DNA as a 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 n^th incoming strand to n^th 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 n^th 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 e_n representing the exchange of the strands n and n+1 of the braid represented as a unitary matrix R acting on V_n\otimes V_n+1.

The basic condition on R is Yang-Baxter equation expressing the defining condition e_ne_n+1e_n= e_n+1e_ne_n+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 A_i 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 n^th strand commute with the generators of braid group which do not act on n^th 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=A_t.

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∫ A_tdt/hbar) coming from the presence of scale potential Φ=A_t 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 of "TGD as Generalized Number Theory".

Chimps remember better than us

Chimpanzees seem to have better memory than us. Mr. Ayumu, a 5-year-old Chimpanzee, can memorize the position of nine Arabic numerals in ascending order represented at computer screen (see the photo of Mr. Ayumy at the blog of Lubos). Something impossible for us. It is of course important to make clear what kind of memory one is testing in these experiments. Also so called idiot savants (sometimes people who are autistic or have severe damage in some brain areas involved with higher cognition) are able to perform memory feats. The essential point is that sensory memory rather than cognitive memory based on abstraction and concepts seems to be in question. In jungle precise sensory memories are a blessing. A severe problem with sensory memories is that you do not know when the event happened and you might be even unable to tell whether the memory mental image correspond to something happening just now.

Conceptual memory in turn saves huge number of bits as compared to direct sensory memory and has probably been one of the reasons for the rapid evolution of human culture. I remember a nice book of Willian Golding about Neanderthals and Homo Sapiens. The basic difference between these two species was supposed to be that Neanderthals had only sensory memories and because they were also unable to cheat, they were therefore doomed to lose the fight for survival.

The chimp making memory recall would literally see the numerals on the mental computer screen rather than remembering conceptually. TGD inspired theory of consciousness explains sensory memories in terms of direct fusion and sharing of sensory mental images associated with recent brain and brain of the geometric past. Sharing involves time like entanglement. Both time-like entanglement and the possibility of subsystems to entangle although systems remain un-entangled are purely TGD based notions and reflect the new view about space-time and quantum theory involving the notion of measurement resolution as a fundamental concept (see this).

DNA as a 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 of "Genes and Memes".

Witten and 3-D quantum gravity

Witten has written together Alex Maloney a new paper about 3-D quantum gravity. They calculate the partition function of pure gravity on an AdS3 space by summing the contributions from classical geometries, including quantum corrections, finding that “the result is not physically sensible”. They propose as a possible cure a sum over complex geometries or sum over some new kind of objects.

In a posting to Kea's blog I considered the basic problems resulting from sticking to sum over histories dogma and discussed TGD approach as an alternative. Just to enjoy the experience of becoming quoted in italics by a theoretician whose ideas I take very seriously, I glue my comments below before continuing.

This sum-over-geometries-dogmatics remains to me one of the strangest mammoth bone in theoretical physics landscape filled by ancient stuff. Sum-over-histories is nothing but an over-abstraction from Hamiltonian time evolution which has remained ill-defined even in single particle quantum mechanics with non-trivial interactions. The only justification was quantum classical correspondence working in non-perturbative situation.

Path integrals lived a high time in my youth and I of course tried also to construct quantum TGD using path integral although the failure was obvious from beginning.

Finally came the realization that the world of classical worlds must be geometrized and the idea that Einstein's geometrization program applies also to quantum theory. An idea which for some very strange reason is still waiting to be discovered by the mainstream. Also the geometrization of fermion statistics followed automatically via anti-communications of space gamma matrices and the connection with HFFs of type II_1 gives for the approach its real technical power. Despite the work of mathematicians like Connes, people simply refuse to realize the power of these algebras and try to understand their physical meaning.

The attempts to construct geometry of world for the classical worlds led also to the idea that Kahler function of world of classical assigns a unique space-time surface to the 3-surface: this just from 4-D general coordinate invariance for basic objects which are 3-D. Bohr orbitology becomes exact part of quantum theory. One could hardly dream of anything more fascinating form of quantum magic. Quantum classical correspondence in this new sense has been the strongest conceptual tool of TGD since then.

After having worked for two decades with this program and learned its amazing power as an idea generator, I cannot but just wonder how irrational a process the evolution of mainstream theoretical physics is. Some authority decides to follow the wrong path just because he does not of anything better and everyone follows. "We do not know of anything better!": isn't this the most often heard justification for string models! This kind of statements from the mouth of theoretical physicist should be criminalized;-)!

After this little bit sour self-quotation in the style of an older statesman who feels of having not quite received the attention he deserves, I continue in a similar self-referential tone by recalling my old postings commenting Witten's ideas. See this and this.

TGD indeed has strong resemblances with Witten's theory. The basic difference is that 3-geometries are replaced by light-like 3-surfaces. They are solutions of vacuum Einstein's equations in a generalized sense. Cosmological constant is however vanishing whereas Witten's geometries have negative cosmological constant which might be responsible for unphysical nature of the resulting theory: conformal field theory should not contain explicit length parameters.

The metric 2-dimensionality of the basic objects implies generalized conformal invariance and almost topological QFT property. Chern-Simons action and its fermionic counterpart are dictated by super-conformal symmetry as a definition of basic dynamics. Light-likeness is a metric notion and makes topological QFT almost topological so that one can speak about masses, etc...

Again I find myself stumbling with these mammoth bones. Colleagues refuse to take seriously a technically completely trivial but physically extremely profound generalization of 2-D conformal invariance although it explains space-time dimension among other things. Perhaps this is about insulted egos: perhaps it is too painful to admit that something went badly wrong for 24 years ago.

