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TGD as a Generalized Number Theory

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Year 2008



Infinite primes and algebraic Brahman Atman identity

The hierarchy of infinite primes (and of integers and rationals) was the first mathematical notion stimulated by TGD inspired theory of consciousness. The construction recipe is equivalent with a repeated second quantization of super-symmetric arithmetic quantum field theory with bosons and fermions labeled by primes such that the many particle states of previous level become the elementary p"../articles/ of new level. The hierarchy of space-time sheets with many particle states of space-time sheet becoming elementary p"../articles/ at the next level of hierarchy and also the hierarchy of n:th order logics are also possible correlates for this hierarchy. For instance, the description of proton as an elementary fermion would be in a well defined sense exact in TGD Universe.

This construction leads also to a number theoretic generalization of space-time point since given real number has infinitely rich number theoretical structure not visible at the level of the real norm of the number a due to the existence of real units expressible in terms of ratios of infinite integers. This number theoretical anatomy suggest kind of number theoretical Brahman=Atman principle stating that the set consisting of number theoretic variants of single point of the imbedding space (equivalent in real sense) is able to represent the points of the world of classical worlds or even quantum states of the Universe . Also a formulation in terms of number theoretic holography is possible.

Just for fun and to test these ideas one can consider a model for the representation of the configuration space spinor fields in terms of algebraic holography. I have considered guesses for this kind of map earlier and it is interesting to find whether additional constraints coming from zero energy ontology and finite measurement resolution might give. The identification of quantum corrections as insertion of zero energy states in time scale below measurement resolution to positive or negative energy part of zero energy state and the identification of number theoretic braid as a space-time correlate for the finite measurement resolution give considerable additional constraints.

  1. The fundamental representation space consists of wave functions in the Cartesian power U8 of space U of real units associated with any point of H. That there are 8 real units rather than one is somewhat disturbing: this point will be discussed below. Real units are ratios of infinite integers having interpretation as positive and negative energy states of a super-symmetric arithmetic QFT at some level of hierarchy of second quantizations. Real units have vanishing net quantum numbers so that only zero energy states defining the basis for configuration space spinor fields should be mapped to them. In the general case quantum superpositions of these basis states should be mapped to the quantum superpositions of real units. The first guess is that real units represent a basis for configuration space spinor fields constructed by applying bosonic and fermionic generators of appropriate super Kac-Moody type algebra to the vacuum state.

  2. What can one say about this map bringing in mind Gödel numbering? Each pair of bosonic and corresponding fermionic generator at the lowest level must be mapped to its own finite prime. If this map is specified, the map is fixed at the higher levels of the hierarchy. There exists an infinite number of this kind of correspondences. To achieve some uniqueness, one should have some natural ordering which one might hope to reflect real physics. The irreps of the (non-simple) Lie group involved can be ordered almost uniquely. For simple group this ordering would be with respect to the sum N=NF+NF,c of the numbers NF resp. NF,c of the fundamental representation resp. its conjugate appearing in the minimal tensor product giving the irrep. The generalization to non-simple case should use the sum of the integers Ni for different factors for factor groups. Groups themselves could be ordered by some criterion, say dimension. The states of a given representation could be mapped to subsequent finite primes in an order respecting some natural ordering of the states by the values of quantum numbers from negative to positive (say spin for SU(2) and color isospin and hypercharge for SU(3)). This would require the ordering of the Cartesian factors of non-simple group, ordering of quantum numbers for each simple group, and ordering of values of each quantum number from positive to negative.

  3. The presence of conformal weights brings in an additional complication. One cannot use conformal as a primary orderer since the number of SO(3)×SU(3) irreps in the super-canonical sector is infinite. The requirement that the probabilities predicted by p-adic thermodynamics are rational numbers or equivalently that there is a length scale cutoff, implies a cutoff in conformal weight. The vision about M-matrix forces to conclude that different values of the total conformal weight n for the quantum state correspond to summands in a direct sum of HFFs. If so, the introduction of the conformal weight would mean for a given summand only the assignment n conformal weights to a given Lie-algebra generator. For each representation of the Lie group one would have n copies ordered with respect to the value of n and mapped to primes in this order.

  4. Cognitive representations associated with the points in a subset, call it P, of the discrete intersection of p-adic and real space-time sheets, defining number theoretic braids, would be in question. Large number of partonic surfaces can be involved and only few of them need to contribute to P in the measurement resolution used. The fixing of P means measurement of N positions of H and each point carries fermion or anti-fermion numbers. A more general situation corresponds to plane wave type state obtained as superposition of these states. The condition of rationality or at least algebraicity means that discrete variants of plane waves are in question.

  5. By the finiteness of the measurement resolution configuration space spinor field decomposes into a product of two parts or in more general case, to their superposition. The part Y+, which is above measurement resolution, is representable using the information contained by P, coded by the product of second quantized induced spinor field at points of P, and provided by physical experiments. Configuration space örbital" degrees of freedom should not contribute since these points are fixed in H.

  6. The second part of the configuration space spinor field, call it Y-, corresponds to the information below the measurement resolution and assignable with the complement of P and mappable to the structure of real units associated with the points of P. This part has vanishing net quantum numbers and is a superposition over the elements of the basis of CH spinor fields and mapped to a quantum superposition of real units. The representation of Y- as a Schrödinger amplitude in the space of real units could be highly unique. Algebraic holography principle would state that the information below measurement resolution is mapped to a Schrödinger amplitude in space of real units associated with the points of P.

  7. This would be also a representation for perceiver-external world duality. The correlation function in which P appears would code for the information appearing in M-matrix representing the laws of physics as seen by conscious entity about external world as an outsider. The quantum superposition of real units would represent the purely subjective information about the part of universe below measurement resolution.

  8. The condition that Y represents a state with vanishing quantum numbers gives additional constraints. The interpretation inspired by finite measurement resolution is that the coordinate h associated with Y corresponds to a zero energy insertion to a positive or negative energy state localizable to a causal diamond inside the upper or lower half of the causal diamond of observer. Below measurement resolution for imbedding space coordinates Y(h) would correspond to a nonlocal operator creating a zero energy state. This would mean that Brahman=Atman would apply to the mini-worlds below the measurement resolution rather than to entire Universe but by algebraic fractality of HFFs this would would not be a dramatic loss.
There is an objection against this picture. One obtains an 8-plet of arithmetic zero energy states rather than one state only. What this strange 8-fold way could mean?

  1. The crucial observation is that hyper-finite factor of type II1 (HFF) creates states for which center of mass degrees of freedom of 3-surface in H are fixed. One should somehow generalize the operators creating local HFF states to fields in H, and an octonionic generalization of conformal field suggests itself. I have indeed proposed a quantum octonionic generalization of HFF extending to an HFF valued field Y in 8-D quantum octonionic space with the property that maximal quantum commutative sub-space corresponds to hyper-octonions . This construction raises X4 M8 and by number theoretic compactification also X4 H in a unique position since non-associativity of hyper-octonions does not allow to identify the algebra of HFF valued fields in M8 with HFF itself.

  2. The value of Y in the space of quantum octonions restricted to a maximal commutative subspace can be expressed in terms of 8 HFF valued coefficients of hyper-octonion units. By the hyper-octonionic generalization of conformal invariance all these 8 coefficients must represent zero energy HFF states. The restriction of Y to a given point of P would give a state, which has 8 HFF valued components and Brahman=Atman identity would map these components to U8 associated with P. One might perhaps say that 8 zero energy states are needed in order to code the information about the H positions of points P.

For background see the chapter Was von Neumann right after all?. See also the article "Topological Geometrodynamics: an Overall View".



Configuration space gamma matrices as hyper-octonionic conformal fields having values in HFF?

The fantastic properties of HFFs of type II1 inspire the idea that a localized version of Clifford algebra of configuration space might allow to see space-time, embedding space, and configuration space as emergent structures.

Configuration space gamma matrices act only in vibrational degrees of freedom of 3-surface. One must also include center of mass degrees of freedom which appear as zero modes. The natural idea is that the resulting local gamma matrices define a local version of HFF of type II1 as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyper-octonions identified as a subspace of complexified octonions. As a matter fact, one can generalize octonions to quantum octonions for which quantum commutativity means restriction to a hyper-octonionic subspace of quantum octonions . Non-associativity is essential for obtaining something non-trivial: otherwise this algebra reduces to HFF of type II1 since matrix algebra as a tensor factor would give an algebra isomorphic with the original one. The octonionic variant of conformal invariance fixes the dependence of local gamma matrix field on the coordinate of HO. The coefficients of Laurent expansion of this field must commute with octonions.

The world of classical worlds has been identified as a union of configuration spaces associated with M4 labeled by points of H or equivalently HO. The choice of quantization axes certainly fixes a point of H (HO) as a point remaining fixed under SO(1,3)×U(2) (SO(1,3)×SO(4)). The condition that hyper-quaternionic inverses of M4 HO points exist suggest a restriction of arguments of the n-point function to the interior of M4.

Associativity condition for the n-point functions forces to restrict the arguments to a hyper-quaternionic plane HQ=M4 of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative sub-space HC of HO. Fixing preferred real and imaginary units means a choice of M2=HC interpreted as a partial choice of quantization axes. This has quite strong implications.

  1. The hyper-quaternionic planes with a fixed choice of M2 are labeled by points of CP2. If the condition M2 T4 characterizes the tangent planes of all points of X4 HO it is possible to map X4 HO to X4 H so that HO-H duality ("number theoretic compactification") emerges. X4 H should correspond to a preferred extremal of Kähler action. The physical interpretation would be as a global fixing of the plane of non-physical polarizations in M8: it is not quite clear whether this choice of polarization need not have direct counterpart for X4 H. Standard model symmetries emerge naturally. The resulting surface in X4 H would be analogous to a warped plane in E3. This new result suggests rather direct connection with super string models. In super string models one can choose the polarization plane freely and one expects also now that the generalized choice M2 M4 M8 of polarization plane can be made freely without losing Poincare invariance with reasonable assumption about zero energy states.

  2. One would like to fix local tangent planes T4 of X4 at 3-D light-like surfaces X3l fixing the preferred extremal of Kähler action defining the Bohr orbit. An additional direction t should be added to the tangent plane T3 of X3l to give T4. This might be achieved if t belongs to M2 and perhaps corresponds to a light-like vector in M2.

  3. Assume that partonic 2-surfaces X belong to dM4 HO defining ends of the causal diamond. This is obviously an additional boundary condition. Hence the points of partonic 2-surfaces are associative and can appear as arguments of n-point functions. One thus finds an explanation for the special role of partonic 2-surfaces and a reason why for the role of light-cone boundary. Note that only the ends of lightlike 3-surfaces need intersect M4 HO. A stronger condition is that the pre-images of light-like 3-surfaces in H belong to M4 HO.

  4. Commutativity condition is satisfied if the arguments of the n-point function belong to an intersection X2M2 HQ and this gives a discrete set of points as intersection of light-like radial geodesic and X2 perhaps identifiable in terms of points in the intersection of number theoretic braids with dH. One should show that this set of points consists of rational or at most algebraic points. Here the possibility to choose X2 to some degree could be essential. As a matter fact, any radial light ray from the tip of light-cone allows commutativity and one can consider the possibility of integrating over n-point functions with arguments at light ray to obtain maximal information. For the pre-images of light-like 3-surfaces commutativity would allow one-dimensional curves having interpretation as braid strands. These curves would be contained in plane M2 and it is not clear whether a unique interpretation as braid strands is possible (how to tell whether the strand crossing another one is infinitesimally above or below it?). The alternative assumption consistent with virtual parton interpretation is that light-like geodesics of X3 are in question.

