m,n) can be made from a braid with m strands: the braid word in question is (σ1..σm-1)n or by (m,n)=(n,m) equivalence from n strands. The construction of infinite primes suggests that also the notion of d-braid makes sense as a collection of d-knots in d+2-space, which move and and define d+1-braid in d+3 space (the additional dimension being defined by time coordinate).

  2. The notion of topological condensate should allow a concrete construction of the pairs of d- and d+2-dimensional manifolds. The 2-D character of the fundamental objects (partons) might indeed make this possible. Also the notion of length scale cutoff fundamental for the notion of topological condensate is a crucial element of the proposed construction.

The concrete construction would proceed as follows.

  1. Consider first the lowest non-trivial level in the hierarchy. One has a collection of 3-D lightlike 3-surfaces X3 i representing ordinary braids. The challenge is to assign to them a 5-D imbedding space in a natural manner. Where do the additional two dimensions come from? The obvious answer is that the new dimensions correspond to the 2-d dimensional partonic 2-surface X2 assignable to the 3-D lightlike surface at which these surfaces have suffered topological condensation. The geometric picture is that X3i grow like plants from ground defined by X2 at 7-dimensional δ M4+×CP2.

  2. The degrees of freedom of X2 should be combined with the degrees of freedom of X3i to form a 5-dimensional space X5. The natural idea is that one first forms the Cartesian products X5i =X3i×X2 and then the desired 5-manifold X5 as their union by posing suitable additional conditions. Braiding means a translational motion of X3i inside X2 defining braid as the orbit in X5. It can happen that X3i and X3j intersect in this process. At these points of the union one must obviously pose some additional conditions.

    Finite (p-adic) length scale resolution suggests that all points of the union at which an intersection between two or more light-like 3-surfaces occurs must be regarded as identical. In general the intersections would occur in a 2-d region of X2 so that the gluing would take place along 5-D regions of X5i and there are therefore good hopes that the resulting 5-D space is indeed a manifold. The imbedding of the surfaces X3i to X5 would define the braiding.

  3. At the next level one would consider the 5-d structures obtained in this manner and allow them to topologically condense at larger 2-D partonic surfaces in the similar manner. The outcome would be a hierarchy consisting of 2n+1-knots in 2n+3 spaces. A similar construction applied to partonic surfaces gives a hierarchy of 2n-knots in 2n+2-spaces.

  4. The notion of length scale cutoff is an essential element of the many-sheeted space-time concept. In the recent context it suggests that d-knots represented as space-time sheets topologically condensed at the larger space-time sheet representing d+2-dimensional imbedding space could be also regarded effectively point-like objects (0-knots) and that their d-knottiness and internal topology could be characterized in terms of additional quantum numbers. If so then d-knots could be also regarded as ordinary colored braids and the construction at higher levels would indeed be very much analogous to that for infinite primes.

For details see the chapter Infinite Primes and Consciousness..

How to represent algebraic complex numbers as geometric objects?

I already told about the idea of representing negative integers and even rationals as p-adic fractals. To gain additional understanding I decided to look at Weekly Finds (Week 102) of John Baez to which Kea gave link. Fascinating reading! Thanks Kea!

The outcome was the realization that the notion of rig used to categorify the subset of algebraic numbers obtained as roots of polynomials with natural number valued coefficients generalizes trivially by replacing natural numbers by p-adic integers. As a consequence one obtains beautiful p-adicization of the generating function F(x) of structure as a function which converges p-adically for any rational x=q for which it has prime p as a positive power divisor.

Effectively this generalization means the replacement of natural numbers as coefficients of the polynomial defining the rig with all rationals, also negative, and all complex algebraic numbers find a category theoretical representation as "cardinalities". These cardinalities have a dual interpretation as p-adic integers which in general correspond to infinite real numbers but are mappable to real numbers by canonical identification and have a geometric representation as fractals as discussed in the previous posting.

1. Mapping of objects to complex numbers and the notion of rig

The idea of rig approach is to categorify the notion of cardinality in such a manner that one obtains a subset of algebraic complex numbers as cardinalities in the category-theoretical sense. One can assign to an object a polynomial with coefficients, which are natural numbers and the condition Z=P(Z) says that P(Z) acts as an isomorphism of the object. One can interpret the equation also in terms of complex numbers. Hence the object is mapped to a complex number Z defining a root of the polynomial interpreted as an ordinary polynomial: it does not matter which root is chosen. The complex number Z is interpreted as the "cardinality" of the object but I do not really understand the motivation for this. The deep further result is that also more general polynomial equations R(Z)= Q(Z) satisfied by the generalized cardinality Z imply R(Z)= Q(Z) as isomorphism. This means that algebra is mapped to isomorphisms.

I try to reproduce what looks the most essential in the explanation of John Baez and relate it to my own ideas but take this as my talk to myself and visit This Week's Finds to learn of this fascinating idea.

  1. Baez considers first the ways of putting a given structure to n-element set. The set of these structures is denoted by Fn and the number of them by Fn. The generating function F(x) = ∑nFnxn packs all this information to a single function.

    For instance, if the structure is binary tree, this function is given by T(x)= ∑nCn-1xn, where Cn-1 are Catalan numbers and n>0 holds true. One can show that T satisfies the formula

    T= X+T2

    since any binary tree is either trivial or decomposes to a product of binary trees, where two trees emanate from the root. One can solve this second order polynomial equation and the power expansion gives the generating function.

  2. The great insight is that one can also work directly with structures. For instance, by starting from the isomorphism T=1+T2 applying to an object with cardinality 1 and substituting T2 with (1+T2)2 repeatedly, one can deduce the amazing formula T7(1)=T(1) mentioned by Kea, and this identity can be interpreted as an isomorphism of binary trees.

  3. This result can be generalized using the notion of rig category (Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, available as math.CT/0212377). In rig category one can add and multiply but negatives are not defined as in the case of ring. The lack of subtraction and division is still the problem and as I suggested in previous posting p-adic integers might resolve the problem.

    Whenever Z is object of a rig category, one can equip it with an isomorphism Z=P(Z) where P(Z) is polynomial with natural numbers as coefficients and one can assign to object "cardinality" as any root of the equation Z=P(Z). Note that set with n elements corresponds to P(Z)= n. Thus subset of algebraic complex numbers receive formal identification as cardinalities of sets. Furthermore, if the cardinality satisfies another equation Q(Z)= R(Z) such that neither polynomial is constant, then one can construct an isomorphism Q(Z)= R(Z). Isomorphisms correspond to equations which is nice!