A further bonus is that 4-D quantum gravity emerges via quantum classical correspondence as a (not 1-1) mapping of quantum physics at 3-D lightlike surfaces to the classical physics in the interior of space-time sheets. This replaces gravitational holography in TGD framework. The construction of space-time sheet assignable to given 3-D lightlike 3-surfaces can be done in quite detailed manner now and an unexpected connection with Higgs fieds emerges.

DNA as a 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 new chapter DNA as Topological Quantum Computer of "Genes and Memes".

About Langlands program

The following represents an off-topic comment to Not-Even-Wrong which I never even considered to post. Peter Woit has a nice posting about Langlands program about which I wrote a chapter for year or so ago with emphasis on what I call number theoretic braids, on the identification of Galois group Gal of algebraic numbers as a permutation group S_∞ for infinite number of objects, and on the connection with hyper-finite factors of type II₁ emerging via the fact that the group algebra of S_∞ is HFF of type II₁.

This background perhaps explains why I had several fleeting impressions of "At some primitive level I might understand this!" while reading Peter's summary. The question about the physical interpretation of the constructs of Geometric Langlands was raised but Peter made with an admirable clarity clear that any comments relating to fundamental physics by non-names would be deleted.

1. Some ideas related to number theoretic Langlands program and their counterparts in TGD framework

The first nice idea of the number theoretic Langlands program is that rationals can be formally regarded as "rational functions" in a discrete space of primes and that the extensions of rationals can be regarded as covering spaces of this "function space" characterized by Galois groups.

In TGD framework this analogy appears in the reverse direction. Infinite rationals and their algebraic extensions (in particular infinite primes) forming an infinite hierarchy can be constructed by an iterated second quantization of an arithmetic quantum field theory as quantum states of a generalized arithmetic QFT. The reverse of analogy means that infinite rationals can be mapped to rational functions.

Langlands program has local and global aspects which in TGD framework would physically correspond to various p-adic physics and real physics (p-adic physics would relate to cognition and intentionality). Note however that p-adic space-time sheets have literally infinite size in the real sense (rational points are common to real and p-adic space-time sheets). Local aspect means that to each prime one can assign a local function field and it corresponds to a p-adic number field Q_p. The elements of Q_p are analogs of formal Laurent series which in general do not converge in real sense. The local Langlands conjecture gives a correspondence between the representations of Gal(Q_p) into a complex Lie group G and complex representations of the corresponding algebraic group ^LG(Q_p) with Q_p coefficients. The global aspect is about Gal(Q) and Langlands relates the representations of Gal(Q) in group G and the dual group ^LG (A_Q), where A_Q denotes adeles.

2. Geometric Langlands program

Witten and others are developing geometric Langlands program which is a generalization of Langlands program from number fields to holomorphic function fields defined at 2-dimensional Riemann surfaces. Now duality corresponds to electric-magnetic duality originally conjectured by Olive and Montonen that one can assign to a gauge theory dual gauge theory formulated in terms of magnetic charges regarded as gauge charges of the dual group ^LG instead of electric charges in G. It would be interesting to relate G and its dual to the groups related to the inclusions of HFFs.

Conformal field theory is a central element of the approach. In particular, disks with punctures appear as a basic notion. Some kind of 4-D theory (twisted 4-D SYM) giving rise to Chern-Simons theory giving rise to 2-D conformal field theory is conjectured to provide representations for groups and their duals.

3. Quantum theory according to TGD

Before comparison with TGD I want to emphasize that quantum theory according to TGD differs from standard quantum theory in many respects.

1. TGD can be formulated using several philosophies about what quantum physics is: quantum physics as Kähler geometry and spinor structure for the world of classical worlds; quantum physics as almost topological QFT; quantum physics as generalized number theory with associativity defining the fundamental dynamical principle; quantum physics reduced to the notion of measurement resolution formulated in terms of inclusions for HFFs of type II₁ (or possibly more general algebras).

2. Kähler geometry of the world of classical worlds provide a geometrization of quantum theory. Kähler function as a functional of light-like 3-surface is defined as Kähler action for the preferred extremal. Fermionic oscillator operators are building blocks of gamma matrices of this space and define super-generators of super-canonical algebra.

3. Zero energy ontology implies radical deviation from the standard QFT framework. S-matrix is generalized to M-matrix defined as the complex square root of density matrix with unitary S-matrix appearing as analog of phase factor. The idea is that quantum theory is square root of statistical physics. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state having interpretation as initial and final states of particle reaction.

4. Measurement resolution is realized in terms of inclusions of HFFs of type II₁ leading to a highly unique identification of M-matrix in terms of Connes tensor product. The non-uniqueness corresponds to statistical physics degrees of freedom. M-matrix has Hermitian operators of the included algebra defining measurement resolution as symmetries. These symmetries should include super-conformal symmetries.

5. Hierarchy of Planck constants realized in terms of book like structure of the imbedding space and introduction of p-adic copies of imbedding space glued to real imbedding space along common rational and algebraic points mean radical generalization of the standard quantum theory framework. Hierarchy of Planck constants brings in dark matter as a hierarchy of macroscopically quantum coherent phases and p-adic physics gives rise to the correlates of intentionality and cognition.

4. Resemblance with quantum TGD identified as almost topological conformal QFT

Geometric Langlands program brings strongly to my uneducated mind TGD as an almost topological QFT with generalized conformal symmetries (light-like 3-surfaces are metrically 2-dimensional). 4-D twisted SYM would be replaced with Kähler action plus holography in the restricted sense that 4-D physics provides a (non-faithful) classical representation of the fundamental 3-D lightlike partonic quantum physics in terms of the preferred extrema of Kähler action having identification as generalized Bohr orbits. This is necessary for quantum measurement theory to make sense at the fundamental level.