To sum up, this picture implies HO-H duality with a choice of a preferred imaginary unit fixing the plane of non-physical polarizations globally, standard model symmetries, and number theoretic braids. The introduction of hyper-octonions could be however criticized: could octonions and quaternions be enough after all? Could HO-H duality be replaced with O-H duality and be interpreted as the analog of Wick rotation? This would mean that quaternionic 4-surfaces in E8 containing global polarization plane E2 in their tangent spaces would be mapped by essentially by the same map to their counterparts in M4×CP2,and the time coordinate in E8 would be identified as the real coordinate. Also light-cones in E8 would make sense as the inverse images of M4.

For background see the chapter Was von Neumann right after all? . See also the article "Topological Geometrodynamics: an Overall View".



DNA as topological quantum computer: XIII

In previous postings I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII 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.

The model of DNA as topological quantum computer was originally motivated by the idea that quantum biology in TGD Universe might teach something about quantum computation like processes possibly taking place in living matter. It turned out that the model of DNA as topological quantum computer began to give lessons about quantum biology. In particular, one must assign 4-color to braid strands represented as flux tubes connecting DNA nucleotides A,T,C,G to lipids of nuclear or cellular membranes. In TGD Universe this color is naturally represented in terms 2 quarks u,d and their antiquarks (scaled up variants of ordinary quarks with large hbar and residing at flux tubes of "wormhole" magnetic fields defining the braid strands).

This sounds definitely something very weird for anyone still inhabiting the simple standard model universe and not gone through 28 year lasting process of discovery starting from the basic idea of TGD and ending up with the recent highly refined picture about how TGD Universe differs from that of standard model. Recall however that the discovery of Barbara Shipman that the patterns of honeybee dance can be understood in terms of the mathematics of color group SU(3) of strong interactions, led her to suggest that quarks are directly involved with cognition and memory. This makes sense since DNA as tqc using 4-colored braids is expected to be closely involved with cognition and memory.

The model led to the prediction that coding regions of DNA might be characterized by a breaking various symmetries at quark level, that is breaking of matter antimatter symmetry, isospin asymmetry, and asymmetry between uuc and ddc type matters (c refers to charge conjugation taking matter into antimatter) could take place at level of coding sequences. Three parameters should characterize this breaking.

I made some sample calculations and found support for the breaking of matter antimatter and symmetry and the generation of anomalous em charge implied by this. Yesterday I learned (thanks go to Dale Trenary for crucial references) that simple basic facts about DNA which can be found from Wikipedia support the proposed vision about symmetry breaking although details were not quite correct.

  1. Chargaff's rules, which I already knew, imply an approximate but not complete matter antimatter symmetry at the level of the entire genome and one can find nice examples about the small breaking. Depending on the explicit corresponds of A,T,C,G with quarks. The breaking of matter antimatter symmetry is quite generally below per cent. The anomalous em charge per nucleon, which depends on scenario (2 options and their charge conjugates) is typically below .1 units per nucleotide. The following table gives representative examples about values of various parameters (anomalous em charge per nucleotide for two options, isospin per nucleotide, quark number per nucleotide, G+C/A+T ratio).

  2. The deviation of C+G/A+T from unity used to classify genomes characterizes the asymmetry between uuc and ddc type matters. C+G/A+T increases as the length of coding sequence increases.

  3. Szybalski's rules state that matter antimatter symmetry and isospin symmetry are broken for coding regions of DNA. The breaking pattern is however more intricate than I had expected. The coding part of the DNA decomposes in fifty-fifty manner into regions in which either matter or antimatter dominates and the directions of transcription and selection of template DNA are different for these regions so that mRNA breaks matter antimatter symmetry and always in the same manner. By the way, I had always thought that the template DNA is always the same. The structural matter antimatter asymmetry of mRNA is obviously translated to a functional asymmetry of DNA. A possible reason is that otherwise DNA would not be stable since it would generate too high anomalous em charge. One can wonder whether the matter antimatter asymmetry for mRNA is compensated by the opposite asymmetry for some other type of RNA inside cell nucleus.

It thus seems that DNA as tqc and the coding of braid color by quarks allows to understand the poorly understood empirical rules about the distribution of codons in DNA. Many fascinating questions and working hypothesis can be considered besides those proposed already earlier.

For details see the chapter DNA as Topological Quantum Computer. See also the brief article here.



DNA as topological quantum computer and ageing

The notion of anomalous em charge is one of the basic implications of the many-sheeted space-time concept. It can be assigned to wormhole contacts carrying fermion and antifermion quantum numbers at their throats. If the classical electromagnetic fields at the two space-time sheets are different, the wormhole contact couples to the difference of vector potentials and behaves like a particle with anomalous charge although the net charge is vanishing.

In the model of DNA as topological quantum computer the braid strands (whose braiding defines tqc program) emanate from DNA nucleotides and end up to the lipids of the nuclear and cellular membranes. They are colored in the sense that one can tell whether the strand arrives from A,T,C or G. This is achieved by representing the braid strand as a wormhole magnetic flux tube with CP conjugate wormhole throats at its ends. To A,T,C,G one assigns a wormhole contact with quark u,d or its antiquark at the "upper" throat and its CP conjugate at the "lower" throat.

There are also symmetries: A and T resp. G and C are mapped to quark and its antiquark so that DNA conjugation corresponds to CP conjugation. Chargaff's rules A≈ T and G≈ A for single DNA strand state that DNA as a whole is matter-antimatter symmetric. A and G are mapped to u,d or their antiquarks and correspond therefore to isospin doublet. This allows to interpret the almost exact A-G and T-C symmetries of the third nucleotide of codon in terms of strong isospin symmetry. Both symmetries can break down for short portions of DNA.

The anomalous em charge of DNA is due to the fact that DNA is negatively charged (2 units of charge per nucleotide due to phosphate) and generates classical em field at the "upper" sheet of wormhole magnetic flux tube. The nearly vanishing Qa for DNA is interpreted as a stability condition for DNA. For long DNA strands Chargaff's rules A≈ T and G≈ A indeed guarantee the vanishing of Qa since A and T resp. C and G correspond to quark and its antiquark. There are four options concerning nucleotide quark correspondence and therefore also the identification of Qa: for one of them one has Qa= [2(A-T)-(G-C)]/3. Integer valuedness allows color singletness for the many quark-antiquark state assignable to DNA strand via the mapping of A,T,C,G to quarks and antiquarks.

Telomeres are of special interests as far as anomalous em charge is considered. Chromosomes are not copied completely in cell replication, and one function of telomeres is to guarantee that the translated part of genome replicates completely for sufficiently many cell divisions. Telomeres consists of 3-20 kilobases long repetitions of TTAGGG, and there is a 100-300 kilobases long repeating sequence between telomere and the rest of the chromosome. Telomeres can form can also 4-stranded structures. Telemere end contains a hair-pin loop as a single stranded part, which prevents the action of DNA repair enzymes on the chromosome end.

Telomerase is a reverse transcriptase enzyme involved with the synthesis of telomeres using RNA strand as a template but since its expression is repressed in many types of human cells, telomere length shortens in each cell replication. In the case of germ cells, stem cells and white blood cells telomerase is expressed and telomere length preserved. Telomere shortening is known to relate to ageing related diseases. On the other hand, overactive telomere expression seems to correlate with cancer.

If telomeres possess braid strands, the compensation of Qa might provide an additional reason for their presence. If this the case and if telomeres are strict multiples of TTAGGG, the shortening of telomeres generates a non-vanishing Qa unless something happens for the active part of DNA too. Color singletness condition should however remain true: the disappearance of 3n multiples of TTAGGG in each replication is the simplest guess for what might happen. In any case, DNA strands would become unstable in cell replication. Qa could be reduced by a partial death of DNA in the sense that some portions of braiding disappear. Also this would induce ill functioning of tqc harware perhaps related to ageing related diseases. Perhaps evolution has purposefully developed this ageing mechanism since eternal life would stop evolution.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer.



On direct testing of quantum consciousness

Quantum entanglement and its reduction in "cognitive" quantum measurement could provide a direct test of quantum consciousness. Andrei Khrennikov [1] has proposed a mathematical formulation of "quantum like" behavior based on his proposal that so called context dependent probabilities could provide alternative description for quantum mechanical interference phenomenon. In quantum theory context would correspond to the choice of quantization axis. Khrennikov has also proposed a modification of Bell inequalities so that they apply on conditional probabilities: this would make it possible to avoid the task of preparing entangled state of brains. The hope is that one could forget completely the microscopic structure of quantum brain and test quantum like behavior by making simple experiments involving just questions to the subject persons and finding whether or not classical rules for conditional probabilities hold true or not.

1. First experiment

Bistable percepts induced by ambiguous figures are especially attractive from the point of view of experimentation. The question would be "Which of the two possible percepts?" and the outcome would be answer to this question. The first experiment reported in [2] was following.

  1. Consider a group S of subject persons. Divide it into two groups U and V containing equally many subject persons. Represent for members of U the question A (bistable percept A). From this one can deduce the probalities p(A=+) and p(A=-). Represent for members of V the question B and and immediately after than the question A (bistable percept A) for those who answered B=-. This experiment gives the conditional probabilities p(A=x/B=y).

  2. The quantity

    cos(θx) = [p(A=x)- p(B=+)p(A=x/B=+) -p(B=-)p(A=x/B=-)]/[2(p(B=+)p(B=-)p(A=x/B=+)p(A=x/B=-)1/2], x=+/-.

    measures the failure of the basic rule

    p(A=x)=p(B=+)p(A=x/B=+) +p(B=-)p(A=x/B=-)

    for classical conditional probability. Note that in quantum theory similar rules applies to transition amplitudes (conditional probability amplitudes) corresponding to the addition of a complete set of states in the inner product between two states (for instance, repeated application of this gives rise to path integral formulation).

  3. One can describe the situation in terms of "quantum like state"

    Ψ(A=x)= [p(B=+)p(A=x/B=+)]1/2 +e+/- [p(B=-)p(A=x/B=-)]1/2

    satisfying p(A=+x) =|Ψ(A=x)|2. If cos(θx) is non-vanishing one can say that that the situation is quantum like. Conte and collaborators conclude that this is indeed the case [2].

2. Second experiment

Second experimental test is more complex and involves generalization of Bell's inequality so that it involves conditional probabilities [1] Let A,B,C=+/- be arbitrary dichotomous random variables satisfying Kolmogorov axioms characterizing classical probability. Then the following analog of Bell inequality can be shown to hold true:

P(A=+,B=+) + P(C=+,B=-)≥ P(A=+,C=+).

In terms of conditional probabilities one has

P(A=+/B=+)/P(B=+) + P(C=+/B=-)/P(C=+) ≥ P(A=+/C=+)/P(C=+).

If the random variables are symmetrically distributed so that one has P(X=+/-)=1/2, for X=A,B,C one obtains

P(A=+/B=+)+P(C=+/B=-)≥ P(A=+/C=+) .

Note that this form of equality is by no means necessary. The symmetric distributions for the random variables would however correspond to maximal entanglement in spin system given best hopes for the violation of the Bell inequality.

  1. The test is following. Consider a group S of subject persons divided into subgroups U and V as above. Pose to the members of U question B and immediately after that question A for those who answered B=+ and question C for those who answered B=-1. For group V represent first the question C and for those who answer C=+ represent the question A. The failure of inequality could regarded as a direct proof for quantum like behavior. That failure does not occur does not of course mean that system is classical but only that the quantal effects are not large enough.

  2. The analogy with Bell's inequality suggest that the questions are analogous to posing the spins of spin pair in spin singlet state to an external magnetic fields determining the quantization axis. The inequality tend to fail when the directions of the magnetic fields for the two spins differ enough. Thus the failure is expected if the questions, in other words ambiguous figures producing bistable percepts differ enough.

3. Criticism and possible improvement of the experiment

In the case of spin pairs the tests of quantum behavior are carried out for the members of spin pair by putting them to magnetic fields having different directions. Now the pair of experiments is made for a single subject person. Hence there is no need to prepare quantum entangled pair of conscious entities.