  4. This is indeed nice that there is something which is not so beautiful as it could be: why should we restrict ourselves to natural numbers as coefficients of P(Z)? Could it be possible to replace them with integers to obtain all complex algebraic numbers as cardinalities? Could it be possible to replace natural numbers by p-adic integers? Oops! I told it! All tension of drama is now lost! Sorry!

2. p-Adic rigs and Golden Object as representation p-adic -1

The notions of generating function and rig generalize to the p-adic context.

  1. The generating function F(x) defining isomorphism Z in the rig formulation converges p-adically for p-adic number containing p as a factor so that the idea that all structures have p-adic counterparts is natural. In the real context the generating function typically diverges and must be defined by analytic continuation. Hence one might even argue that p-adic numbers are more natural in the description of structures assignable to finite sets than reals.

  2. For rig one considers only polynomials P(Z) (Z corresponds to the generating function F) with coefficients which are natural numbers. Any p-adic integer can be however interpreted as a non-negative integer: natural number if it is finite and "super-natural" number if it is infinite. Hence can generalize the notion of rig by replacing natural numbers by p-adic integers. The rig formalism would thus generalize to arbitrary polynomials with integer valued coefficients so that all complex algebraic numbers could appear as cardinalities of category theoretical objects. Even rational coefficients are allowed. This is highly natural number theoretically.

  3. For instance, in the case of binary trees the solutions to the isomorphism condition T=p+T2 giving T= [1+/- (1-4p)1/2]/2 and T would be complex number [p+/-(1-4p)1/2]/2. T(p) can be interpreted also as a p-adic number by performing power expansion of square root: this super-natural number can be mapped to a real number by the canonical identification and one obtains also the set theoretic representations of the category theoretical object T(p) as a p-adic fractal. This interpretation of cardinality is much more natural than the purely formal interpretation as a complex number. This argument applies completely generally. The case x=1 discussed by Baez gives T= [1+/-(-3)1/2]/2 allows p-adic representation if -3==p-3 is square mod p. This is the case for p=7 for instance.

  4. John Baez poses also the question about the category theoretic realization of Golden Object, his big dream. In this case one would have Z= G= -1+G2=P(Z). The polynomial on the right hand side does not conform with the notion of rig since -1 is not a natural number. If one allows p-adic rigs, x=-1 can be interpreted as a p-adic integer (p-1)(1+p+...), positive and infinite and "super-natural", actually largest possible p-adic integer in a well defined sense. A further condition is that Golden Mean converges as a p-adic number: this requires that sqrt(5) must exist as a p-adic number: (5=1+4)1/2 certainly converges as power series for p=2 so that Golden Object exists 2-adically. By using quadratic resiprocity theorem of Euler, one finds that 5 is square mod p only if p is square mod 5. To decide whether given p is Golden it is enough to look whether p mod 5 is 1 or 4. For instance, p=11, 19, 29, 31 (M5) are Golden. Mersennes Mk,k=3,7,127 and Fermat primes are not Golden. One representation of Golden Object as p-adic fractal is the p-adic series expansion of [1/2+/-51/2]/2 representable geometrically as a binary tree such that there are 0< xn+1≤p branches at each node at height n if n:th p-adic coefficient is xn. The "cognitive" p-adic representation in terms of wavelet spectrum of classical fields is discussed in the previous posting.

  5. It would be interesting to know how quantum dimensions of quantum groups assignable to Jones inclusions relate to the generalized cardinalities. The root of unity property of quantum phase (qn+1=1) suggests Q=Qn+1=P(Q) as the relevant isomorphism. For Jones inclusions the cardinality q =exp(i2π/n) would not be however equal to quantum dimension d(n)= 4cos2(π/n).

For details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness.

Is it possible to have a set with -1 elements?

I find Kea's blog interesting because it allows to get some grasp about very different styles of thinking of a mathematician and physicist. For mathematician it is very important that the result is obtained by a strict use of axioms and deduction rules. Physicist (at least me: I dare to count me as physicist) is a cognitive opportunist: it does not matter how the result is obtained by moving along axiomatically allowed paths or not, and the new result is often more like a discovery of a new axiom and physicist is ever-grateful for Gödel for giving justification for what sometimes admittedly degenerates to a creative hand-waving. For physicist ideas form a kind of bio-shere and the fate of the individual idea depends on its ability to survive, which is determined by its ability to become generalized, its consistency with other ideas, and ability to interact with other ideas to produce new ideas.

During last days we have had a little bit of discussion inspired by the problem related to the categorification of basic number theoretical structures. I have learned from Kea that sum and product are natural operations for objects of category but that subtraction and division are problematic. I dimly realize that this relates to the fact that negative numbers and inverses of integers do not have a realization as a number of elements for any set. The naive physicist inside me asks immediately: why not go from statics to dynamics and take operations (arrows with direction) as objects: couldn't this allow to define subtraction and division? Is the problem that the axiomatization of group theory requires something which purest categorification does not give? Or aren't the numbers representable in terms of operations of finite groups not enough? In any case cyclic groups would allow to realize roots of unity as operations (Z2 would give -1).

I also wonder in my own simplistic manner why the algebraic numbers might not somehow result via the representations of permutation group of infinite number of elements containing all finite groups and thus Galois groups of algebraic extensions as subgroups? Why not take the elements of this group as objects of the basic category and continue by building group algebra and hyper-finite factors of type II1 isomorphic to spinors of world of classical worlds, and...yes-yes-yes, I must stop!

This discussion led me to ask what the situation is in the case of p-adic numbers. Could it be possible to represent the negative and inverse of p-adic integer, and in fact any p-adic number, as a geometric object? In other words, does a set with -1 or 1/n elements exist? If this were in some sense true for all p-adic number fields, then all this wisdom combined together might provide something analogous to the adelic representation for the norm of a rational number as product of its p-adic norms.

Of course, this representation might not help to define p-adics or reals categorically but might help to understand how p-adic cognitive representations defined as subsets for rational intersections of real and p-adic space-time sheets could represent p-adic number as the number of points of p-adic fractal having infinite number of points in real sense but finite in the p-adic sense. This would also give a fundamental cognitive role for p-adic fractals as cognitive representations of numbers.