Punctures, whose interpretation remains unclear in the Langlands program, associate themselves naturally with the intersections of the initial and final partonic 2-surface of particle reaction with the generalized number theoretic braids (strands can now fuse) defined by the orbits of the minima of Higgs expectation identified as a generalized eigenvalue of the modified Dirac operator (certain holomorphic function on partonic 2-surface by the properties of modified Dirac). Partons correspond to quantum states created by applying fermionic fields at the points of the number theoretic braid so that one has a concrete physical interpretation. TGD as a generalized number theory vision of course relies on the idea that fundamental physics provides a representation for number theory understood in some very general sense.

Physically punctures correspond to the lowest step in a dimensional hierarchy. Light-like 3-surface decomposes to cells bounded by 2-D surfaces such that each 3- region is independent dynamical unit. One has effectively discretized 3-D physics. The resulting 2-D surfaces (partons) obey conformal field theory separately and a collection of 1-D curves serves as causal determinants for them. Number theoretic universality in turn forces to select Higgs minima as a subset of points common to real and p-adic partonic 2-surfaces.

What is interesting that the number theoretical braids emerge in the TGD based proposal for the formulation of the number theoretic Langlands program based on Gal(Q)= S_∞ identification and relates also naturally to the conformal field approach appearing in the geometric Langlands. Could number theoretic braids allow to unify number theoretic and geometric Langlands as the unification of number fields and rational function fields provided by the notion of infinite prime suggests? This I cannot of course answer. These are just ideas inspired by the physics of TGD. It would be flattering if some real mathematician would consider these ideas seriously. I dare to believe that since TGD seems to be a working physical theory it could help also to discover "working" mathematics.

The work of Kanarev and Mizuno about cold fusion in electrolysis

The article of Kanarev and Mizuno [1] reports findings supporting the occurrence of cold fusion in NaOH and KOH hydrolysis. The situation is different from standard cold fusion where heavy water D₂O is used instead of H₂O.

1. One can understand the cold fusion reactions reported by Mizuno as nuclear reactions in which part of what I call dark proton string having negatively charged color bonds (essentially a zoomed up variant of ordinary nucleus with large Planck constant) suffers a phase transition to ordinary matter and experiences ordinary strong interactions with the nuclei at the cathode. In the simplest model the final state would contain only ordinary nuclear matter.

2. Negatively charged color bonds could correspond to pairs of quark and antiquark or to pairs of color octet electron and antineutrino having mass of order 1 MeV. Also quantum superpositions of quark and lepton pairs can be considered. Note that TGD predicts that leptons can have colored excitations and production of neutral leptopions formed from them explains the anomalous production of electron-positron pairs associated with heavy ion collisions near Coulomb wall.

3. The so called H_1.5O anomaly of [2] can be understood if 1/4 of protons of water forms dark lithium nuclei or heavier nuclei formed as sequences of these just as ordinary nuclei are constructed as sequences of ⁴He and lighter nuclei in nuclear string model. The results force to consider the possibility that nuclear isotopes unstable as ordinary matter can be stable dark matter. In the formation of these sequence the negative electronic charge of hydrogen atoms goes naturally to the color bonds. The basic interaction would generate charge quark pair (or a pair of color octet electron and antineutrino or a quantum superposition of quark and lepton pair) plus color octet neutrino. By lepton number conservation each electron pair would give rise to a color singlet particle formed by two color octet neutrinos and defining the analog of leptobaryon. Di-neutrino would leave the system unless unless it has large enough mass. Neutrino mass scale .1 eV gives for the Compton time scale the estimate .1 attoseconds which would suggest that di-neutrinos do not leak out. Recall that attosecond is the time scale in which H_1.5O behavior prevails.

4. The data of Mizuno requires that the protonic strings have net charge of three units and by em stability have neutral color bonds at ends and negatively charged bonds in between. Dark variants of Li isotopes would be in question. The so called lithium problem of cosmology (the observed abundance of lithium is by a factor 2.5 lower than predicted by standard cosmology [3]) can be resolved if lithium nuclei transform partially to dark lithium nuclei.

5. Biologically important ions K⁺, Cl^-, Ca⁺⁺ appear in cathode in plasma electrolysis and would be produced in cold nuclear reactions of dark Li nuclei of water and Na⁺. This suggests that cold nuclear reactions occur also in living cell and produce metabolic energy. There exists evidence for nuclear transmutations in living matter [4]. In particular, Kervran claims that it is very difficult to understand where the Ca in egg shells comes from. Cell membrane would provide the extremely strong electric field perhaps creating the plasma needed for cold nuclear reactions somewhat like in plasma electrolysis.

6. The model is consistent with the model for cold fusion of deuterium nuclei [5]. In this case nuclear reaction would however occur on the "dark side". The absence of He from reaction products can be understood if the D nuclei in Pd target are transformed by weak interactions between D and Pd nuclei to their neutral counterparts analogous to di-neutrons. Neutral color bond could transform to negatively charged one by the exchange of W⁺ boson of a scaled version of weak interactions with the range of interaction given by atomic length scale. Also exchange of charge ρ meson of scaled down variant of QCD could affect the same thing. This interaction might be at work also for ordinary nuclei in condensed matter and ordinary nuclei could contain protons and negatively charged color bonds neutrons. The difference in mass would be very small since the quarks have mass of order MeV.

The model leads also to a new understanding of ordinary [6] and plasma electrolysis of water [7], and allows to identify hydrogen bond as dark OH bond.

1. The model for plasma hydrolysis relies on the observation of Kanarev that the energy of OH bonds in water is reduce from about 8 eV to a value around .5 eV which corresponds to the fundamental metabolic energy quantum resulting in dropping of proton from atomic k=137 space-time sheet and also to a typical energy of hydrogen bond. This suggests the possibility that hydrogen bond is actually a dark OH bond. From 1/hbar-proportionality of perturbative contribution of Coulomb energy for bond one obtains that dark bond energy scales as 1/hbar so that dark OH bond could be in question. In Kanarev's plasma electrolysis the temperature is between .5-1 eV and thermal radiation could induce producing 2H₂+O₂ by the splitting of the dark OH bonds. One could have hbar=2⁴×hbar₀. Also in the ordinary electrolysis the OH bond energy is reduced by a factor of order 2 which suggest that in this case one has hbar=2×hbar₀.