The use of ensemble is the problematic aspect of experiments. Human beings are extremely complex systems and one can argue that it is impossible to prepare an ensemble of identical systems in cognitive sense. A possible manner to avoid the problem would be the replacement of ensembles with a time series of experiments performed for a single subject person. In both experiments one could perform the two kinds of experiments many times to single subject person and deduce various probabilities and cos(θ) from the outcome of the experiments.

4. Interpretation in terms of zero energy ontology and DNA as tqc

The discussions with Elio Conte led to the realization that in zero energy ontology the experiments differ from the corresponding experiments for two-spin system only in that space-like entanglement is replaced with time like entanglement. The experiment would be essentially a measurement of probabilities defined by the matrix elements of M-matrix defining the generalization of S-matrix. Hence Bell's inequalities and their generalizations should apply in genuine quantum sense. By performing the experiments for a single subject person as time series one might be therefore able study whether quantum consciousness in the sense of TGD makes sense.

Quantum consciousness approach however requires that bistable percepts have genuine microscopic quantum states as their physical correlates. This is not assumed in the approach of Khrennikov.

  1. If the vision about DNA as topological quantum computer makes sense, the question to the answer "Which of the two possible percepts?" can be regarded as a qubit which is some function of a large number of qubits and same function irrespective of the ambiguous figure used. This could hold quite generally, at least for a given sensory modality. The qubits appearing as arguments of this function are determined by the sensory input defined by the ambiguous figure. The ambiguous figure would take the role of magnetic field determining the directions of quantization axes for a large collection of qubits appearing as arguments of the Boolean function (one cannot exclude the possibility that neuronal synchrony forces all these axes to have same direction). Qubit could correspond to spin or some spin like variable. The quantization axes could correspond in this case to the direction of external magnetic field acting on 1-gate of tqc.

  2. Qubit could be replaced with an n-state system: this would require a generalization of the Bell inequalities. The model of DNA as tqc suggests that qubit might be replaced with qutrit defined by a quark triplet (third quark with vanishing color isospin would correspond to ill-defined truth value. The inability of subject persons to identify the percept always indeed encourages to consider this option. Color group SU(3) (SO(3) subset SU(3)) defines the set of possible quantization axes as points of the flag manifold F= SU(3)/U(1)× U(1) (SO(3)/SO(2)= S2). Quantization axes would be determined by the direction of color magnetic field in color Lie algebra and sensory input would define a sequence of 1-gates at the lipids ends of the braid strands, and realized as color rotations of the flux tube defining braid strand. This hypothesis would conform with the proposal of Barbara Shipman that honeybee dance that quarks are in some mysterious manner involved with cognition [3].

For background see that chapter DNA as Topological Quantum Computer.

References

[1] A. Khrennikov (2004), Bell's inequality for conditional probabilities as a test for quantum like behaviour of mind, arXiv:quant-ph/0402169.

[2] E. Conte, O. Todarello, A. Federici, J. P. Zbilut (2008), Minds States Follow Quantum Mechanics During Perception and Cognition of Ambigious Figures: A Final Experimental Confirmation, arXiv:0802.1835v1 [physics.gen-ph].

[3] B. Shipman (1998), The geometry of momentum mappings on generalized flag manifolds, connections with a dynamical system, quantum mechanics and the dance of honeybee.

B. Shipman (1998), On the geometry of certain isospectral sets in the full Kostant-Toda lattice.

B. Shipman (1998), A symmetry of order two in the full Kostant-Toda lattice.



DNA as topological quantum computer: XII

In previous postings I, II, III, IV, V, VI, VII, VIII, IX, X, XI 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.

One of the challenges is the realization of single particle gates representing U(2) rotation of the qubit. The first thing to come mind was that U(2) corresponds to U(2) rotation induced by magnetic field and electric fields. Yesterday I realized much more elegant realization in terms of SU(3) rotation, where SU(3) is color group associated with strong interactions. Soon I remembered that there is direct evidence for the prediction that color SU(3) is associated with tqc and thus cognition: something that does not come first in mind! I have myself written text about the strange finding of topologist Barbara Shipman suggesting that quarks are in some mysterious manner involved with honeybee dance and proposed an interpretation.

1. The realization of 1-gate in terms of ordinary rotations

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, 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.

A possible realization for single particle gate- characterized by 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.

One can criticize this model. The introduction of magnetic pulses does not look an attractive idea and seems to require additional structures besides magnetic flux tubes (MEs?). It would be much nicer to assign the magnetic field with the flux tubes defining the braid strands. The rotation of magnetic field would however require changing the direction of braid strands. This does not look natural. Could one do without this rotation by identifying spin like degree of freedom in some other manner? This is indeed possible.

2. The realization of 1-gate in terms of color rotations

TGD predicts a hierarchy of copies of scaled up variants of both weak and color interactions and these play a key role in TGD inspired model of living matter. Both weak isospin and color isospin could be considered as alternatives for the ordinary spin as a realization of qubit in TGD framework. Below color isospin is discussed. Below color isospin is discussed but one could consider also a realization in terms of nuclei and their exotic counterparts differing only by the replacement of neutral color bond between nuclei of nuclear string with a charged one. Charge entanglement between nuclei would guarantee overall charge conservation.

  1. Each space-time sheet of braid strands contains quark and antiquark at its ends. Color isospin and hypercharge label their states. Two of the quarks of the color triplet form doublet with respect to color isospin and the third is singlet and has different hyper charge Y. Hence qubit could be realized in terms of color isospin I3 instead of ordinary spin but third quark would be inert in the Boolean sense. Qubit could be also replaced with qutrit and isospin singlet could be identified as a statement with ill-defined truth value. Trits are used also in ordinary computers. In TGD framework finite measurement resolution implies fuzzy qubits and the third state might relate to this fuzziness.

  2. Magnetic flux tubes are also color magnetic flux tubes carrying non-vanishing classical color gauge field in the case that they are non-vacuum extremals. The holonomy group of classical color field is an Abelian subgroup of the U(1)× U(1) Cartan subgroup of color group. Classical color magnetic field defines the choice of quantization axes for color quantum numbers. For instance, magnetic moment is replaced with color magnetic moment and this replacement is in key role in simple model for color magnetic spin spin splittings between spin 0 and 1 mesons as well as spin 1/2 and 3/2 baryons.

  3. There is a symmetry breaking of color symmetry to subgroup U(1)I3× U(1)Y and color singletness is in TGD framework replaced by a weaker condition stating that physical states have vanishing net color quantum numbers. This makes possible the measurement of color quantum numbers in the manner similar to that for spin. For instance, color singlet formed by quark and antiquark with opposite color quantum numbers can in the measurement of color quantum numbers of quark reduce to a state in which quark has definite color quantum numbers. This state is a superposition of states with vanishing Y and I3 in color singlet and color octet representations. Strong form of color confinement would not allow this kind of measurement.

  4. Color rotation in general changes the directions of quantization axis of I3 and Y and generates a new state basis. Since U(1)× U(1) leaves the state basis invariant, the space defined by the choices of quantization axes is 6-dimensional flag manifold F=SU(3)/U(1)×U(1). In contrast to standard model, color rotations in general do not leave classical electromagnetic field invariant since classical em field is a superposition of color invariant induced Kähler from and color non-invariant part proportional classical Z0 field. Hence, although the magnetic flux tube retains its direction and shape in M4 degrees of freedom, its electromagnetic properties are affected and this is visible at the level of classical electromagnetic interactions.

  5. If color isospin defines the qubit or qutrit in topological quantum computation, color quantum numbers and the flag manifold F should have direct relevance for cognition. Amazingly, there is a direct experimental support for this! Years ago topologist Barbara Shipman made the intriguing observation that honeybee dance can be understood in terms of a model involving the flag manifold F (see this, this, and this). This led her to propose that quarks are in some mysterious manner involved with the honeybee dance. My proposal was that color rotations of the space-time sheets associated with neurons represent geometric information: sensory input would be coded to color rotations defining the directions of quantization axes for I3 and Y. Subsequent state function reduction would provide conscious representations in terms of trits characterizing for instance sensory input symbolically.

I introduced also the notions of geometric and sensory qualia corresponding to the two choices involved with the quantum measurement: the choice of quantization axes performed by the measurer and the "choice" of final state quantum numbers in state function reduction. In the case of honeybee dance geometric qualia could code information about the position of the food source. The changes of color quantum numbers in quantum jump were identified as visual colors. In state function reduction one cannot speak about change of quantum numbers but about their emergence. Therefore one must distinguish between color qualia and the conscious experience defined by the emergence of color quantum numbers: the latter would have interpretation as qutrit.

To sum up, this picture suggests that 1-gates of DNA tqc (understood as "dance of lipids") are defined by color rotations of the ends of space-like braid strands and at lipids. The color rotations would be induced by sensory and other inputs to the system. Topological quantum computation would be directly related to conscious experience and sensory and other inputs would fix the directions of the color magnetic fields.

For details see the chapter DNA as Topological Quantum Computer.



Prime Hilbert spaces and infinite primes

Kea told in her blog about a result of quantum information science which seems to provide an additional reason why for p-adic physics.

Suppose that one has N-dimensional Hilbert space which allows N+1 mutually unbiased basis. This means that the moduli squared for the inner product of any two states belonging to different basis equals to 1/N. If one knows all transition amplitudes from a given state to all states of all N+1 mutually unbiased basis, one can fully reconstruct the state. For N=pn dimensional N+1 unbiased basis can be found and the article of Durt gives an explicit construction of these basis by applying the properties of finite fields. Thus state spaces with pn elements - which indeed emerge naturally in p-adic framework - would be optimal for quantum tomography. For instance, the discretization of one-dimensional line with length of pn units would give rise to pn-D Hilbert space of wave functions.

The observation motivates the introduction of prime Hilbert space as as a Hilbert space possessing dimension which is prime and it would seem that this kind of number theoretical structure for the category of Hilbert spaces is natural from the point of view of quantum information theory. One might ask whether the tensor product of mutually unbiased bases in the general case could be constructed as a tensor product for the bases for prime power factors. This can be done but since the bases cannot have common elements the number of unbiased basis obtained in this manner is equal to M+1, where M is the smallest prime power factor of N. It is not known whether additional unbiased bases exists.

1. Hierarchy of prime Hilbert spaces characterized by infinite primes

The notion of prime Hilbert space provides a new interpretation for infinite primes, which are in 1-1 correspondence with the states of a supersymmetric arithmetic QFT. The earlier interpretation was that the hierarchy of infinite primes corresponds to a hierarchy of quantum states. Infinite primes could also label a hierarchy of infinite-D prime Hilbert spaces with product and sum for infinite primes representing unfaitfully tensor product and direct sum.

  1. At the lowest level of hierarchy one could interpret infinite primes as homomorphisms of Hilbert spaces to generalized integers (tensor product and direct sum mapped to product and sum) obtained as direct sum of infinite-D Hilbert space and finite-D Hilbert space. (In)finite-D Hilbert space is (in)finite tensor product of prime power factors. The map of N-dimensional Hilbert space to the set of N-orthogonal states resulting in state function reduction maps it to N-element set and integer N. Hence one can interpret the homomorphism as giving rise to a kind of shadow on the wall of Plato's cave projecting (shadow quite literally!) the Hilbert space to generalized integer representing the shadow. In category theoretical setting one could perhaps see generalize integers as shadows of the hierarchy of Hilbert spaces.

  2. The interpretation as a decomposition of the universe to a subsystem plus environment does not seem to work since in this case one would have tensor product. Perhaps the decomposition could be to degrees of freedom to those which are above and below measurement resolution. Perhaps one should try to interpret physically the process of transferring degrees of freedom from tensor product to direct sum.