1. How to construct a set with -1 elements?

The basic observation is that p-adic -1 has the representation


As a real number this number is infinite or -1 but as a p-adic number the series converges and has p-adic norm equal to 1. One can also map this number to a real number by canonical identification taking the powers of p to their inverses: one obtains p in this particular case. As a matter fact, any rational with p-adic norm equal to 1 has similar power series representation.

The idea would be to represent a given p-adic number as the infinite number of points (in real sense) of a p-adic fractal such that p-adic topology is natural for this fractal. This kind of fractals can be constructed in a simple manner: from this more below. This construction allows to represent any p-adic number as a fractal and code the arithmetic operations to geometric operations for these fractals.

These representations - interpreted as cognitive representations defined by intersections of real and p-adic space-time sheets - are in practice approximate if real space-time sheets are assumed to have a finite size: this is due to the finite p-adic cutoff implied by this assumption and the meaning a finite resolution. One can however say that the p-adic space-time itself could by its necessarily infinite size represent the idea of given p-adic number faithfully.

This representation applies also to the p-adic counterparts of algebraic numbers in case that they exist. For instance, roughly one half of p-adic numbers have square root as ordinary p-adic number and quite generally algebraic operations on p-adic numbers can give rise to p-adic numbers so that also these could have set theoretic representation. For p mod 4=1 also sqrt(-1) exists: for instance, for p=5: 22=4=-1 mod 5 guarantees this so that also imaginary unit and complex numbers would have a fractal representation. Also many transcendentals possess this kind of representation. For instance exp(xp) exists as a p-adic number if x has p-adic norm not larger than 1. log(1+xp) also.

Hence a quite impressive repertoire of p-adic counterparts of real numbers would have representation as a p-adic fractal for some values of p. Adelic vision would suggest that combining these representations one might be able to represent quite a many real numbers. In the case of π I do not find any obvious p-adic representation (for instance sin(π/6)=1/2 does not help since the p-adic variant of the Taylor expansion of π/6;=arcsin(1/2) does not converge p-adically for any value of p). It might be that there are very many transcendentals not allowing fractal representation for any value of p.

2. Conditions on the fractal representations of p-adic numbers

Consider now the construction of the fractal representations in terms of rational intersections of real real and p-adic space-time sheets. The question is what conditions are natural for this representation if it corresponds to a cognitive representation is realized in the rational intersection of real and p-adic space-time sheets obeying same algebraic equations.

  1. Pinary cutoff is the analog of the decimal cutoff but is obtained by dropping away high positive rather than negative powers of p to get a finite real number: example of pinary cutoff is -1=(p-1)(1+p+p2+...)→ (p-1)(1+p+p2). This cutoff must reduce to a fractal cutoff meaning a finite resolution due to a finite size for the real space-time sheet. In the real sense the p-adic fractal cutoff means not forgetting details below some scale but cutting out all above some length scale. Physical analog would be forgetting all frequencies below some cutoff frequency in Fourier expansion.

    The motivation comes from the fact that TGD inspired consciousness assigns to a given biological body there is associated a field body or magnetic body containing dark matter with large hbar and quantum controlling the behavior of biological body and so strongly identifying with it so as to belief that this all ends up to a biological death. This field body has an onion like fractal structure and a size of at least order of light-life: at least 100 happy light years in my own case is my optimistic expectation. Of course, also larger onion layers could be present and would represent those levels of cognitive consciousness not depending on the sensory input on biological body: some altered states of consciousness could relate to these levels. In any case, the larger the magnetic body, the better the numerical skills of the p-adic mathematician;-).

  2. Lowest pinary digits of x= x0+x1p+x2p2+..., xn<p must have the most reliable representation since they are the most significant ones. The representation must be also highly redundant to guarantee reliability. This requires repetitions and periodicity. This is guaranteed if the representation is hologram like with segments of length pn with digit xn represented again and again in all segments of length pm, m>n.

  3. The TGD based physical constraint is that the representation must be realizable in terms of induced classical fields assignable to the field body hierarchy of an intelligent system interested in artistic expression of p-adic numbers using its own field body as instrument. As a matter, sensory and cognitive representations are realized at field body in TGD Universe and EEG is in a fundamental role in building this representation. By p-adic fractality fractal wavelets are the most natural candidate. The fundamental wavelet should represent the p different pinary digits and its scaled up variants would correspond to various powers of p so that the representation would reduce to a Fourier expansion of a classical field.

3. Concrete representation

Consider now a concrete candidate for a representation satisfying these constraints.

  1. Consider a p-adic number

    y= pn0x, x= ∑ xnpn, n≥n0=0.

    If one has representation for a p-adic unit x the representation of is by a purely geometric fractal scaling of the representation by pn. Hence one can restrict the consideration to p-adic units.

  2. To construct the representation take a real line starting from origin and divide it into segments with lengths 1, p, p2,.. In TGD framework this scalings come actually as powers of p1/2 but this is just a technical detail.

  3. It is natural to realize the representation in terms of periodic field patterns. One can use wavelets with fractal spectrum pnλ0 of "wavelet lengths", where λ0 is the fundamental wavelength. Fundamental wavelet should have p different patterns correspond to the p values of pinary digit as its structures. Periodicity guarantees the hologram like character enabling to pick n:th digit by studying the field pattern in scale pn anywhere inside the field body.

  4. Periodicity guarantees also that the intersections of p-adic and real space-time sheets can represent the values of pinary digits. For instance, wavelets could be such that in a given p-adic scale the number of rational points in the intersection of the real and p-adic space-time sheet equals to xn. This would give in the limit of an infinite pinary expansion a set theoretic realization of any p-adic number in which each pinary digit xn corresponds to infinite copies of a set with xn elements and fractal cutoff due to the finite size of real space-time sheet would bring in a finite precision. Note however that p-adic space-time sheet necessarily has an infinite size and it is only real world realization of the representation which has finite accuracy.

  5. A concrete realization for this object would be as an infinite tree with xn+1 ≤ p branches in each node at level n (xn+1 is needed in order to avoid the splitting tree at xn=0). In 2-adic case -1 would be represented by an infinite pinary tree. Negative powers of p correspond to the of the tree extending to a finite depth in ground.