2. The transformation of OH bonds to their dark counterparts requires energy and this energy would come from dark nuclear reactions. The liberated (dark) photons could kick protons from (dark) atomic space-time sheets to smaller space-time sheets and remote metabolism would provide the energy for the transformation of OH bond. The existence of dark hydrogen bonds with energies differing by integer scaling is predicted and powers of 2 are favored. It is known that at least two hydrogen bonds for which energies differ by factor 2 exist in ice [8].

3. In plasma electrolysis the increase of the input voltage implies a mysterious reduction of the electron current with the simultaneous increase of the size of the plasma region near the cathode. The electronic charge must go somewhere and the natural place are negative color bonds connecting dark protons to dark lithium isotopes. The energy liberated in cold nuclear reactions would create plasma by ionizing hydrogen atoms which in turn would generate more dark protons fused to dark lithium isotopes and increase the rate of energy production by dark nuclear reactions. This means a positive feedback loop analogous to that occurring in ordinary nuclear reactions.

The model explains also the burning of salt water discovered by Kanzius [9] as a special case of plasma electrolysis since the mechanism does not necessitate the presence of either anode, cathode, or electron current.

1. The temperature of the flame is estimated to be 1500 C. The temperature in water could be considerably higher and 1500 C defines a very conservative estimate. Hydrolysis would be preceded by the transformation of HO bonds to hydrogen bonds and dark nuclear reactions would provide the energy. Again positive feedback loop should be created. Dark radio wave photons would transform to microwave photons and together with nuclear energy production would keep the water at the temperature corresponding to the energy of.017 eV (for conservative estimate T=.17 eV in water) so that dark OH bonds would break down thermally.

2. For T=1500 C the energy of dark OH bond (hydrogen bond) would be very low, around .04 eV for hbar=180×hbar₀ and nominal value 8 eV OH bond energy (this is not far from the energy assignable to the membrane resting potential) from the condition that dark radio wave frequency 13.65 MHz corresponds to the microwave frequency needed to heat water by the rotational excitation of water molecules.

3. Visible light would result as dark protons drop from k=165 space-time sheet to any larger space-time sheet or from k=164 to k=165 space-time sheet (2 eV radiation). 2 eV photons would explain the yellow color in the flame (not red as I have claimed earlier). The red light present in Kanarev's experiment can be also understood since there is entire series E(n)= E_∞× (1-2^-n) of energies corresponding to transitions to space-time sheets with increasing p-adic length scale. For k=165 n<6 corresponds to red or infrared light and n>5 to yellow light.

4. There is no detectable or perceivable effect on hand by the radio wave radiation. The explanation would be that dark hydrogen bonds in cellular water correspond to a different values of Planck constant. One should of course check whether the effect is really absent.

For more details see the chapter Nuclear String Hypothesis.

References

[1] Cold fusion by plasma electrolysis of water, Ph. M. Kanarev and T. Mizuno (2002),
http://www.guns.connect.fi/innoplaza/energy/story/Kanarev/coldfusion/.

[2] M. Chaplin (2005), Water Structure and Behavior,
http://www.lsbu.ac.uk/water/index.html.
For 41 anomalies see http://www.lsbu.ac.uk/water/anmlies.html.
For the icosahedral clustering see http://www.lsbu.ac.uk/water/clusters.html.
J. K. Borchardt(2003), The chemical formula H₂O - a misnomer, The Alchemist 8 Aug (2003).
R. A. Cowley (2004), Neutron-scattering experiments and quantum entanglement, Physica B 350 (2004) 243-245.
R. Moreh, R. C. Block, Y. Danon, and M. Neumann (2005), Search for anomalous scattering of keV neutrons from H₂O-D₂O mixtures, Phys. Rev. Lett. 94, 185301.

[3] C. Charbonnel and F. Primas (2005), The lithium content of the Galactic Halo stars.
See also Lithium.

[4]C. L. Kervran (1972), Biological transmutations, and their applications in chemistry, physics, biology, ecology, medicine, nutrition, agriculture, geology, Swan House Publishing Co.
P. Tompkins and C. Bird (1973), The secret life of plants, Harper and Row, New York.

[5] Cold fusion is back at the American Chemical Society.
See also Cold fusion - hot news again .

[6] Electrolysis of water.

[7] P. Kanarev (2002), Water is New Source of Energy, Krasnodar.

[8] J-C. Li and D.K. Ross (1993), Evidence of Two Kinds of Hydrogen Bonds in Ices. J-C. Li and D.K. Ross, Nature, 365, 327-329.

[9] Burning salt water.

The Lie-algebra of symmetries of M-matrix forms an infinite-dimensional Jordan algebra

The notion of Jordan algebra was born as a mathematization of algebra of observables. Also Lie algebra can be seen as this kind of mathematization of the notion of observable. The linear combinations of Hermitian operators with real coefficients are Hermitian and define an algebra under the product A*B= (AB+BA)/2, which is commutative and non-associative but satisfies the weaker associativity condition (xy)(xx)= x(y(xx)).

There exists four infinite families of Jordan algebras plus one exceptional Jordan algebra. The finite-dimensional real, complex, and quaternionic matrix algebras with product defined as above are Jordan algebras. Also the Euclidian gamma matrix algebra defined by Euclidian inner product and with real coefficients is Jordan algebra and known as so called spin factor: now the commutativity is not put in by hand. The exceptional Jordan algebra consists of a real linear space of Hermitian 3× 3 matrices with octonionic coefficients and with symmetrized product.