  3. The construction of these Hilbert spaces would reduce to that of infinite primes. The analog of the fermionic sea would be infinite-D Hilbert space which is tensor product of all prime Hilbert spaces Hp with given prime factor appearing only once in the tensor product. One can "add n bosons" to this state by replacing of any tensor factor Hp with its n+1:th tensor power. One can "add fermions" to this state by deleting some prime factors Hp from the tensor product and adding their tensor product as a finite-direct summand. One can also "add n bosons" to this factor.

  4. At the next level of hierarchy one would form infinite tensor product of all infinite-D prime Hilbert spaces obtained in this manner and repeat the construction. This can be continued ad infinitum and the construction corresponds to abstraction hierarchy or a hierarchy of statements about statements or a hierarchy of n:th order logics. Or a hierarchy of space-time sheets of many-sheeted space-time. Or a hierarchy of p"../articles/ in which certain many-particle states at the previous level of hierarchy become p"../articles/ at the new level (bosons and fermions). There are many interpretations.

  5. Note that at the lowest level this construction can be applied also to Riemann Zeta function. ζ would represent fermionic vacuum and the addition of fermions would correspond to a removal of a product of corresponding factors ζp from ζ and addition of them to the resulting truncated ζ function. The addition of bosons would correspond to multiplication by a power of appropriate ζp. At zeros of ζ the modified zeta functions reduce to their fermionic parts. The analog of ζ function at the next level of hierarchy would be product of all these modified ζ functions and probably fails to exist as a smooth function since the product would typically converge to either zero or infinity.

2. Hilbert spaces assignable to infinite integers and rationals make also sense

  1. Also infinite integers make sense since one can form tensor products and direct sums of infinite primes and of corresponding Hilbert spaces. Also infinite rationals exist and this raises the question what kind of state spaces inverses of infinite integers mean.

  2. Zero energy ontology suggests that infinite integers correspond to positive energy states and their inverses to negative energy states. Zero energy states would be always infinite rationals with real norm which equals to real unit.

  3. The existence of these units would give for a given real number an infinite rich number theoretic anatomy so that single space-time point might be able to represent quantum states of the entire universe in its anatomy (number theoretical Brahman=Atman).

    Also the world of classical worlds (light-like 3-surfaces of the imbedding space) might be imbeddable to this anatomy so that basically one would have just space-time surfaces in 8-D space and configuration space would have representation in terms of space-time based on generalized notion of number. Note that infinitesimals around a given number would be replaced with infinite number of number-theoretically non-equivalent real units multiplying it.

3. Should one generalize the notion of von Neumann algebra?

Especially interesting are the implications of the notion of prime Hilbert space concerning the notion of von Neumann algebra -in particular the notion of hyper-finite factors of type II1 playing a key role in TGD framework. Does the prime decomposition bring in additional structure? Hyper-finite factors of type II1 are canonically represented as infinite tensor power of 2×2 matrix algebra having a representation as infinite-dimensional fermionic Fock oscillator algebra and allowing a natural interpretation in terms of spinors for the world of classical worlds having a representation as infinite-dimensional fermionic Fock space.

Infinite primes would correspond to something different: a tensor product of all p×p matrix algebras from which some factors are deleted and added back as direct summands. Besides this some factors are replaced with their tensor powers.

Should one refine the notion of von Neumann algebra so that one can distinguish between these algebras as physically non-equivalent? Is the full algebra tensor product of this kind of generalized hyper-finite factor and hyper-finite factor of type II1 corresponding to the vibrational degrees of freedom of 3-surface and fermionic degrees of freedom? Could p-adic length scale hypothesis - stating that the physically favored primes are near powers of 2 - relate somehow to the naturality of the inclusions of generalized von Neumann algebras to HFF of type II1?

For background see that chapter Infinite Primes and Consciousness.



Summary of possible symmetries of DNA suggested by the model of topological quantum computation

The following gives a list of possible symmetries of DNA inspired by the identification of braid color.

1. Color confinement in strong form

The states of quarks and anti-quarks associated with DNA both wormhole wormhole throats of braided (living) DNA strand can be color singlets and have thus integer valued anomalous em charge. The resulting prediction depends on the assignment of quarks and antiquarks to A,T,C,G which in principle should be determined by the minimization of em interaction energy between quark and nucleotide. For instance 2(A-T)-(G-C) mod 3=0 for a piece of living DNA which could make possible color singletness. As a matter fact, color singletness conditions are equivalent for all possible for braid color assignments. This hypothesis might be weakened. For instance, it could hold true only for braided parts of DNA and this braiding are dynamical. It could also hold for entire braid with both ends included only: in this case it does not pose any conditions on DNA. Questions: Do all living DNA strands satisfy this rule? Are only the double stranded parts of DNA braided and satisfy the rule. What about loops of hairpins?

2. Matter antimatter asymmetry at quark level

A←→ T and G←→ C corresponds to charge conjugation at the level of quarks (quark ←→ antiquark). Chargaff's rules states A≈ T and C≈ G for long DNA strands and mean matter-antimatter symmetry in the scale of DNA strand. Double strand as a whole is matter anti-matter symmetric. Matter-antimatter asymmetry is realized functionally at the level of DNA double strand in the sense that only DNA strand is transcribed. The study of some examples shows that genes defined as transcribed parts of DNA do not satisfy Chargaff's rule. This inspires the hypothesis about the breaking of matter antimatter symmetry. Genes have non-vanishing net A-T and C-G and therefore also net Qa with sign opposite to that in control regions. Just as the Universe is matter-antimatter asymmetric, also genes would be matter-antimatter asymmetric.

3. Isospin symmetry at quark level

A←→ G and T←→ A correspond change of anomalous em charge by 1 unit and these operations respect color confinement condition. Local modifications of DNA inducing these changes should be preferred. The identification for the symmetries A←→ G and T←→ A for the third nucleotide of code is as isospin symmetries. For the vertebrate mitochondrial code the symmetry exact and for nuclear code slightly broken.

4. Matter antimatter asymmetry and isospin symmetries for the first two nucleotides

The first two nucleotides of the codon dictate to a high degree which amino-acid is coded. This inspires the idea that 3-code has emerged as fusion of 1- and 2-codes in some sense. There are two kinds of 2-codons. The codons of type A have fractional em charge and net quark number (consisting of either matter or antimatter at quark level) and are not able to form color singlets. The codons of type B have integer em charge and vanishing quark number (consisting of matter and antimatter) and are able to form color singlets. The 2-codons of type A (resp. B) are related by isospin rotations and there should be some property distinguishing between types A and B. There indeed is: if 2-codon is matter-antimatter asymmetric, 1-codon is not and vice versa.

  1. For almost all type A codons the amino-acid coded by the codon does not depend on the last nucleotide. There are two exceptions in the case of the nuclear code: (leu,leu,phe,phe) and (ile,ile,ile,met). For human mitochondrial code one has (ile,ile,ile,ile) and thus only one exception to the rule. The breaking of matter-antimatter symmetry for the third nucleotide is thus very small.

  2. For codons of type B the 4-columns code always for two doublets in the case of vertebrate mitochondrial code so that for codons with vanishing net quark number the breaking of matter-antimatter symmetry for the third nucleotide is always present.

5. Em stability

Anomalous em charge Qa vanishes for DNA and perhaps also mRNA strand containing also the G cap and poly-A tail which could compensate for the Qa of the transcribed region so that

2(A-T)-(G-C)≈ 0

or some variant of it holds true. Chargaff's rules for long DNA strands imply the smallness of Qa.

6. Summary of testable working hypothesis

Following gives a summary of testable working hypothesis related to the isospin symmetry and color singletness. The property of having integer valued/vanishing Qa is referred to as property P.

    >

  1. Gene plus control region and also DNA repeats should have property P. Transcribed and control regions of gene have Qa with opposite signs.

  2. Transposons, repeating regions, the overhangs associated with the cut and paste of transposon, and the DNA strands resulting in cutting should have property P. This could explain why transposons can paste themselves to AT and GC (Qa=0) rich repeating regions of DNA. The points at which DNA can be cut should differ by a DNA section having property P. This gives precise predictions for the points at which transposons and pieces of viral DNA can join and could have implications for genetic engineering.

  3. If also mRNA is braided, it has property P. This can be only true if the poly-A tail compensates for the non-vanishing Qa associated with the translated region.

  4. Living hairpins should have property P. If only double helix parts of hairpins are braided, the prediction is trivially true by the palindrome property. tRNA or at least parts of it could be braided. Braids could end to the nuclear membrane or mRNA or to the amino-acid attachable to tRNA. For stem regions Qa is integer valued. The fact that the nucleotide of the anticodon corresponding to the third nucleotide of codon can base pair with several nucleotides of mRNA suggests that I(nositol) can have Qa opposite to that of A,T,C and U opposite to that of A,G. For 2-anticodon the pairing would be unique. This would give a lot of freedom to achieve property P in weak sense for tRNA. Braid structure for tRNA + amino-acid could be different that for tRNA alone and also in the translation the braid structure could change.

  5. Also aminoacids could be braided. Qa could vary and correspond to Qa for one of the codons coding for it. The aminoacid sequences of catalysts attaching to DNA strand should have opposite Qa for each codon-aminoacid pair so that aminoacid would attach only to the codons coding for it.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer.



Transposons and anomalous em charge

TGD based model of tqc relies on colored braids with the color of braid in one-one correspondence with nucleotides A,T,C,G and represented by 2 quarks and 2 anti-quarks. The basic prediction of the braid concept is anomalous em charge defined as the net quark charge assignable to DNA space-time sheets of DNA sequence. This notion makes sense also for more general molecules possessing braids. Transposons provide an especially simple manner to test the hypothesis that anomalous em charge is integer valued (quarks can form color singlet) or even vanishing (by stability).

Transposons (see this and the article of D. F. Voytas (2008), Fighting fire with fire, Nature vol 451, January) are moving and copying genes. Moving genes cut from initial position and past to another position of double strand. Copying genes copy themselves first to RNA and them to a full DNA sequence which is then glued to the double strand by cut and paste procedure. They were earlier regarded as mere parasites but now it is known that their transcription is activated under stress situations so that they help DNA to evolve. In tqc picture their function would be to modify tqc hardware. For copying transposons the cutting of DNA strand occurs usually at different points for DNA and cDNA so that "sticky ends" result ("overhang" and its complement) (see ). Often the overhang has four nucleotides. The copied transposon have ends which are reversed conjugates of each other so that transposons are palindromes as are also DNA hairpins. This is suggestive of the origin of transposons./p>

In order to avoid boring repetitions let us denote by "satisfy P" for having having integer valued (or even vanishing) Qa. The predictions are following:

  1. The double strand parts associated with the segments of DNA produced by cutting should satisfy P.
  2. The cutting of DNA should take place only at positions separated by segments satisfying P.
  3. The overhangs should satisfy P.
  4. Transposons should satisfy P.
In the example mentioned here, the overhang is CTAG and has vanishing Qa.

It is known that transposons - repeating regions itself - tend to attach to the repeating regions of DNA.

  1. There are several kinds of repeating regions. 6-10 base pair long sequences can be repeated in untranslated regions up to 105 times and whole genes can repeat themselves 50-104 times.
  2. Repeats are classified into tandems (say TTAGGG associated with telomeres), interspersed repetitive DNA (nuclear elements), and transposable repeat elements. Interspersed nuclear elements (INEs) are classified LINEs (long), SINEs (short), TLTRs (Transposable elements with Long Terminal Repeats), and DNA transposons themselves.
  3. LINEs contain AT rich regions. SINEs known as alus (about 280 bps) contain GC rich regions whereas mariner elements (about 80 bps) are flanked by TA pairs. LTRs have length 300-1000 bps. DNA transposons are flanked with two short inverted repeat sequences flanking the reading frame: "inverted" refers to the palindrome property already mentioned.