For details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness.

Intronic portions of genome code for RNA: for what purpose?

The last issue of New Scientist contains an article about the discovery that only roughly one half of DNA expresses itself as aminoacid sequences. The article is published in Nature. The Encyclopedia of DNA Elements (ENCODE) project has quantified RNA transcription patterns and found that while the "standard" RNA copy of a gene gets translated into a protein as expected, for each copy of a gene cells also make RNA copies of many other sections of DNA. In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene.

In particular, intron portions ("junk DNA", the portion of which increases as one climbs up in evolutionary hierarchy) are transcribed to RNA in large amounts. What is also interesting that the RNA fragments correspond to pieces from several genes which raises the question whether there is some fundamental unit smaller than gene.

None of the extra RNA fragments gets translated into proteins, so the race is on to discover just what their function is. TGD proposal is that it gets braided and performs a lot of topological quantum computation (see this). Topologically quantum computing RNA fits nicely with replicating number theoretic braids associated with light-like orbits of partonic 2-surfaces and with their spatial "printed text" representations as linked and knotted partonic 2-surfaces giving braids as a special case (see this). An interesting question is how printing and reading could take place. Is it something comparable to what occurs when we read consciously? Is the biological portion of our conscious life identifiable with this reading process accompanied by copying by cell replication and as secondary printing using aminoacid sequences?

This picture conforms with TGD view about pre-biotic evolution. Plasmoids [1], which are known to share many basic characteristics assigned with life, came first: high temperatures are not a problem in TGD Universe since given frequency corresponds to energy above thermal energy for large enough value of hbar. Plasmoids were followed by RNA, and DNA and aminoacid sequences emerged only after the fusion of 1- and 2-letter codes fusing to the recent 3-letter code. The cross like structure of tRNA molecules carries clear signatures supporting this vision. RNA would be still responsible for roughly half of intracellular life and perhaps for the core of "intelligent life".

I have also proposed that this expression uses memetic code which would correspond to Mersenne M127=2127-1 with 2126 codons whereas ordinary genetic code would correspond to M7=27-1 with 26 codons. Memetic codons in DNA representations would consist of sequences of 21 ordinary codons. Also representations in terms of field patterns with duration of .1 seconds (secondary p-adic time scale associated with M127 defining a fundamental biorhythm) can be considered.

A hypothesis worth of killing would be that the DNA coding for RNA has memetic codons scattered around genome as basic units. It is interesting to see whether the structure of DNA could give any hints that memetic codon appears as a basic unit.

  1. In a "relaxed" double-helical segment of DNA, the two strands twist around the helical axis once every 10.4 base pairs of sequence. 21 genetic codons correspond 63 base pairs whereas 6 full twists would correspond to 62.4 base pairs.

  2. Nucleosomes are fundamental repeating units in eukaryotic chromatin possessing what is known as 10 nm beads-on-string structure. They repeat roughly every 200 base pairs: integer number of genetic codons would suggest 201 base pairs. 3 memetic codons makes 189 base pairs. Could this mean that only a fraction p≈ 12/201, which happens to be of same order of magnitude as the portion of introns in human genome, consists of ordinary codons? Inside nucleosomes the distance between neighboring contacts between histone and DNA is about 10 nm, the p-adic length scale L(151) associated with the Gaussian Mersenne (1+i)151-1 characterizing also cell membrane thickness and the size of nucleosomes. This length corresponds to 10 codons so that there would be two contacts per single memetic codon in a reasonable approximation. In the example of Wikipedia nucleosome corresponds to about 146=126+20 base pairs: 147 base pairs would make 2 memetic codons and 7 genetic codons.

    The remaining 54 base pairs between histone units + 3 ordinary codons from histone unit would make single memetic codon. That only single memetic codon is between histone units and part of the memetic codon overlaps with histone containing unit conforms with the finding that chromatin accessibility and histone modification patterns are highly predictive of both the presence and activity of transcription start sites. This would leave 4 genetic codons and 201 base pairs could decompose as memetic codon+2 genetic codons+memetic codon+2 genetic codons. The simplest possibility is however that memetic codons are between histone units and histone units consist of genetic codons. Note that memetic codons could be transcribed without the straightening of histone unit occurring during the transcription leading to protein coding.

[1] E. Lozneanu and M. Sanduloviciu (2003), Minimal-cell system created in laboratory by self-organization, Chaos, Solitons and Fractals, Volume 18, Issue 2, October, p. 335. See also Plasma blobs hint at new form of life, New Scientist vol. 179 issue 2413 - 20 September 2003, page 16.

For details see the new chapter Topological Quantum Computation in TGD Universe.

Farey sequences, Riemann hypothesis, tangles, and TGD

Farey sequences allow an alternative formulation of Riemann Hypothesis and subsequent pairs in Farey sequence characterize so called rational 2-tangles. In TGD framework Farey sequences relate very closely to dark matter hierarchy, which inspires "Platonia as the best possible world in the sense that cognitive representations are optimal" as the basic variational principle of mathematics. This variational principle supports RH.

Possible TGD realizations of tangles, which are considerably more general objects than braids, are considered. One can assign to a given rational tangle a rational number a/b and the tangles labelled by a/b and c/d are equivalent if ad-bc=+/-1 holds true. This means that the rationals in question are neighboring members of Farey sequence. Very light-hearted guesses about possible generalization of these invariants to the case of general N-tangles are made.

For more details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness.

Quantum quandaries

For long time it has been clear that category theory might provide a fundamental formulation of quantum TGD. The problem has been that category theory seems to postulate quite too many objects. The reading of Quantum Quandaries by John Baez helped to see the situation in all its simplicity.

  1. Topological Quantum Field theories have extremely simple formulation as a functor from the category of cobordisms (topological evolutions between n-1-manifolds by connecting n-manifold) to the category of Hilbert spaces assignable to n-1-manifolds.

  2. Since light-like partonic 3-surfaces correspond to almost topological QFT, with the overall important "almost" coming just from the light-likeness in the induced metric, the theory is non-trivial physically and nothing of the beauty of TQFT as a functor is lost. Cobordisms are however replaced by what I have christened Feynmann cobordisms generalizing the Feynman diagrams to 3-D context: the ends of light-like 3-manifolds meet at the vertices which correspond to 2-dimensional partonic surfaces.