1. The notion of finite measurement resolution leads to infinite-dimensional Jordan algebra in TGD framework

Hermitian operators of N subset M, where N and M are hyperfinite factors of type II₁ and N specifies the measurement resolution, act as maximal symmetries of M-matrix so that finite measurement resolution corresponds to an infinite-D symmetry group and Jordan algebra corresponds now to operators whose action has no detectable physical effect rather than algebra of observables. A so called Hermitian Jordan algebra is in question. Of course, also Lie-algebra commutator i(AB-BA) defines a Hermitian operator in N.

The maximal symmetries of M-matrix mean that the Hermitian generators of the algebra define a generalization of finite-dimensional Jordan algebra. The condition that all Hermitian operators involved are finite-dimensional brings in mind the definition of the permutation group S_∞ as consisting of finite permutations only and also the definition of infinite-dimensional Clifford algebra. Thus the natural interpretation of the algebra in question would be as maximal possible dynamical gauge symmetry implied by the finite measurement resolution. The active symmetries would be analogous to global gauge transformations and act non-trivially on all tensor factors in tensor product representation as a tensor product of 2× 2 Clifford algebras.

Quaternionic Jordan algebra is natural in TGD framework since 2× 2 Clifford algebra reduces to complexified quaternions and contains as sub-algebras real and complex Jordan algebras. Also Clifford algebra of world of classical worlds is a generalized Jordan algebra.

2. Octonions and TGD

There are intriguing hints that octonions might be important for TGD.

1. U(1), SU(2), and SU(3) are the factors of standard model gauge group and also the natural symmetries of minimal Jordan algebras relying on complex numbers, quaternions, and octonions. These symmetry groups relate also naturally to the geometry of CP₂.

2. 8-D Clifford algebra allows also octonionic representation.

3. The idea that one could make HFF of type II₁ a genuine local algebra analogous to gauge algebra can be realized only if the coordinate is non-associative since otherwise the coordinate can be represented as a tensor factor represented by a matrix algebra. Octonionic coordinate means an exception and would make 8-D imbedding space unique in that it would allow local version of HFF of type II₁.

4. These observations partially motivate a nebulous concept that I have christened HO-H duality (see this) - admittedly a rather speculative idea -stating that TGD can be formulated alternatively using hyper-octonions (subspace of complexified octonions with Minkowskian signature of metric) as the imbedding space and assuming that the dynamics is determined by the condition that space-time surfaces are hyper-quaternionic or co-hyper-quaternionic (and thus associative or co-associative). Associativity condition would determine the dynamics.

3. What about the octonionic Jordan algebra?

The question is therefore whether also 3×3 octonionic Jordan algebra might have some role in TGD framework.

1. Suppose for a moment that the above interpretation for the Hermitian operators as elements of a sub-factor N defining the measurement resolution generalizes also to the case of octonionic state space and operators represented as octonionic matrices. Also the direct sums of octonion valued matrices belonging to the octonionic Jordan algebra define a Jordan algebra and included algebras would now correspond to direct sums for copies of this Jordan algebra. One could perhaps say that the gauge symmetries associated with octonionic N would reduce to the power SU(3)_oⁿ= SU(3)_o× SU(3)_o×... of the octonionic SU(3) acting on the fundamental triplet representation.

2. Triplet character is obviously problematic and one way out could be projectivization leading to the octonionic counterpart of CP₂. Octonionic scalings should not affect the physical state so that physical states as octonionic rays would correspond to octonionic CP_n. It is not however possible to realize the linear superposition of quantum states in CP_n. The octonionic (quaternionic) counterpart of CP₂ would be 2× 8-dimensional and U(2)_o would act as a matrix multiplication in this space. Realizing associativity (commutativity) condition for 2× 8 spinors defined by octonionic CP₂ by replacing octonions with quaternions (complex numbers) would give 2× 4-dimensional (2× 2-dimensional) space.

3. The first question is whether CP₂ as a factor of imbedding space could somehow relate to the octonionic Jordan algebra. Could one think that this factor relates to the configuration space degrees of freedom assignable to CP₂ rather than Clifford algebra degrees of freedom? That color does not define spin like quantum numbers in TGD would conform with this. Note that the partial waves associated S² associated with light-cone boundary would correspond naturally to SU(2) and quaternionic algebra.

4. Second question is whether the HFF of type II₁ could result from its possibly existing octonionic generalization by these two steps and whether the reduction of the octonionic symmetries to complex situation would give SU(3)× SU(3)... reducing to U(2)× U(2)× .... The Lie-algebra of symmetries of M-matrix forms a Jordan algebra.

For a background see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

Comments about E8 theory of Garrett Lisi

I have been a week in travel and during this time there has been a lot of fuss about the E₈ theory proposed by Garrett Lisi in physics blogs such as Not-Even-Wrong and Reference Frame, in media, and even New Scientist wrote about the topic. I have been also asked to explain whether there is some connection between Lisi's theory and TGD.

1. Objections against Lisi’s theory

The basic claim of Lisi is that one can understand the particle spectrum of standard model in terms of the adjoint representation of a noncompact version of E₈ group.

There are several objections against E₈ gauge theory interpretation of Lisi.

1. Statistics does not allow to put fermions and bosons in the same gauge multiplet. Also the identification of graviton as a part of a gauge multiplet seems very strange if not wrong since there are no roots corresponding to a spin 2 two state.

2. Gauge couplings come out wrong for fermions and one must replace YM action with an ad hoc action.

3. Poincare invariance is a problem. There is no clear relationship with the space-time geometry so that the interpretation of spin as E₈ quantum numbers is not really justified.