AT and CG have vanishing Qa so that their presence in LINEs and SINEs would make the cutting and pasting easy allowing to understand why transposons favor these regions. Viruses are known to contain long repeating terminal sequences (LTR). One could also check whether DNA decomposes to regions satisfying P and surrounded by repeating sequences which satisfy P separately or as whole as in the case DNA transposons.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer.



What selected the bio-molecules?

The extremely low probabilities for the selection of bio-molecules from a super-astrophysical number of alternatives represents one of the bottleneck problems of biology relying on the prevailing view about biochemistry. The notion of braid could resolve this problem.

Suppose that the presence of braids distinguishes between living and dead matter, that the four nucleotides are mapped to colored braid strands (that is to 2 quarks + 2 anti-quarks), and that a given amino-acid is mapped in a non-deterministic manner to one of the 3-braids associated with the DNA triplets coding for it. Braids could be associated besides DNA, amino-acids, and lipids also to other bio-molecules and define more general analogs of genetic codes as correspondences between bio-molecules able to react.

The implication would be that the step of catalytic reactions bringing together the catalyst and reactants would occur by a temporary reduction of Planck constant only for subsets of bio-molecules connected by braid strands and the pattern of braid strands involved would define the geometro-dynamical pattern of the reaction. The outcome would be a selection of very restricted subsets of bio-molecules able to form reaction networks and of reaction pathways. This would imply Darwinian selection of subsets of bio-molecules able to co-exist and dramatically enhance the probability for the emergence of life as we know it.

One challenge is to predict what kind of braids can begin from a given bio-molecule, say nucleotide or amino-acid. The physicist's guess would be that the (electromagnetic only?) interaction energy between bio-molecule and given pattern of wormhole contacts having quark and anti-quark at its throats should select the preferred braids as minima of the interaction energy. How closely the presence of hydrogen bonds (and of conjectured "half hydrogen bonds") relates to this is also an interesting question.

For the model of DNA as topological quantum computer see the chapter DNA as Topological Quantum Computer.



Structure and function of tRNA in braid picture

The recent beautiful results (for a popular summary see [pwpop]) about programming of bio-molecular self assembly combined with the earlier model for the pre-biotic evolution inspire interesting insights about the role of braiding in translation. According to the TGD based model of pre-biotic evolution [prebio], 3-code should have resulted as a fusion of 1- and 2- codes to 3-code involving fusion of tRNA1 and tRNA2 to tRNA. Second hypothesis is that during RNA era the function of tRNA2 was to generate RNA2 double strand from single RNA strand and that amino-acids catalyzed this process. The considerations that follow strongly suggest that tRNA1 was involved with a non-deterministic generation of new RNA sequences essential for the evolution. After the establishment of 3-code these two process fused to a deterministic process generating amino-acid sequences. RNA era could still continue inside cell and play an important role in evolution.

A. Structure of tRNA molecule

The structure of tRNA- although more complex than that of hairpin- has much common with that of hairpins. Therefore it is interesting to look this structure from the point of view of TGD. For instance, one can find whether the notions of braiding, anomalous em charge and quark color could provide additional insights about the structure and function of tRNA. The shape of the tRNA molecule [tRNA] in 2-D representation is that of cruciform.

  1. tRNA molecule can be seen as single RNA strand just as hairpin. The five stems are double strands analogous to the necks of the hairpin. Strand begins at 5' end of the acceptor stem directed upwards. The second strand of acceptor stem continues as a toehold ending to 3' end of tRNA. The toehold has at its end ACC to which the amino-acid (rather than conjugate DNA) attaches.

  2. tRNA molecule contains three arms with hairpin structure. A arm containing the anticodon is directed downwards. D and T arms are horizontal and directed to left and right. Between T arm and A arm there is additional variable hairpin like structure but with highly degenerate loop is degenerate. It has emerged during evolution.

  3. The structure of tRNA minus anticodon depends on anti-codon which conforms with the fact T and D arms are related to the binding of amino-acid so that their nucleotide composition correlates with that of anticodon.

B. Wobble base pairing

The phenomenon of wobble base pairing [wobble] is very important. There are only about 40 tRNA molecules instead of 61 which means that one-to-one map between mRNA nucleotides and tRNA conjugate nucleotides is not possible. Crick suggests that so called wobble base pairing resolves the problem. What happens that the first nucleotide of anticodon is either A, G, U, or I(nosine) [inosine]. The base-pairings for third nucleotide are {A-U, G-C, U-{A,G}, I-{U,A,C}. The explanation for the non unique base pairing in the case of U is that its geometric configuration is quite not the same as in ordinary RNA strand. I is known to have 3-fold base pairing.

Minimization of the number of tRNAs requiring that only three mRNA codons act as stopping signs predicts that the number of tRNAs is 40.

  1. It is convenient to classify the 4-columns of code table according to whether all four codons code for same amino-acid ((T,C,A,G)→ X), whether 4-column decomposes into two dublets: [(T,C),(A,G)]→ [X,Y], or whether it decomposes to triplet and singlet ([(T,C,A),G]→ [ile,met]). There are also the 4-columns containing stop codon: [(U,C),(A,G)]→ [(tyr,tyr),(stop,stop)] and [(U,C),A,G]→ [(cys,sys),stop,trp]. Mitochondrial code has full A-G and T-C symmetries whereas for vertebrate nuclear code 3 4-columns break this symmetry.

  2. Consider first 4-columns for which the doublet symmetry is broken. [tyr,tyr, top,stop] column must correspond to first tRNA nucleotide which is A or G (tyr). The absence of anti-codons containing U implies stop codon property. For [cys,sys,stop,trp] one must have A,G and C but U is not allowed. ile-met column can correspond to tRNAs with I and C as the first nucleotide.

  3. For 4-columns coding for two doublet amino-acids the minimal set of first tRNA codons is {A,G,U}. For completely symmetric 4-columns the minimal set of tRNA codons is {I,U}. Thus {A,G,U,I} would replace {A,G,U,C}.

  4. There are 9 completely symmetric 4-columns making 18 tRNAs, 5 doublet pairs making 15 tRNAs, ile-met giving 2 tRNAs, and the columns containing stopping codons giving 5 tRNAs. Altogether this gives 18+15+2+5= 40. Also the deviations from the standard code can be understood in terms of the properties of tRNA.

C. Wobble base pairing in TGD framework

Consider first the interpretation of wobble base pairing in TGD framework assuming the braiding picture and the mapping of nucleotides to quarks. The completely symmetric 4-columns correspond to unbroken isospin and matter-antimatter symmetries. 4-columns decomposing into doublets result from the breaking of matter-antimatter symmetry at quark level. ile-met column corresponds to the breaking of both symmetries. The base pairings of I obviously break both symmetries.

The non-unique based pairing of U and I means that they cannot correspond to a unique quark or anti-quark in braiding U pairs with both A and G so that the braid strands starting from these RNA nucleotides must both be able to end to tRNA U. Hence tRNA U is not sensitive to the isospin of the quark. This non-uniqueness could relate to the assumed anomalous geometric character of the binding of U codon to tRNA sequence. The braid strands beginning from U, A, and C must be able to end up to I so that I can discriminate only between {U,C,A} and G.

D. Anomalous em charge and color singletness hypothesis for tRNA

One can test also whether the vanishing of anomalous em charge of tRNA leads to testable predictions. One can also try understand translation process in terms of the braiding dynamics. One must distinguish between the states of tRNA alone and tRNA + amino-acid for which braidings are expected to be different.

Before continuing it must be made clear that braiding hypothesis is far from being precisely formulated. One question is whether the presence of the braiding could distinguish between matter in vivo and vitro. For instance, the condition that anomalous em charge is integer valued or vanishing for DNA hairpins in vivo gives strong condition on the loop of the hairpin but or hairpins in vitro there would be no such conditions. Second point is that amino-acids and I and U in tRNA1 could carry variable anomalous em charge allowing rather general compensation mechanism.

D.1 tRNA without amino-acid

  1. The minimal assumption is that braiding hypothesis applies only to the stem regions of tRNA in this case. In this case the strands can indeed begin from strand and end up to conjugate strand. The possibility of color singletness and vanishing of total anomalous em charge are automatically satisfied for the stem regions as a whole in absence of non-standard base pairings. In general the acceptor stem contains however G*U base pair which is matter-antimatter symmetric but breaks isospin symmetry and gives unit anomalous charge for the acceptor stem. Also other stems can contain G*U , U*G pairings as also P*G and L*U pairings (P and L denote amino-acids Pro and Leu). The study of some concrete examples [tRNAseqs] shows that single G*U bond is possible so that anomalous em charge can be non-vanishing but integer valued for double strand part of tRNA. Suppose that a given amino-acid can have anomalous of any codon coding for it. If P in G*P pair has the anomalous em charge of the codon CCG, G*P pair has vanishing anomalous em charge. If L corresponds to CUA the value of anomalous em charge is integer.

  2. The anomalous em charge in general fails to vanish for the loops of hairpins. For the braids possibly associated with the loops of tRNA the strands can only end up to tRNA itself or nuclear membrane. If there are no braid strands associated with these regions, there is no color or anomalous em charge to be canceled so that the situation trivializes. On the other hand, in the case of tRNA I and U associated with the first nucleotide of the anticodon of tRNA can have a varying value of anomalous em charge. Therefore integer valued em charge and color singletness become possible for tRNA. tRNA can also contain aminoacids. If the aminoacids can carry a varying anomalous em charge with a spectrum corresponding to its values for DNA codons coding it, also they could help to stabilize tRNA by canceling the anomalous em charge.

D.2 tRNA plus amino-acid

  1. Amino-acyl tRNA synthetase, which is the catalyst inducing the fusion of amino-acid with ACC stem [tRNA], could have braid strands to both amino-acid and tRNA and have regions with opposite anomalous em charges compensating separately that of amino-acid and of the active part of tRNA. The required correlation of amino-acid with anticodon would suggest that both D and T loops and A-loop are included. The simplest option is however that the anticodon is connected by braid to amino-acid so that braiding would define the genetic code at the fundamental level and the many-to-one character of genetic code would reflect the 1-to-many character of amino-acid-quark triplet correspondence. This hypothesis is easy to kill: for the portion of catalyst attaching to a given portion of DNA strand amino-acids and codons should have opposite anomalous em charges: Qa(amino)=-Qa(codon).

  2. After the catalysis involving reduction of hbar amino-acid and tRNA would form a system with a vanishing net anomalous em charge but with a braiding structure more complex than that before the fusion.

  3. In the translation process the braiding structure of tRNA- amino-acid system should re-organize: the braid strands connecting anticodon with amino-acid are transformed to braid strands connecting it to mRNA codon with a subsequent reduction of hbar of braid strands bringing tRNA into the vicinity of mRNA. In the transcription the anticodon-codon braiding would be replaced with amino-acid-mRNA braiding forcing formation of the amino-acid sequence. It will be later found that the simpler option without this step corresponds to the earlier hypothesis according to which amino-acids acted originally as catalysts for the formation of RNA double strand.

  4. tRNA is basically coded by genes which suggests that the general symmetries of the genetic code apply to to the variants of tRNA associated with same anticodon. Hence the variants should result from each other by isospin splits and modifications such as permutations of subsequent nucleotides and addition of AT and CG pairs not changing overall color and isospin properties. Also anomalous base pairs X*Y can be added provide their net anomalous em charge vanishes.

  5. tRNA has a complex tertiary (3-D) structure [tertiary] involving base pairing of distant nucleotides associated with the roots of the stem regions where tRNA twists sharply. This pairing could involve formation of braid strands connecting the nucleotides involved. The reduction of Planck constant for these strands could be an essential element of the formation of the tertiary structure.