  3. Also the counterparts of ordinary string diagrams having interpretation as ordinary cobordisms are possible but have nothing to do with particle reactions: the particle simply decomposes into several pieces and spinor fields propagate along different routes. This is the space-time correlate for what happens in double slit experiment when photon travels along two different paths simultaneously.

  4. The intriguing results is that for n<4-dimensional cobordisms unitary S-matrix exists only for trivial cobordisms. I wonder whether string theorists have considered the possible catastrophic consequences concerning the non-perturbative dream about the unique stringy S-matrix. In the zero energy ontology of TGD S-matrix appears as time-like entanglement coefficients and need not be unitary. I have already proposed that p-adic thermodynamics and thermodynamics in general could be regarded as an exact part of quantum theory in this framework and the basic mathematics of hyper-finite factors provides strong technical support for this idea. It could be that one cannot require unitarity in the case of Feynman cobordisms and that only the condition that S-matrix for a product of Feynman cobordisms is a product of S-matrices for composites. Hence the time parameter in S-matrix can be replaced with complex time parameter with imaginary part in the role of temperature without losing the product structure. p-Adic thermodynamics and particle massication might be topological necessities in this framework.

For more details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness.

Platonism, Constructivism, and Quantum Platonism

I have been trying to understand how Category Theory and Set Theory relate to quantum TGD inspired view about fundamentals of mathematics. I managed to clarify my thoughts about what these theories are by reading the article Structuralism, Category Theory and Philosophy of Mathematics by Richard Stefanik (Washington: MSG Press, 1994). The reactions to postings in Kea's blog and email correspondence with Sampo Vesterinen have been very stimulating and inspired the attempt to represent TGD based vision about the unification of mathematics, physics, and consciousness theory in a more systematic manner.

The basic ideas behind TGD vision are following. One cannot understand mathematics without understanding mathematical consciousness. Mathematical consciousness and its evolution must have direct quantum physical correlates and by quantum classical correspondence these correlates must appear also at space-time level. Quantum physics must allow to realize number as a conscious experience analogous to a sensory quale. In TGD based ontology there is no need to postulate physical world behind the quantum states as mathematical entities (theory is the reality). Hence number cannot be any physical object, but can be identified as a quantum state or its label and its number theoretical anatomy is revealed by the conscious experiences induced by the number theoretic variants of particle reactions. Mathematical systems and their axiomatics are dynamical evolving systems and physics is number theoretically universal selecting rationals and their extensions in a special role as numbers, which can can be regarded elements of several number fields simultaneously.

For details see the last section of the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness or the article Platonism, Constructivism, and Quantum Platonism.

Sierpinski topology and quantum measurement theory with finite measurement resolution

I have been trying to understand whether category theory might provide some deeper understanding about quantum TGD, not just as a powerful organizer of fuzzy thoughts but also as a tool providing genuine physical insights. Kea is also interested in categories but in much more technical sense. Her dream is to find a category theoretical formulation of M-theory as something, which is not the 11-D something making me rather unhappy as a physicist with second foot still deep in the muds of low energy phenomenology.

Kea talks about topos, n-logos,... and their possibly existing quantum variants. I have used to visit Kea's blog in the hope of stealing some category theoretic intuition. It is also nice to represent comments knowing that they are not censored out immediately if their have the smell of original thought: this is quite too often the case in alpha male dominated blogs. It might be that I had luck this morning!

1. Locales, frames, Sierpinski topologies and Sierpinski space

Kea mentioned the notions of locale and frame . In Wikipedia I learned that complete Heyting algebras, which are fundamental to category theory, are objects of three categories with differing arrows. CHey, Loc and its opposite category Frm (arrows reversed). Complete Heyting algebras are partially ordered sets which are complete lattices. Besides the basic logical operations there is also algebra multiplication. From Wikipedia I learned also that locales and the dual notion of frames form the foundation of pointless topology. These topologies are important in topos theory which does not assume the axiom of choice.

So called particular point topology assumes a selection of single point but I have the physicist's feeling that it is otherwise rather near to pointless topology. Sierpinski topology is this kind of topology. Sierpinski topology is defined in a simple manner: set is open only if it contains a given point p. The dual of this topology defined in the obvious sense exists also. Sierpinski space consisting of just two points 0 and 1 is the universal building block of these topologies in the sense that a map of an arbitrary space to Sierpinski space provides it with Sierpinski topology as the induced topology. In category theoretical terms Sierpinski space is the initial object in the category of frames and terminal object in the dual category of locales. This category theoretic reductionism looks highly attractive to me.

2. Particular point topologies, their generalization, and finite measurement resolution

Pointless, or rather particular point topologies might be very interesting from physicist's point of view. After all, every classical physical measurement has a finite space-time resolution. In TGD framework discretization by number theoretic braids replaces partonic 2-surface with a discrete set consisting of algebraic points in some extension of rationals: this brings in mind something which might be called a topology with a set of particular algebraic points.

Perhaps the physical variant for the axiom of choice could be restricted so that only sets of algebraic points in some extension of rationals can be chosen freely. The extension would depend on the position of the physical system in the algebraic evolutionary hierarchy defining also a cognitive hierarchy. Certainly this would fit very nicely to the formulation of quantum TGD unifying real and p-adic physics by gluing real and p-adic number fields to single super-structure via common algebraic points.

There is also a finite measurement resolution in Hilbert space sense not taken into account in the standard quantum measurement theory based on factors of type I. In TGD framework one indeed introduces quantum measurement theory with a finite measurement resolution so that complex rays becomes included hyper-finite factors of type II1 (HFF, see this).

  • Could topology with particular algebraic points have a generalization allowing a category theoretic formulation of the quantum measurement theory without states identified as complex rays?

  • How to achieve this? In the transition of ordinary Boolean logic to quantum logic in the old fashioned sense (von Neuman again!) the set of subsets is replaced with the set of subspaces of Hilbert space. Perhaps this transition has a counterpart as a transition from Sierpinski topology to a structure in which sub-spaces of Hilbert space are quantum sub-spaces with complex rays replaced with the orbits of subalgebra defining the measurement resolution. Sierpinski space {0,1} would in this generalization be replaced with the quantum counterpart of the space of 2-spinors. Perhaps one should also introduce q-category theory with Heyting algebra being replaced with q-quantum logic.