4. Finite-dimensional representations of non-compact E₈ are non-unitary. Non-compact gauge groups are however not possible since one would need unitary infinite-dimensional representations which would change the physical interpretation completely. Note that also Lorentz group has only infinite-D unitary representations and only the extension to Poincare group allows to have fields transforming according to finite-D representations.

5. The prediction of three fermion families is nice but one can question the whole idea of putting particles with mass scales differing by a factor of order 10¹² (top and neutrinos) into same multiplet. For some reason colleagues stubbornly continue to see fundamental gauge symmetries where there seems to be no such symmetry. Accepting the existence of a hierarchy of mass scales seems to be impossible for a theoretical physicistin main main stream although fractals have been here for decades.

6. Also some exotic particles not present in standard model are predicted: these carry weak hyper charge and color (6-plet representation) and are arranged in three families.

2. Three attempts to save Lisi’s theory

To my opinion, the shortcomings of E₈ theory as a gauge theory are fatal but the possibility to put gauge bosons and fermions of the standard model to E₈ multiplets is intriguing and motivatse the question whether the model could be somehow saved by replacing gauge theory with a theory based on extended fundamental objects possessing conformal invariance.

1. In TGD framework H-HO duality allows to consider Super-Kac Moody algebra with rank 8 with Cartan algebra assigned with the quantized coordinates of partonic 2-surface in 8-D Minkowski space M⁸ (identifiable as hyper-octonions HO). The standard construction for the representations of simply laced Kac-Moody algebras allows quite a number of possibilities concerning the choice of Kac-Moody algebra and the non-compact E₈ would be the maximal choice.

2. The first attempt to rescue the situation would be the identification of the weird spin 1/2 bosons in terms of supersymmetry involving addition of righthanded neutrino to the state giving it spin 1. This options does not seem to work.

3. The construction of representations of non-simply laced Kac-Moody algebras (performed by Goddard and Olive at eighties) leads naturally to the introduction of fermionic fields for algebras of type B, C, and F: I do not know whether the construction has been made for G₂. E₆, E₇, and E₈ are however simply laced Lie groups with single root length 2 so that one does not obtain fermions in this manner.

4. The third resuscitation attempt is based on fractional statistics. Since the partonic 2-surfaces are 2-dimensional and because one has a hierarchy of Planck constants, one can have also fractional statistics. Spin 1/2 gauge bosons could perhaps be interpreted as anyonic gauge bosons meaning that particle exchange as permutation is replaced with braiding homotopy. If so, E₈ would not describe standard model particles and the possibility of states transforming according to its representations would reflect the ability of TGD to emulate any gauge or Kac-Moody symmetry.

   The standard construction for simply laced Kac-Moody algebras might be generalized considerably to allow also more general algebras and fractionization of spin and other quantum numbers would suggest fractionization of roots. In stringy picture the symmetry group would be reduced considerably since longitudinal degrees of freedom (time and one spatial direction) are unphysical. This would suggest a symmetry breaking to SO(1,1)× E₆ representations with ground states created by tachyonic Lie allebra generators and carrying mass squared –2 in suitable units. In TGD framework the tachyonic conformal weight can be compensated by super-canonical conformal weight so that massless states getting their masses via Higgs mechanism and p-adic thermodynamics would be obtained.

3. Could supersymmetry rescue the situation?

E₈ is unique among Lie algebras in that its adjoint rather than fundamental representation has the smallest dimension. One can decompose the 240 roots of E₈ to 112 roots for which two components of SO(7,1) root vector are +/- 1 and to 128 vectors for which all components are +/- 1/2 such that the sum of components is even. The latter roots Lisi assigns to fermionic states. This is not consistent with spin and statistics although SO(3,1) spin is half-integer in M⁸ picture.

The first idea which comes in mind is that these states correspond to super-partners of the ordinary fermions. In TGD framework they might be obtained by just adding covariantly constant right-handed neutrino or antineutrino state to a given particle state. The simplest option is that fermionic super-partners are complex scalar fields and sbosons are spin 1/2 fermions. It however seems that the super-conformal symmetries associated with the right-handed neutrino are strictly local in the sense that global super-generators vanish. This would mean that super-conformal super-symmetries change the color and angular momentum quantum numbers of states. This is a pity if indeed true since super-symmetry could be broken by different p-adic mass scale for super partners so that no explicit breaking would be needed.

4. Could Kac Moody variant of E₈ make sense in TGD?

One can leave gauge theory framework and consider stringy picture and its generalization in TGD framework obtained by replacing string orbits with 3-D light-like surfaces allowing a generalization of conformal symmetries.

H-HO duality is one of the speculative aspects of TGD. The duality states that one can either regard imbedding space as H=M⁴×CP₂ or as 8-D Minkowski space M⁸ identifiable as the space HO of hyper-octonions which is a subspace of complexified octonions. Spontaneous compactification for M⁸ described as a phenomenon occurring at the level of Kac-Moody algebra would relate HO-picture to H-picture which is definitely the fundamental picture. For instance, standard model symmetries have purely number theoretic meaning in the resulting picture.

The question is whether the non-compact E₈ could be replaced with the corresponding Kac Moody algebra and act as a stringy symmetry. Note that this would be by no means anything new. The Kac-Moody analogs of E₁₀ and E₁₁ algebras appear in M-theory speculations. Very little is known about these algebras. Already E_n, n>8 is infinite-dimensional as an analog of Lie algebra. The following argument shows that E₈ representations do not work in TGD context unless one allows anyonic statistics.