E. Triplet code as a fusion of singlet and doublet codes?

In [prebio] I have discussed the hypothesis that the standard 3-code has emerged as a fusion of 1-codes with 4 1-codons and 2-code with 16 2-codons. It is interesting to see whether this model is consistent with the braid picture.

E.1 tRNA as fusion of tRNA1 and tRNA1

The earlier proposal was that the fusion of 1- and 2-code to 3-code meant (at least) the fusion of tRNA1 and tRNA2 to form a more complex tRNA of 3-code. This process would have involved fusion of 1- and 2-anticodons of tRNA. The visual inspection of tRNA shows that tRNA1 and tRNA2 could have been simple RNA hairpins during pre-biotic evolution. The variable loop associated with the T arm has indeed emerged during evolution and its function is believed to relate to the stability of tRNA [tRNA]. For instance, the anomalous em charge of the variable loop could compensate for the net em anomalous charge of amino-acid-tRNA system.

tRNA1 is identifiable as a piece of tRNA extending from 5' end to the first nucleotide (wobble nucleotide) of the anticodon. tRNA2 would contain at its 5'-end 2-codon and plus T arm and second half of the acceptor stem. The simpler structure of D-arm (in particular, the stem involves only 3 codon pairs) conforms with this view.

The emergence of tRNA anticodon as a fusion of 1-anticodon and 2-anti-codons could explain the wobble base pairing. The inverse assignment {U→ A, C→ G, {A,G}→ U, {U,A,C}→ I} deduced from the the number 40 of tRNAs and assigning unique 1-codon to only G could be interpreted as a non-deterministic correspondence generating new RNA sequences from existing ones.

E.2 The change of the role of amino-acids in the transition from pre-biotic to biotic evolution

In [prebio] it was proposed that during RNA era amino-acids catalyzed the replication of 2-RNA to its conjugate and that at some state the role of amino-acids and 2-anti-codons changed and instead of conjugate of 2-RNA strand amino-acid sequence was generated. In braiding picture this transition could be understood as a phase transition changing the dynamics of braiding.

  1. Before the transition the amino-acid-2-anticodon braid generated in the formation of tRNA2- amino-acid complex was replaced with 2-anticodon-RNA braid and amino-acid catalyzing the formation of RNA-conjugate strand pair.

  2. In the transition a new step emerged: amino-acid began to form a braid with RNA codon and amino-acid sequence instead of conjugate RNA strand was generated in the process. Note that the number of amino-acids could have been larger than 16 before the transition since several amino-acids could have catalyzed same pairing of 2-codon with its 2-anticodon.

Contrary to the assumption of the original more complex model [prebio], tRNA1 and tRNA2 would have acted on same RNA sequences. Before the transition to 3-code tRNA2 and amino-acids would have been responsible for the formation of double strands of RNA (tqc at RNA level requires the presence of double strands). tRNA1 would have taken care of non-deterministic generation of new RNA sequences driving the evolution during RNA era. There is evidence that centrosomes have their RNA based code and this code might correspond to 2-codon code and involve also the non-deterministic 1-code.

The objection is that the resulting RNA sequences contain A, G, U, and I and are analogous to conjugates of RNA sequences rather than being proper RNA sequences. A possible way out of the problem is to build a conjugate of this sequence using tRNA2. The problem is that if I base pairs with A,T, or C, ne obtains only the codons T,C,A. If U pairs with A and G as in the case of 1-code, also G is obtained. The presence of G*U pairs in tRNA2 suggests that these pairings were indeed present. The presence of I in the tRNA1 induced RNA sequences might prevent their interpretation as genuine RNA sequences, which would imply conjugation symmetry of RNA.

The objection is that the resulting RNA sequences contain A, G, U, and I and are analogous to conjugates of RNA sequences rather than being proper RNA sequences. A possible way out of the problem is to build a conjugate of this sequence using tRNA1 again. Since I pairs with A,T, or C and U with A and G and G with G and A with U all nucleotides appear in the resulting sequence. The anomalous G*U base pairs in tRNA could be seen as remnants of RNA era. The presence of I in the tRNA1 induced RNA sequences might prevent their interpretation as genuine RNA sequences, which would imply conjugation symmetry of RNA.

There is an additional argument supporting the idea that the coding of amino-acids emerges only after the formation of 3-code. If the 2-code would have coded for amino-acids before the fusion of the codes, the fusion should have involved also the fusion of corresponding RNA sequences in order to guarantee that the resulting 3-RNA sequence still codes for the amino-acids coded by 2-RNA sequences plus some new ones. This kind of fusion is not too plausible although I have considered this possibility in the earlier model [prebio].

F. Was the counterpart of cell membrane present during RNA era?

Topological quantum computation should have taken place already during RNA era. This suggest that the counterpart of the cell membrane was present already at that time. Quite recently it was reported that DNA duplexes of length 6 to 20 base pairs can join to longer cylinders which in turn form liquid crystals and that the liquid crystal phase separates from the phase formed by single DNA strands. Long strands had been already earlier known to form liquid crystals. This encourages to think that also RNA duplexes are able to self-organize in this manner so that the analog of cell nucleus containig RNA double helices as genetic material could have existed already during RNA era.

The nuclear membranes could have consisted of either ordinary RNA or its variant consisting of A,T,G,I produced by tRNA1. The latter option would allow to distinguish between coding RNA and RNA used as building block of various structures. The sequences consisting of 30 RNA base pairs would correspond to the thickness of cell membrane and to the codon of M61 code. Lipid layer of thickness 5 nm would correspond to roughly 16 base pairs and to the codon assignable to M17.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer.



Programming of bio-molecular self assembly pathways from TGD point of view

There is an interesting work about programming bio-molecular self assembly pathways [Y. Peng Yin et al (2007), Programming biomolecular self-assembly pathways, Nature 451, 318-322 (17 January 2008)]. The catalytic self assembly of complexes of nuclei acids is carried out automatically by a program represented implicitly as a mixture of linear DNA strand acting as catalyst and so called hairpin DNAs containing three nucleation sites at, bt, ct - so called toeholds.

A. Key ideas

The basic idea is that a set of bio-molecular reactions can be programmed to occur in a desired order by using a generalization of lock and key mechanism. The simplest self assembly pathway can be specified by a collection of keys and locks. In the beginning there is only one key and the this key fits to only one door, which leads into a room with several doors. The lock eats the key but gives one or more keys. If the room contains several doors to which the keys fits, the reaction corresponds to addition of several branches to the already existing reaction product. By continuing in this manner one eventually ends up to the last room and at the last step the lock gives back the original key so that it can act as a catalyst.

The translation of this idea to a program defining self assembly pathway is following.

  1. DNA hairpin [stemloop] defines key structural element of the self-assembly program. Hairpin is a single-stranded DNA strand in meta-stable configuration having form A+B+C such that B forms a loop and C is a palindrome [palindrome]. The formal expression for palindromy is C= At*: this means that C read backwards (Ct) is conjugate A* of A implying that A and C running in opposite direction can form a double strand (duplex) by hydrogen bonding. As catalytic a* acting as key forms a double strand with a, the hairpin molecule opens to a linear DNA molecule and energy is liberated. In this process original key is lost but the two other toe-holds bt and ct contained by the hairpin become available as keys. Each hairpin in the mixture of catalyst and hairpin molecules has its own lock and two keys.

  2. The process of opening new doors continues until all hairpin molecules are used. The key given by the last lock must be catalyst strand a*. The outcome is a molecule consisting of pieces of DNA strands and can possess a very complex topology. For instance, the formation trees and star like structures can be easily programmed.

  3. To run this program one needs only an optimal mixture of catalyst molecule and hairpin DNA molecules. In the applications discussed in hairpins have length of order 10 nm which corresponds to p-adic length scale L(151) defining also cell membrane thickness. That L(151) corresponds also to the length of 30-nucleotide sequence defining the codon of the code associated with Mersenne prime M61=261-1 might not be an accident. The simplest applications are autocatalytic formation of DNA duplex molecules and of branched junctions, nucleated dendritic growth, and autonomous locomotion of a bipedal walker.

The basic idea in the realization of the autonomous motion of bipedal walker is to cheat the walker to follow a track marked by food. The walker literally eats the food and receives in this manner the metabolic energy needed to make the step to the next piece of food. The menu contains two kinds of hairpins as foods: hairpins A attached regularly along the desired path of the walker (second DNA strand) and hairpins B but not attached to the strand. The front leg l of the walker attaches to A and this catalyzes the formation of the duplex A*B as a waste and the liberated metabolic energy allows to make a step in which hind leg becomes the front leg.

B. TGD view about the situation

The possibility to program the self-assembly relies on the almost deterministic realization of the lock and key mechanism. The presence of braid strands could make this possible.

  1. Consider first the hypothesis about the cancelation of anomalous DNA charge. The palindromic character of A means that the neck of the hairpin has vanishing anomalous em charge and also vanishing color charge is possible. Hence palindromes are favored in TGD Universe. The circular piece B is not in general color singlet. It could have braid strands connecting it to it to some other DNA or nuclear membrane but this is not necessary. Same applies to the toehold at at the end of the other strand of neck.

  2. The attachment of the lock to key could be seen as a process in which a braid consisting of magnetic flux tubes connecting lock and key strands (DNA and its conjugate) is formed spontaneously and followed by a phase transition reducing hbar contracting the flux tubes and in this manner guiding the key to the lock.

For a more detailed exposition and background see the chapter DNA as Topological Quantum Computer.



DNA as topological quantum computer: XI

In previous postings I, II, III, IV, V, VI, VII, VIII, IX, X 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.

Since the representation in the book and in previous postings is bottom-up and not well-organized, it is perhaps worth of providing a summary about the model in both bottom-up (very briefly) and top-to-bottom manner.

1. Bottom-up approach

I ended up with the third model in bottom-up manner and this representation is followed also in the text. The model which looks the most plausible one relies on two specific ideas.

  1. Sharing of labor means conjugate DNA would do tqc and DNA would "print" the outcome of tqc in terms of RNA yielding aminoacids in the 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.

  2. 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.

2. Top-down approach

One ends up to the model also in top-down manner.

  1. Darwinian selection for which standard theory of self-organization provides a model, should apply also to tqc programs. Tqc programs should correspond to asymptotic self-organization patterns selected by dissipation in the presence of metabolic energy feed. The spatial and temporal pattern of the metabolic energy feed characterizes the tqc program - or equivalently - sub-program call.

  2. Since braiding characterizes the tqc program, the self-organization pattern should correspond to a hydrodynamical flow or a pattern of magnetic field inducing the braiding. Braid strands must correspond to magnetic flux tubes of the magnetic body of DNA. If each nucleotide is transversal magnetic dipole it gives rise to transversal flux tubes, which can also connect to the genome of another cell.

  3. The output of tqc sub-program is probability distribution for the outcomes of state function reduction so that the sub-program must be repeated very many times. It is represented as four-dimensional patterns for various rates (chemical rates, nerve pulse patterns, EEG power distributions,...) having also identification as temporal densities of zero energy states in various scales. By the fractality of TGD Universe there is a hierarchy of tqcs corresponding to p-adic and dark matter hierarchies. Programs (space-time sheets defining coherence regions) call programs in shorter scale. If the self-organizing system has a periodic behavior each tqc module defines a large number of almost copies of itself asymptotically. Generalized EEG could naturally define this periodic pattern and each period of EEG would correspond to an initiation and halting of tqc. This brings in mind the periodically occurring sol-gel phase transition inside cell near the cell membrane.

  4. Fluid flow must induce the braiding which requires that the ends of braid strands must be anchored to the fluid flow. Recalling that lipid mono-layers of the cell membrane are liquid crystals and lipids of interior mono-layer have hydrophilic ends pointing towards cell interior, it is easy to guess that DNA nucleotides are connected to lipids by magnetic flux tubes and hydrophilic lipid ends are stuck to the flow.