3. Fuzzy quantum logic as counterpart for Sierpinksi space

This program, which I formulated only after this section had been written, might indeed make sense (ideas never learn to emerge in the logical order of things;-)). The lucky association was to the ideas about fuzzy quantum logic realized in terms of quantum 2-spinor that I had developed a couple of years ago. Fuzzy quantum logic would reflect the finite measurement resolution. I just list the pieces of the argument.

Spinors and qbits: Spinors define a quantal variant of Boolean statements, qbits. One can however go further and define the notion of quantum qbit, qqbit. I indeed did this for couple of years ago (the last section in Was von Neumann Right After All?).

Q-spinors and qqbits: For q-spinors the two components a and b are not commuting numbers but non-Hermitian operators. ab= qba, q a root of unity. This means that one cannot measure both a and b simultaneously, only either of them. aa+ and bb+ however commute so that probabilities for bits 1 and 0 can be measured simultaneously. State function reduction is not possible to a state in which a or b gives zero! The interpretation is that one has q-logic is inherently fuzzy: there are no absolute truths or falsehoods. One can actually predict the spectrum of eigenvalues of probabilities for say 1. q-Spinors bring in mind strongly the Hilbert space counterpart of Sierpinski space. One would however expect that fuzzy quantum logic replaces the logic defined by Heyting algebra.

Q-locale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by sub-spaces of Hilbert space in the conventional quantum logic. Q-openness would be defined by identifying quantum spinors as the initial object, q-Sierpinski space. a (resp. b for dual category) would define q-open set in this space. Q-open sets for other quantum spaces would be defined as inverse images of a (resp. b) for morphisms to this space. Only for q=1 one could have the q-counterpart of rather uninteresting topology in which all sets are open and every map is continuous.

Q-locale and HFFs: The q-Sierpinski character of q-spinors would conform with the very special role of Clifford algebra in the theory of HFFs, in particular, the special role of Jones inclusions to which one can assign spinor representations of SU(2). The Clifford algebra and spinors of the world of classical worlds identifiable as Fock space of quark and lepton spinors is the fundamental example in which 2-spinors and corresponding Clifford algebra serves as basic building brick although tensor powers of any matrix algebra provides a representation of HFF.

Q-measurement theory: Finite measurement resolution (q-quantum measurement theory) means that complex rays are replaced by sub-algebra rays. This would force the Jones inclusions associated with SU(2) spinor representation and would be characterized by quantum phase q and bring in the q-topology and q-spinors. Fuzzyness of qqbits of course correlates with the finite measurement resolution.

Q-n-logos: For other q-representations of SU(2) and for representations of compact groups (see appendix of this) one would obtain something which might have something to do with quantum n-logos, quantum generalization of n-valued logic. All of these would be however less fundamental and induced by q-morphisms to the fundamental representation in terms of spinors of the world of classical worlds. What would be however very nice that if these q-morphisms are constructible explicitly it would become possible to build up q-representations of various groups using the fundamental physical realization - and as I have conjectured (see this) - McKay correspondence and huge variety of its generalizations would emerge in this manner.

The analogs of Sierpinski spaces: The discrete subgroups of SU(2), and quite generally, the groups Zn associated with Jones inclusions and leaving the choice of quantization axes invariant, bring in mind the n-point analogs of Sierpinski space with unit element defining the particular point. Note however that n≥3 holds true always so that one does not obtain Sierpinski space itself. Could it be that all of these n preferred points belong to any open set? Number theoretical braids identified as subsets of the intersection of real and p-adic variants of algebraic partonic 2-surface define second candidate for the generalized Sierpinski space with set of preferred points. Recall that the generalized imbedding space related to the quantization of Planck constant is obtained by gluing together coverings of M4×CP2→ M4×CP2/Ga×Gb along their common points. The topology in question would mean that if some point in the covering belongs to an open set, all of them do so. The interpretation could be that the points of fiber form a single inseparable quantal unit.

For more details see the chapter Was von Neumann Right After All?.

Jones inclusions and construction of S-matrix and U matrix

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

1. S-matrix

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

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

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

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

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

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

2. U-matrix

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

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

3. Relation to TGD inspired theory of consciousness

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

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

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

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

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

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

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

For more details see the chapter Was von Neumann Right After All?.

About infinite primes, points of the world of classical worlds, and configuration space spinor fields

The idea that configuration space CH of 3-surfaces, "the world of classical worlds", could be realized in terms of number theoretic anatomies of single space-time point using the real units formed from infinite rationals, is very attractive.

The correspondence of CH points with infinite primes and thus with infinite number of real units determined by them realizing Platonia at single space-time point, can be understood if one assume that the points of CH correspond to infinite rationals via their mapping to hyper-octonion real-analytic rational functions conjectured to define foliations of HO to hyper-quaternionic 4-surfaces inducing corresponding foliations of H.

The correspondence of CH spinors with the real units identified as infinite rationals with varying number theoretical anatomies is not so obvious. It is good to approach the problem by making questions.

  1. How the points of CH and CH spinors at given point of CH correspond to various real units? Configuration space Hamiltonians and their super-counterparts characterize modes of configuration space spinor fields rather than only spinors. Does this mean that only ground states of super-conformal representations, which are expected to correspond elementary p"../articles/, correspond to configuration space spinors and are coded by infinite primes?

  2. How do CH spinor fields (as opposed to CH spinors) correspond to infinite rationals? Configuration space spinor fields are generated by elements of super-conformal algebra from ground states. Should one code the matrix elements of the operators between ground states and creating zero energy states in terms of time-like entanglement between ground states represented by real units and assigned to the preferred points of H characterizing the tips of future and past light-cones and having also interpretation as arguments of n-point functions?

The argument represented in detail in TGD as a Generalized Number Theory III: Infinite Primes is in a nutshell following.

  1. CH itself and CH spinors are by super-symmetry characterized by ground states of super-conformal representations and can be mapped to infinite rationals defining real units Uk multiplying the eight preferred H coordinates hk whereas configuration space spinor fields correspond to discrete analogs of Schrödinger amplitudes in the space whose points have Uk as coordinates. The 8-units correspond to ground states for an 8-fold tensor power of a fundamental super-conformal representation or to a product of representations of this kind.