1. In TGD framework space-time dimension is D=8. The speculative hypothesis of HO-H duality (see this) inspired by string model dualities states that the descriptions based on the two choices of imbedding space are dual. One can start from 8-D Cartan algebra defined by quantized M⁸ coordinates regarded as fields at string orbit just as in string model. A natural constraint is that the symmetries act as isometries or holonomies of the effectively compactified M⁸. The article Octonions of John Baez discusses exceptional Lie groups and shows that compact form of E₈ appears as isometry group of 16-dimensional octo-octonionic projective plane E₈/(Spin(16)Z₂): the analog of CP₂ for complexified octonions. There is no 8-D space allowing E₈ as an isometry group. Only SO(1,7) can be realized as the maximal Lorentz group with 8-D translational invariance.

2. In HO picture some Kac Moody algebra with rank 8 acting on quantized M⁸ coordinates defining stringy fields is natural. The charged generators of this algebra are constructible using the standard recipe involving operators creating coherent states and their conjugates obtained as operator counterparts of plane waves with momenta replaced by roots of the simply laced algebra in question and by normal ordering.

3. Poincare group has 4-D maximal Cartan algebra and this means that only 4 Euclidian dimensions remain. Lorentz generators can be constructed in standard manner in terms of Kac-Moody generators as Noether currents.

4. The natural Kac-Moody counterpart for spontaneous compactification to CP₂ would be that these dimensions give rise to the generators of electroweak gauge group identifiable as a product of isometry and holonomy groups of CP₂ in the dual H-picture based on M⁴×CP₂. Note that in this picture electroweak symmetries would act geometrically in E⁴ whereas in CP₂ picture they would act only as holonomies.

Could one weaken the assumption that Kac-Moody generators act as symmetries and that spin-statistics relation would be satisfied?

1. The hierarchy of Planck constants relying on the generalization of the notion of imbedding space breaks Poincare symmetry to Lorentz symmetry for a given sector of the world of classical worlds for which one considers light-like 3-surfaces inside future and past directed light cones. Translational invariance is obtained from the wave function for the position of the tip of the light cone in M⁴. In this kind of situation one could consider even E₈ symmetry as a dynamical symmetry.

2. The hierarchy of Planck constants involves a hierarchy of groups and fractional statistics at the partonic 2-surface with rotations interpreted as braiding homotopies. The fractionization of spin allows anyonic statistics and could allow bosons with anyonic half-odd integer spin. Also more general fractional spins are possible so that one can consider also more general algebras than Kac-Moody algebras by allowing roots to have more general values. Quantum versions of Kac-Moody algebras would be in question. This picture would be consistent with the view that TGD can emulate any gauge algebra with 8-D Cartan algebra and Kac-Moody algebra dynamically. This vision was originally inspired by the study of the inclusions of hyper-finite factors of type II₁. Even higher dimensional Kac-Moody algebras are predicted to be possible.

3. It must be emphasized that these considerations relate in TGD framework to Super-Kac Moody algebra only. The so called super-canonical algebra is the second quitessential part of the story. In particular, color is not spinlike quantum number for quarks and quark color corresponds to color partial waves in the world of classical worlds or more concretely, to the rotational degrees of freedom in CP₂ analogous to ordinary rotational degrees of freedom of rigid body. Arbitrarily high color partial waves are possible and also leptons can move in triality zero color partial waves and there is a considerable experimental evidence for color octet excitations of electron and muon but put under the rug.

5. Can one interpret three fermion families in terms of E₈ in TGD framework?

The prediction of three fermion generations by E₈ picture must be taken very seriously. In TGD three fermion generations correspond to three lowest genera g=0,1,2 (handle number) for which all 2-surfaces have Z₂ as global conformal symmetry (hyper-ellipticity). One can assign to the three genera a dynamical SU(3) symmetry. They are related by SU(3) triality, which brings in mind the triality symmetry acting on fermion generations in E₈ model. SU(3) octet and singlet bosons correspond to pairs of light-like 3-surfaces defining the throats of a wormhole contact and since their genera can be different one has color singlet and octet bosons. Singlet corresponds to ordinary bosons. Color octet bosons must be heavy since they define neutral currents between fermion families.

The three E₈ anyonic boson families cannot represent family replication since these symmetries are not local conformal symmetries: it obviously does not make sense to assign a handle number to a given point of partonic 2-surface! Also bosonic octet would be missing in E₈ picture.

One could of course say that in E₈ picture based on fractional statistics, anyonic gauge bosons can mimic the dynamical symmetry associated with the family replication. This is in spirit with the idea that TGD Universe is able to emulate practically any gauge - or Kac-Moody symmetry and that TGD Universe is busily mimicking also itself.

To sum up, the rank 8 Kac-Moody algebra - emerging naturally if one takes HO-H duality seriously - corresponds very naturally to Kac-Moody representations in terms of free stringy fields for Poincare-, color-, and electro-weak symmetries. One can however consider the possibility of anyonic symmetries and the emergence of non-compact version of E₈ as a dynamical symmetry, and TGD suggests much more general dynamical symmetries if TGD Universe is able to act as the physics analog of the Universal Turing machine.

For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts of "TGD as Generalized Number Theory".

References

[1] G. Lisi (2007), An exceptionally simple theory of everything,

[2] Z. Merali (1007), Is mathematical pattern the theory of everything?, New Scientist issue 2630.

[3] E₈ .

[4] J. Baez (2002), The Octonions.

Cosmic rays above GKZ bound from distant galactic nuclei

Lubos tells about the announcement of Pierre Auger Collaboration relating to ultrahigh energy cosmic rays. I glue below a popular summary of the findings.

Scientists of the Pierre Auger Collaboration announced today (8 Nov. 2007) that active galactic nuclei are the most likely candidate for the source of the highest-energy cosmic rays that hit Earth. Using the Pierre Auger Observatory in Argentina, the largest cosmic-ray observatory in the world, a team of scientists from 17 countries found that the sources of the highest-energy particles are not distributed uniformly across the sky. Instead, the Auger results link the origins of these mysterious particles to the locations of nearby galaxies that have active nuclei in their centers. The results appear in the Nov. 9 issue of the journal Science.