  5. The topology of the braid traversing cell membrane cannot not affected by the hydrodynamical flow. Hence braid strands must be split during tqc. This also induces the desired magnetic isolation from the environment. Halting of tqc reconnects them and make possible the communication of the outcome of tqc.

  6. There are several problems related to the details of the realization. How nucleotides A,T,C,G are coded to strand color and what this color corresponds to? The prediction that wormhole contacts carrying quark and anti-quark at their ends appear in all length scales in TGD Universe resolves the problem. How to split the braid strands in a controlled manner? High Tc super conductivity provides the mechanism: braid strand can be split only if the supra current flowing through it vanishes. A suitable voltage pulse induces the supra-current and its negative cancels it. The conformation of the lipid controls whether it it can follow the flow or not. How magnetic flux tubes can be cut without breaking the conservation of the magnetic flux? The notion of wormhole magnetic field saves the situation now: after the splitting the flux returns back along the second space-time sheet of wormhole magnetic field.

To sum up, it seems that essentially all new physics involved with TGD based view about quantum biology enter to the model in crucial manner.

For details see the chapter DNA as Topological Quantum Computer.



About the arrow of psychological time and notion of self: once again!

Quantum classical correspondence predicts that the arrow of subjective time is somehow mapped to that for the geometric time. The detailed mechanism for how the arrow of psychological time emerges has however remained open. Also the notion of self is problematic. I have proposed two alternative notions of self and have not been able to choose between them. A further question is what happens during sleep: do we lose consciousness or is it that we cannot remember anything about this period? The work with the model of topological quantum computation (see previous posting) has led to an overall view allowing to select the most plausible answer to these questions. But let us be cautious!

A. Two earlier views about how the arrow of psychological time emerges

The basic question how the arrow of subjective time is mapped to that of geometric time. The common assumption of all models is that quantum jump sequence corresponds to evolution and that by quantum classical correspondence this evolution must have a correlate at space-time level so that each quantum jump replaces typical space-time surface with a more evolved one.

  1. The earliest model assumes that the space-time sheet assignable to observer ("self") drifts along a larger space-time sheet towards geometric future quantum jump by quantum jump: this is like driving car in a landscape but in the direction of geometric time and seeing the changing landscape. There are several objections.

    1. Why this drifting?

    2. If one has a large number of space-time sheets (the number is actually infinite) as one has in the hierarchy the drifting velocity of the smallest space-time sheet with respect to the largest one can be arbitrarily large (infinite).

    3. It is alarming that the evolution of the background space-time sheet by quantum jumps, which must be the quintessence of quantum classical correspondence, is not needed at all in the model.

  2. Second model relies on the idea that intentional action -understood as p-adic-to-real phase transition for space-time sheets and generating zero energy states and corresponding real space-time sheets - proceeds as a kind of wave front towards geometric future quantum jump by quantum jump. Also sensory input would be concentrated on this kind of wave front. The difficult problem is to understand why the contents of sensory input and intentional action are localized so strongly to this wave front and rather than coming from entire life cycle.

There are also other models but these two are the ones which come into my mind first.

B. The third option

The third explanation for the arrow of psychological time - which I have considered earlier but only half-seriously - began to look very elegant during last night. This option is actually favored by Occam's razor since it uses only the assumption that space-time sheets are replaced by more evolved ones in each quantum jump. Also the model of tqc favors it.

  1. The simplest assumption is that evolution in a reasonable approximation means shifting of the field patterns backwards in geometric time by some amount per quantum jump. This makes sense since the shift with respect to M4 time coordinate is an exact symmetry of extremals of Kähler action. It is also an excellent approximate symmetry for the preferred extremals of Kähler action and thus for maxima of Kähler function spoiled only by the presence of light-cone boundaries. This shift occurs for both the perceiver space-time sheet and perceived space-time sheet representing external world: both perceiver and percept change.

  2. Both the landscape and observer space-time sheet remain in the same position in imbedding space but both are modified by this shift in each quantum jump. The perceiver experiences this as a motion in 4-D landscape. Perceiver (Mohammed) would not drift to the geometric future (the mountain) but geometric future (the mountain) would effectively come to the perceiver (Mohammed)!

  3. There is an obvious analogy with Turing machine: what is however new is that the tape effectively comes from the geometric future and Turing machine can modify the entire incoming tape by intentional action. This analogy might be more than accidental and could provide a model for quantum Turing machine operating in TGD Universe. This Turing machine would be able to change its own program as a whole by using the outcomes of the computation already performed.

  4. The concentration of the sensory input and the effects of conscious motor action to a narrow interval of time (.1 seconds typically, secondary p-adic time scale associated with the largest Mersenne M127 defining p-adic length scale which is not completely super-atronomical) can be understood as a concentration of sensory/motor attention to an interval with this duration: the space-time sheet representing sensory "me" would have this temporal length and "me" definitely corresponds to a zero energy state.

  5. The fractal view about topological quantum computation strongly suggests an ensemble of almost copies of sensory "me" scattered along my entire life cycle and each of them experiencing my life as a separate almost copy. My childhood is still sensorily lived but has moved about 57 years backwards in geometric time and would live the year 1897 but enjoy all techno conveniences of the year 1950!

  6. The model of geometric and subjective memories would not be modified in an essential manner: memories would result when "me" is connected with my almost copy in the geometric past by braid strands or massless extremals (MEs) or their combinations (ME parallel to magnetic flux tube is the analog of Alfwen wave in TGD).

C. Can one choose between the two variants for the notion of self?

I have considered two different notions of "self" and it is interesting to see whether this picture might allow to choose between them.

  1. In the original variant of the theory "self" corresponds to a sequence of quantum jumps. "Self" would result through a binding of quantum jumps to single "string" in close analogy and actually in a concrete correspondence with the formation of bound states. Each quantum jump has a fractal structure: unitary process is followed by a sequence of state function reductions and preparations proceeding from long to short scales. Selves can have sub-selves and one has self hierarchy. The questionable assumption is that self remains conscious only as long as it is able to avoid entanglement with environment.

  2. According to the newer variant of theory, quantum jump has a fractal structure so that there are quantum jumps within quantum jumps: this hierarchy of quantum jumps within quantum jumps would correspond to the hierarchy of dark matters labelled by the values of Planck constant. Each fractal structure of this kind would have highest level (largest Planck constant) and this level would corresponds to the self. What might be called irreducible self would corresponds to a quantum jump without any sub-quantum jumps (no mental images). The quantum jump sequence for lower levels of dark matter hierarchy would create the experience of flow of subjective time.

    It would be nice to reduce the notion of self hierarchy to that of fractal quantum jump in the sense of dark matter hierarchy but there is an objection. Does this concept really make sense? Fractality is a geometric notion and subjective time does not reduce to the geometry. It is also not quite clear whether the reasonable looking idea about the role of entanglement can be kept.

The older variant of self looks more attractive if one accepts the new model for the arrow of psychological time.

  1. Entire Universe performs the quantum jump and there is an infinite fractal hierarchy of scales associated with quantum jump and state function reduction/state preparation part of quantum jump proceeds as a sequence from long to short scales. One cannot assign any finite geometric duration to a given step in this sequence since the geometric duration assignable to the entire quantum jump would in this case be automatically infinite. In this framework our life cycle would most naturally correspond to a sequence of quantum jumps.

  2. The simplest guess for the interval of geometric time assignable to single quantum jump is as CP2 time. p-Adic time scales define alternative and perhaps more attractive identification. The larger the value of p-adic prime p, the faster the psychological time would flow and faster the experienced rate of evolution would be. Also the hierarchy of Planck constants suggests a hierarchy of these times and the concentration of attention to to dark matter levels would make the flow of psychological time much faster. The model of tqc suggests that each period of EEG rhythm corresponds to single quantum jump for corresponding "me" in un-entangled self-state.

  3. The ability to avoid entanglement with environment would be essential for the original notion of self. One can of however ask whether the assumption about the loss of consciousness in entanglement - that is during sleep - is really necessary. One could however argue that if consciousness is really lost during sleep, we could not have the deep conviction that we existed yesterday. Furthermore, during topological quantum computation entanglement is absent and thus this state should correspond to conscious experience. Night time is however the best time for tqc since sensory input and motor action do not take metabolic resources and we certainly do problem solving during sleep. Thus we should be conscious at some level during sleep and perform quite a long tqc. Perhaps we are!

    Could it be that we do not remember anything about the period of sleep because our attention is directed elsewhere and memory recall uses only copies of "me" assignable to brain manufacturing standardized mental images? Perhaps the communication link to the mental images during sleep experienced at dark levels of existence is lacking or sensory input and motor activities of busy westeners do not allow to use metabolic energy to build up this kind of communications. Hence one can seriously ask, whether self is actually eternal with respect to the subjective time and whether entangling with some system means only diving into the ocean of consciousness as someone has expressed it. We would be Gods as also quantum classical correspondence in the reverse direction requires (p-adic cognitive space-time sheets have literally infinite size in both temporal and spatial directions). This would be the most optimistic view that one can imagine.

This arguments look nice but more arguments are needed to exclude the model of self as single quantum jump. D. What after biological death?

Could the new option allow to speculate about the course of events at the moment of death? Certainly this particular sensory "me" would effectively meet the geometro-temporal boundary of the biological body: sensory input would cease and there would be no biological body to use anymore. "Me" might lose its consciousness (if it can!). "Me" has also other mental images than sensory ones and these could begin to dominate the consciousness and "me" could direct its attention to space-time sheets corresponding to much longer time scale, perhaps even to that of life cycle, giving a summary about the life.

What after that? The Tibetan Book of Dead gives some inspiration. A western "me" might hope (and even try use its intentional powers to guarantee) that quantum Turing tape brings in a living organism, be it human or cat or dog or at least some little bug. If this "me" is lucky, it could direct its attention to it and become one of the very many sensory "me's" populating this particular 4-D biological body. There would be room for a newcomer unlike in the alternative models. A "me" with Eastern/New-Ageish traits could however direct its attention permanently to the dark space-time sheets and achieve what might she might call enlightment.

For details see the chapter DNA as Topological Quantum Computer.



DNA as topological quantum computer: X

In previous postings I, II, III, IV, V, VI, VII, VIII, IX 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.

Many problems of quantum computation in standard sense might relate to a wrong view about quantum theory. If TGD Universe is the physical universe, the situation would improve in many respects. There is the new fractal view about quantum jump and observer as "self"; there is p-adic length scale hierarchy and hierarchy of Planck constants as well as self hierarchy; there is a new view about entanglement and the possibility of irreducible entanglement carrying genuine information and making possible quantum superposition of fractal quantum computations and quantum parallel dissipation; there is zero energy ontology, the notion of M-matrix allowing to understand quantum theory as a square root of thermodynamics, the notion of measurement resolution allowing to identify M-matrix in terms of Connes tensor product; there is also the notion of magnetic body providing one promising realization for braids in tqc, etc... Taking the risk of boring the reader by repeating things that I have already said I will summarize these new aspects TGD below.

There is also a second motivation. Quantum TGD and TGD inspired theory of consciousness involve quite a bundle of new ideas and the continual checking of internal consistency by writing it through again and again is of utmost importance. The following considerations can be also seen as this kind of checking. I can only represent apologies to the benevolent reader: this is a work in progress.

A. Fractal hierarchies

Fractal hierarchies are the essence of TGD. There is hierarchy of space-time sheets labelled by preferred p-adic primes. There is hierarchy of Planck constants reflecting a book like structure of the generalized imbedding space and identified in terms of a hierarchy of dark matters. These hierarchies correspond at the level of conscious experience to a hierarchy of conscious entities -selves: self experiences its sub-selves as mental images.