  2. General states are coded by quantum entangled states defined as entangled states of positive and negative energy ground states with entanglement coefficients defined by the product of operators creating positive and negative energy states represented by the units. Normal ordering prescription makes the mapping unique.

  3. The condition that various symmetries have number theoretical correlates leads to rather detailed view about the map of ground states to real units. As a matter fact one ends up with a detailed view about number theoretical realization of fundamental symmetries of standard model.

  4. It seems that quantal generalization of the fundamental associativity and commutativity conditions might be needed in the sense that quantum states are superpositions over all possible associations associated with a given hyper-octonionic prime. Only infinite integers identifiable as many particle states would reduced to infinite rational integers mappable to rational functions of hyper-octonionic coordinate with rational coefficients. Infinite primes could be genuinely hyper-quaternionic. This would imply automatically color confinement but would allow colored partons.
For more details see the chapter Infinite Primes and Consciousness.

Updated vision about infinite primes

I have updated the chapter about infinite primes so that it conforms with the recent general view about number theoretic aspects of quantum TGD. A lot of obsoletia have been thrown away and new insights have emerged.

  1. In particular, the identification of the mapping of infinite primes to space-time surfaces is fixed by associativity condition so that it only yields 4-D surfaces rather than a hierarchy of 4n-D surfaces of 8n-D imbedding spaces. This observation was actually trivial but had escaped my attention.

  2. What is especially fascinating is that configuration space and configuration space spinor fields might be represented in terms of the number theoretical anatomy of imbedding space points. Configuration space spinor fields associated with a given sub-configuration space labelled by a preferred point of imbedding space (this includes tip of lightcone) would be analogs of ordinary wave functions defined in the space of points which are identical in the real sense. One can say that physics in a well-defined sense reduces to space-time level after all.

I attach below the abstract of the revised chapter Infinite Primes and Consciousness.

Infinite primes are besides p-adicization and the representation of space-time surface as a hyper-quaternionic sub-manifold of hyper-octonionic space, basic pillars of the vision about TGD as a generalized number theory and will be discussed in the third part of the multi-chapter devoted to the attempt to articulate this vision as clearly as possible.

1. Why infinite primes are unavoidable

Suppose that 3-surfaces could be characterized by p-adic primes characterizing their effective p-adic topology. p-Adic unitarity implies that each quantum jump involves unitarity evolution U followed by a quantum jump. Simple arguments show that the p-adic prime characterizing the 3-surface representing the entire universe increases in a statistical sense. This leads to a peculiar paradox: if the number of quantum jumps already occurred is infinite, this prime is most naturally infinite. On the other hand, if one assumes that only finite number of quantum jumps have occurred, one encounters the problem of understanding why the initial quantum history was what it was. Furthermore, since the size of the 3-surface representing the entire Universe is infinite, p-adic length scale hypothesis suggest also that the p-adic prime associated with the entire universe is infinite.

These arguments motivate the attempt to construct a theory of infinite primes and to extend quantum TGD so that also infinite primes are possible. Rather surprisingly, one can construct what might be called generating infinite primes by repeating a procedure analogous to a quantization of a super symmetric quantum field theory. At given level of hierarchy one can identify the decomposition of space-time surface to p-adic regions with the corresponding decomposition of the infinite prime to primes at a lower level of infinity: at the basic level are finite primes for which one cannot find any formula.

2. Two views about the role of infinite primes and physics in TGD Universe

Two different views about how infinite primes, integers, and rationals might be relevant in TGD Universe have emerged.

a) The first view is based on the idea that infinite primes characterize quantum states of the entire Universe. 8-D hyper-octonions make this correspondence very concrete since 8-D hyper-octonions have interpretation as 8-momenta. By quantum-classical correspondence also the decomposition of space-time surfaces to p-adic space-time sheets should be coded by infinite hyper-octonionic primes. Infinite primes could even have a representation as hyper-quaternionic 4-surfaces of 8-D hyper-octonionic imbedding space.

b) The second view is based on the idea that infinitely structured space-time points define space-time correlates of mathematical cognition. The mathematical analog of Brahman=Atman identity would however suggest that both views deserve to be taken seriously.

3. Infinite primes and infinite hierarchy of second quantizations

The discovery of infinite primes suggested strongly the possibility to reduce physics to number theory. The construction of infinite primes can be regarded as a repeated second quantization of a super-symmetric arithmetic quantum field theory. Later it became clear that the process generalizes so that it applies in the case of quaternionic and octonionic primes and their hyper counterparts. This hierarchy of second quantizations means enormous generalization of physics to what might be regarded a physical counterpart for a hierarchy of abstractions about abstractions about.. The ordinary second quantized quantum physics corresponds only to the lowest level infinite primes. This hierarchy can be identified with the corresponding hierarchy of space-time sheets of the many-sheeted space-time.

One can even try to understand the quantum numbers of physical p"../articles/ in terms of infinite primes. In particular, the hyper-quaternionic primes correspond four-momenta and mass squared is prime valued for them. The properties of 8-D hyper-octonionic primes motivate the attempt to identify the quantum numbers associated with CP2 degrees of freedom in terms of these primes. The representations of color group SU(3) are indeed labelled by two integers and the states inside given representation by color hyper-charge and iso-spin.

4. Infinite primes as a bridge between quantum and classical

An important stimulus came from the observation stimulated by algebraic number theory. Infinite primes can be mapped to polynomial primes and this observation allows to identify completely generally the spectrum of infinite primes whereas hitherto it was possible to construct explicitly only what might be called generating infinite primes.

This in turn led to the idea that it might be possible represent infinite primes (integers) geometrically as surfaces defined by the polynomials associated with infinite primes (integers).

Obviously, infinite primes would serve as a bridge between Fock-space descriptions and geometric descriptions of physics: quantum and classical. Geometric objects could be seen as concrete representations of infinite numbers providing amplification of infinitesimals to macroscopic deformations of space-time surface. We see the infinitesimals as concrete geometric shapes!

5. Various equivalent characterizations of space-times as surfaces

One can imagine several number-theoretic characterizations of the space-time surface.