Active Galactic Nuclei (AGN) are thought to be powered by supermassive black holes that are devouring large amounts of matter. They have long been considered sites where high-energy particle production might take place. They swallow gas, dust and other matter from their host galaxies and spew out particles and energy. While most galaxies have black holes at their center, only a fraction of all galaxies have an AGN. The exact mechanism of how AGNs can accelerate particles to energies 100 million times higher than the most powerful particle accelerator on Earth is still a mystery.

About million cosmic ray events have been recorded and 80 of them correspond to particles with energy above the so called GKZ bound, which is .54 × 10¹¹ GeV. Electromagnetically interacting particles with these energies from distant galaxies should not be able to reach Earth. This would be due to the scattering from the photons of the microwave background. About 20 particles of this kind however comes from the direction of distant active galactic nuclei and the probability that this is an accident is about 1 per cent. Particles having only strong interactions would be in question. The problem is that this kind of particles are not predicted by the standard model (gluons are confined).

1. What does TGD say about the finding?

TGD provides an explanation for the new kind of particles.

1. The original TGD based model for the galactic nucleus is as a highly tangled cosmic string (in TGD sense of course, see this). Much later it became clear that also TGD based model for black-hole is as this kind of string like object near Hagedorn temperature (see this and this). Ultrahigh energy particles could result as decay products of a decaying split cosmic string as an extremely energetic galactic jet. Kind of cosmic fire cracker would be in question. Originally I proposed this decay as an explanation for the gamma ray bursts. It seems that gamma ray bursts however come from thickened cosmic strings having weaker magnetic field and much lower energy density (see this).

2. TGD predicts particles having only strong interactions (see this). I have christened these particles super-canonical quanta. These particles correspond to the vibrational degrees of freedom of partonic 2-surface and are not visible at the quantum field theory limit for which partonic 2-surfaces become points.

2. What super-canonical quanta are?

Super-canonical quanta are created by the elements of super-canonical algebra, which creates quantum states besides the super Kac-Moody algebra present also in super string model. Both algebras relate closely to the conformal invariance of light-like 3-surfaces.

1. The elements of super-canonical algebra are in one-one correspondence with the Hamiltonians generating symplectic transformations of δM⁴₊× CP₂. Note that the 3-D light-cone boundary is metrically 2-dimensional and possesses degenerate symplectic and Kähler structures so that one can indeed speak about symplectic (canonical) transformations.

2. This algebra is the analog of Kac-Moody algebra with finite-dimensional Lie group replaced with the infinite-dimensional group of symplectic transformations (see this). This should give an idea about how gigantic a symmetry is in question. This is as it should be since these symmetries act as the largest possible symmetry group for the Kähler geometry of the world of classical worlds (WCW) consisting of light-like 3-surfaces in 8-D imbedding space for given values of zero modes (labelling the spaces in the union of infinite-dimensional symmetric spaces). This implies that for the given values of zero modes all points of WCW are metrically equivalent: a generalization of the perfect cosmological principle making theory calculable and guaranteing that WCW metric exists mathematically. Super-canonical generators correspond to gamma matrices of WCW and have the quantum numbers of right handed neutrino (no electro-weak interactions). Note that a geometrization of fermionic statistics is achieved.

3. The Hamiltonians and super-Hamiltonians have only color and angular momentum quantum numbers and no electro-weak quantum numbers so that electro-weak interactions are absent. Super-canonical quanta however interact strongly.

3. Also hadrons contain super-canonical quanta

One can say that TGD based model for hadron is at space-time level kind of combination of QCD and old fashioned string model forgotten when QCD came in fashion and then transformed to the highly unsuccessful but equally fashionable theory of everything.

1. At quantum level the energy corresponding to string tension explaining about 70 per cent of proton mass corresponds to super-canonical quanta (see this). Supercanonical quanta allow to understand hadron masses with a precision better than 1 per cent.

2. Super-canonical degrees of freedom allow also to solve spin puzzle of the proton: the average quark spin would be zero since same net angular momentum of hadron can be obtained by coupling quarks of opposite spin with angular momentum eigen states with different projection to the direction of quantization axis.

3. If one considers proton without valence quarks and gluons, one obtains a boson with mass very nearly equal to that of proton (for proton super-canonical binding energy compensates quark masses with high precision). These kind of pseudo protons might be created in high energy collisions when the space-time sheets carrying valence quarks and super-canonical space-time sheet separate from each other. Super-canonical quanta might be produced in accelerators in this manner and there is actually experimental support for this from Hera (see this).

4. The exotic particles could correspond to some p-adic copy of hadron physics predicted by TGD and have very large mass smaller however than the energy. Mersenne primes M_n= 2ⁿ-1 define excellent candidates for these copies. Ordinary hadrons correspond to M₁₀₇. The protons of M₃₁ hadron physics would have the mass of proton scaled up by a factor 2^(107-31)/2=2³⁸≈ 2.6×10¹¹. Energy should be above 2.6 × 10¹¹ GeV and is above .54 × 10¹¹ GeV for the particles above the GKZ limit. Even super-canonical quanta associated with proton of this kind could be in question. Note that CP₂ mass corresponds roughly to about 10¹⁴ proton masses.

5. Ideal blackholes would be very long highly tangled string like objects, scaled up hadrons, containing only super-canonical quanta. Hence it would not be surprising if they would emit super-canonical quanta. The transformation of supernovas to neutron stars and possibly blackholes would involve the fusion of hadronic strings to longer strings and eventual annihilation and evaporation of the ordinary matter so that only super-canonical matter would remain eventually. A wide variety of intermediate states with different values of string tension would be possible and the ultimate blackhole would correspond to highly tangled cosmic string. Dark matter would be in question in the sense that Planck constant could be very large.