  1. The approach based on octonions and quaternions suggests that space-time surfaces correspond to associative, or equivalently, hyper-quaternionic surfaces of hyper-octonionic imbedding space HO. Also co-associative, or equivalently, co-hyper-quaternionic surfaces are possible. These foliations can be mapped in a natural manner to the foliations of H=M^4\times CP_2 by space-time surfaces which are identified as preferred extremals of the Kähler action (absolute minima or maxima for regions of space-time surface in which action density has definite sign). These views are consistent if hyper-quaternionic space-time surfaces correspond to so called Kähler calibrations.

  2. Hyper-octonion real-analytic surfaces define foliations of the imbedding space to hyper-quaternionic 4-surfaces and their duals to co-hyper-quaternionic 4-surfaces representing space-time surfaces.

  3. Rational infinite primes can be mapped to rational functions of n arguments. For hyper-octonionic arguments non-associativity makes these functions poorly defined unless one assumes that arguments are related by hyper-octonion real-analytic maps so that only single independent variable remains. These hyper-octonion real-analytic functions define foliations of HO to space-time surfaces if b) holds true.

The challenge of optimist is to prove that these characterizations are equivalent.

6. The representation of infinite primes as 4-surfaces

The difficulties caused by the Euclidian metric signature of the number theoretical norm forced to give up the idea that space-time surfaces could be regarded as quaternionic sub-manifolds of octonionic space, and to introduce complexified octonions and quaternions resulting by extending quaternionic and octonionic algebra by adding imaginary units multiplied with √{-1. This spoils the number field property but the notion of prime is not lost. The sub-space of hyper-quaternions resp.-octonions is obtained from the algebra of ordinary quaternions and octonions by multiplying the imaginary part with √-1. The transition is the number theoretical counterpart for the transition from Riemannian to pseudo-Riemannian geometry performed already in Special Relativity.

The commutative √-1 relates naturally to the algebraic extension of rationals generalized to an algebraic extension of rational quaternions and octonions and conforms with the vision about how quantum TGD could emerge from infinite dimensional Clifford algebra identifiable as a hyper-finite factor of type II1.

The notions of hyper-quaternionic and octonionic manifold make sense but it is implausible that H=M4× CP2 could be endowed with a hyper-octonionic manifold structure. Indeed, space-time surfaces are assumed to be hyper-quaternionic or co-hyper-quaternionic 4-surfaces of 8-dimensional Minkowski space M8 identifiable as the hyper-octonionic space HO. Since the hyper-quaternionic sub-spaces of HO with a fixed complex structure are labelled by CP2, each (co)-hyper-quaternionic four-surface of HO defines a 4-surface of M4× CP2. One can say that the number-theoretic analog of spontaneous compactification occurs.

Any hyper-octonion analytic function HO--> HO defines a function g: HO--> SU(3) acting as the group of octonion automorphisms leaving a selected imaginary unit invariant, and g in turn defines a foliation of HO and H=M4× CP2 by space-time surfaces. The selection can be local which means that G2 appears as a local gauge group.

Since the notion of prime makes sense for the complexified octonions, it makes sense also for the hyper-octonions. It is possible to assign to infinite prime of this kind a hyper-octonion analytic polynomial P: HO--> HO and hence also a foliation of HO and H=M4× CP2 by 4-surfaces. Therefore space-time surface could be seen as a geometric counterpart of a Fock state. The assignment is not unique but determined only up to an element of the local octonionic automorphism group G2 acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane. In fact, a map HO--> S6 characterizes the choice since SO(6) acts effectively as a local gauge group.

The construction generalizes to all levels of the hierarchy of infinite primes if one poses the associativity requirement implying that hyper-octonionic variables are related by hyper-octonion real-analytic maps, and produces also representations for integers and rationals associated with hyper-octonionic numbers as space-time surfaces. By the effective 2-dimensionality naturally associated with infinite primes represented by real polynomials 4-surfaces are determined by data given at partonic 2-surfaces defined by the intersections of 3-D and 7-D light-like causal determinants. In particular, the notions of genus and degree serve as classifiers of the algebraic geometry of the 4-surfaces. The great dream is of course to prove that this construction yields the solutions to the absolute minimization of Kähler action.

7. Generalization of ordinary number fields: infinite primes and cognition

Both fermions and p-adic space-time sheets are identified as correlates of cognition in TGD Universe. The attempt to relate these two identifications leads to a rather concrete model for how bosonic generators of super-algebras correspond to either real or p-adic space-time sheets (actions and intentions) and fermionic generators to pairs of real space-time sheets and their p-adic variants obtained by algebraic continuation (note the analogy with fermion hole pairs).

The introduction of infinite primes, integers, and rationals leads also to a generalization of real numbers since an infinite algebra of real units defined by finite ratios of infinite rationals multiplied by ordinary rationals which are their inverses becomes possible. These units are not units in the p-adic sense and have a finite p-adic norm which can be differ from one. This construction generalizes also to the case of hyper- quaternions and -octonions although non-commutativity and in case of octonions also non-associativity pose technical problems to which the reduction to ordinary rational is simplest cure which would however allow interpretation as decomposition of infinite prime to hyper-octonionic lower level constituents. Obviously this approach differs from the standard introduction of infinitesimals in the sense that sum is replaced by multiplication meaning that the set of real units becomes infinitely degenerate.

Infinite primes form an infinite hierarchy so that the points of space-time and imbedding space can be seen as infinitely structured and able to represent all imaginable algebraic structures. Certainly counter-intuitively, single space-time point is even capable of representing the quantum state of the entire physical Universe in its structure. For instance, in the real sense surfaces in the space of units correspond to the same real number 1, and single point, which is structure-less in the real sense could represent arbitrarily high-dimensional spaces as unions of real units.

One might argue that for the real physics this structure is completely invisible and is relevant only for the physics of cognition. On the other hand, one can consider the possibility of mapping the configuration space and configuration space spinor fields to the number theoretical anatomies of a single point of imbedding space so that the structure of this point would code for the world of classical worlds and for the quantum states of the Universe. Quantum jumps would induce changes of configuration space spinor fields interpreted as wave functions in the set of number theoretical anatomies of single point of imbedding space in the ordinary sense of the word, and evolution would reduce to the evolution of the structure of a typical space-time point in the system. Physics would reduce to space-time level but in a generalized sense. Universe would be an algebraic hologram, and there is an obvious connection both with Brahman=Atman identity of Eastern philosophies and Leibniz's notion of monad.

For more details see the revised chapter Infinite Primes and Consciousness.

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