Fractal hierarchies mean completely new element in the model for quantum computation. The decomposition of quantum computation to a fractal hierarchy of quantum computations is one implication of this hierarchy and means that each quantum computation proceeds from longer to shorter time scales Tn=2-nT0 as a cascade like process such that at each level there is a large number of quantum computations performed with various values of input parameters defined by the output at previous level. Under some additional assumptions to be discussed later this hierarchy involves at a given level a large number of replicas of a given sub-module of tqc so that the output of single fractal sub-module gives automatically probabilities for various outcomes as required.

B. Irreducible entanglement and possibility of quantum parallel quantum computation

The basic distinction from standard measurement theory is irreducible entanglement not reduced in quantum jump.

B.1 NMP and the possibility of irreducible entanglement

Negentropy Maximimization Principle states that entanglement entropy is minimized in quantum jump. For standard Shannon entropy this would lead to a final state which corresponds to a ray of state space. If entanglement probabilities are rational -or even algebraic - one can replace Shannon entropy with its number theoretic counterpart in which p-adic norm of probability replaces the probability in the argument of logarithm: log(pn)→ log(Np(pn). This entropy can have negative values. It is not quite clear whether prime p should be chosen to maximize the number theoretic negentropy or whether p is the p-adic prime characterizing the light-like partonic 3-surface in question.

Obviously NMP favors generation of irreducible entanglement which however can be reduced in U process. Irreducible entanglement is something completely new and the proposed interpretation is in terms of experience of various kinds of conscious experiences with positive content such as understanding.

Quantum superposition of unitarily evolving quantum states generalizes to a quantum superposition of quantum jump sequences defining dissipative time evolutions. Dissipating quarks inside quantum coherent hadrons would provide a basic example of this kind of situation.

B.2 Quantum parallel quantum computations and conscious experience

The combination of quantum parallel quantum jump sequences with the fractal hierarchies of scales implies the possibility of quantum parallel quantum computations. In ordinary quantum computation halting selects single computation but in the recent case arbitrarily large number of computations can be carried out simultaneously at various branches of entangled state. The probability distribution for the outcomes is obtained using only single computation.

One would have quantum superposition of space-time sheets (assignable to the maxima of Kähler function) each representing classically the outcome of a particular computation. Each branch would correspond to its own conscious experience but the entire system would correspond to a self experiencing consciously the outcome of computation as intuitive and holistic understanding, abstraction. Emotions and emotional intellect could correspond to this kind of non-symbolic representation for the outcome of computation as analogs for collective parameters like temperature and pressure.

B.3 Delicacies

There are several delicacies involved.

  1. The above argument works for factors of type I. For HFFs of type II1 the finite measurement resolution characterized in terms of the inclusion Nsubset M mean is that state function reduction takes place to N-ray. There are good reasons to expect that the notion of number theoretic entanglement negentropy generalizes also to this case. Note that the entanglement associated with N is below measurement resolution.

  2. In TGD inspired theory of consciousness irreducible entanglement makes possible sharing and fusion of mental images. At space-time level the space-time sheets corresponding to selves are disjoint but the space-time sheets topologically condensed at them are joined typically by what I call join along boundaries bonds identifiable as braid strands (magnetic flux quanta). In topological computation with finite measurement resolution this kind of entanglement with environment would be below the natural resolution and would not be a problem.

  3. State function reduction means quantum jump to an eigen state of density matrix. Suppose that density matrix has rational elements. Number theoretic vision forces to ask whether the quantum jump to eigen state is possible if the eigenvalues of ρ do not belong to the algebraic extension of rationals and p-adic numbers used. If not, then one would have number theoretically irreducible entanglement depending on the algebraic extension used. If the eigenvalues actually define the extension there would be no restrictions: this option is definitely simpler.

  4. Fuzzy quantum logic (see this) brings also complications. What happens in the case of quantum spinors that spin ceases to be observable and one cannot reduce the state to spin up or spin down. Rather, one can measure only the eigenvalues for the probability operator for spin up (and thus for spin down) so that one has fuzzy quantum logic characterized by quantum phase. Inclusions of HFFs are characterized by quantum phases and a possible interpretation is that the quantum parallelism related to the finite measurement resolution could give rise to fuzzy qubits. Also the number theoretic quantum parallelism implied by number theoretic NMP could effectively make probabilities as operators. The probabilities for various outcomes would correspond to outcomes of quantum parallel state function reductions.

C.Connes tensor product defines universal entanglement

Both time-like entanglement between quantum states with opposite quantum numbers represented by M-matrix and space-like entanglement reduce to Connes tensor dictated highly uniquely by measurement resolution characterized by inclusion of HFFs of type II1

C.1 Time-like and space-like entanglement in zero energy ontology

If hyper-finite factors of II1 are all that is needed then Connes tensor product defines universal S-matrix and the most general situation corresponds to a direct sum of them. M-matrix for each summand is product of Hermitian square root of density matrix and unitary S-matrix multiplied by a square root of probability having interpretation as analog for Boltzmann weight or probability defined by density matrix (note that it is essential to have Tr(Id)=1 for factors of type II1. If factor of type I are present situation is more complex. This means that quantum computations are highly universal and M-matrices are characterized by the inclusion N subset M in each summand defining measurement resolution. Hermitian elements of N act as symmetries of M-matrix. The identification of the reducible entanglement characterized by Boltzmann weight like parameters in terms of thermal equilibrium would allow interpret quantum theory as square root of thermodynamics.

If the entanglement probabilities defined by S-matrix and assignable to N rays do not belong to the algebraic extension used then a full state function reduction is prevented by NMP. Ff the generalized Boltzmann weights are also algebraic then also thermal entanglement is irreducible. In p-adic thermodynamics for Virasoro generator L0 and using some cutoff for conformal weights the Boltzmann weights are rational numbers expressible using powers of p-adic prime p.

C.2 Effects of finite temperature

Usually finite temperature is seen as a problem for quantum computation. In TGD framework the effect of finite temperature is to replace zero energy states formed as pairs of positive and negative energy states with a superposition in which energy varies.

One has an ensemble of space-time sheets which should represent nearly replicas of the quantum computation. There are two cases to be considered.

  1. If the thermal entanglement is reducible then each space-time sheet gives outcome corresponding to a well defined energy and one must form average over these outcomes.

  2. If thermal entanglement is irreducible each space-time sheet corresponds to a quantum superposition of space-time sheets, and if the outcome is represented classically as rates and temporal field patterns, it should reflect thermal average of the outcomes as such.

If the degrees of freedom assignable to topological quantum computation do not depend on the energy of the state, thermal width does not affect at all the relevant probabilities. The probabilities are actually affected even in the case of tqc since 1-gates are not purely topological and the effects of temperature in spin degrees of freedom are unavoidable. If T grows the probability distribution for outcomes flattens and it becomes difficult to select the desired outcome as that appearing with maximal probability.

D. Possible problems related to quantum computation

At least following problems are encountered in quantum computation.

  1. How to preserve quantum coherence for a sufficiently long time so that unitary evolution can be achieved?

  2. The outcome of calculation is always probability distribution: for instance, the output with maximum probability can correspond to the result of computation. The problem is how to replicate the computation with a sufficient accuracy. Or more precisely, how to produce replicas of the hardware of quantum computer defined in terms of classical physics?

  3. How to isolate the quantum computer from the external world during computation and despite this feed in the inputs and extract the outputs?

D.1 The notion of coherence region in TGD framework

In standard framework one can speak about coherence in two senses. At the level of Schrödinger amplitudes one speaks about coherence region inside which it makes sense to speak about Schrödinger time evolution. This notion is rather defined.

In TGD framework coherence region is identifiable as inside which modified Dirac equation holds true. Strictly speaking, this region corresponds to a light-like partonic 3-surface whereas 4-D space-time sheet corresponds to coherence region for classical fields. p-Adic length scale hierarchy and hierarchy of Planck constants means that arbitrarily large coherence regions are possible.

The precise definition for the notion of coherence region and the presence of scale hierarchies imply that the coherence in the case of single quantum computation is not a problem in TGD framework. De-coherence time or coherence time correspond to the temporal span of space-time sheet and a hierarchy coming in powers of two for a given value of Planck constant is predicted by basic quantum TGD. p-Adic length scale hypothesis and favored values of Planck constant would naturally reflect this fundamental fractal hierarchy.

D.2 De-coherence of density matrix and replicas of tqc

Second phenomenological description boils down to the assumption that non-diagonal elements of the density matrix in some preferred basis (involving spatial localization of p"../articles/) approach to zero. The existence of more or less faithful replicas of space-time sheet in given scale allows to identify the counterpart of this notion in TGD context. De-coherence would mean a loss of information in the averaging of M-matrix and density matrix associated with these space-time sheets.

Topological computations are probabilistic. This means that one has a collection of space-time sheets such that each space-time sheet corresponds to more or less same tqc and therefore same M-matrix. If M is too random (in the limits allowed by Connes tensor product), the analog of generalized phase information represented by its "phase" - S-matrix - is useless.

In order to avoid de-coherence in this sense, the space-time sheets must be approximate copies of each other. Almost copies are expected to result by dissipation leading to asymptotic self-organization patterns depending only weakly on initial conditions and having also space-time correlate. Obviously, the role of dissipation in eliminating effects of de-coherence in tqc would be something new. The enormous symmetries of M-matrix, the uniqueness of S-matrix for given resolution and parameters characterizing braiding, fractality, and generalized Bohr orbit property of space-time sheets, plus dissipation give good hopes that almost replicas can be obtained.

D.3 Isolation and representations of the outcome of tqc

The interaction with environment makes quantum computation difficult. In the case of topological quantum computation this interaction corresponds to the formation of braid strands connecting the computing space-time sheet with space-time sheets in environment. The environment is four-dimensional in TGD framework and an isolation in time direction might be required. The space-time sheets responsible for replicas of tqc should not be connected by light-like braids strands having time-like projections in M4.

Length scale hierarchy coming in powers of two and finite measurement resolution might help considerably. Finite measurement resolution means that those strands which connect space-time sheets topologically condensed to the space-time sheets in question do not induce entanglement visible at this level and should not be affect tqc in the resolution used.

Hence only the elimination of strands responsible for tqc at given level and connecting computing space-time sheet to space-time sheets at same level in environment is necessary and would require magnetic isolation. Note that super-conductivity might provide this kind of isolation. This kind of elimination could involve the same mechanism as the initiation of tqc which cuts the braid strands so the initiation and isolation might be more or less the same thing.

Strands reconnect after the halting of tqc and would make possible the communication of the outcome of computation along strands by using say em currents in turn generating generalized EEG, nerve pulse patterns, gene expression, etc... halting and initiation could be more or less synonymous with isolation and communication of the outcome of tqc.

D.4 How to express the outcome of quantum computation?

The outcome of quantum computation is basically a representation of probabilities for the outcome of tqc. There are two representations for the outcome of tqc. Symbolic representation which quite generally is in terms of probability distributions represented in terms "classical space-time" physics. Rates for various processes having basically interpretation as geometro-temporal densities would represent the probabilities just as in case of particle physics experiment. For tqc in living matter this would correspond to gene expression, neural firing, EEG patterns,...

A representation as a conscious experience is another (and actually the ultimate) representation of the outcome. It need not have any symbolic counterpart since it is felt. Intuition, emotions and emotional intelligence would naturally relate to this kind of representation made possible by irreducible entanglement. This representation would be based on fuzzy qubits and would mean that the outcome would be true or false only with certain probability. This unreliability would be felt consciously.

In the proposed model of tqc the emergence of EEG rhythm (say theta rhythm) and correlated firing patterns would correspond to the isolation at the first half period of tqc and random firing at second half period to the sub-sequent tqc:s at shorter time scales coming as negative powers of 2. The fractal hierarchy of time scales would correspond to a hierarchy of frequency scales for generalized EEG and power spectra at these scales would give information about the outcome of tqc. Synchronization would be obviously an essential element in this picture and could be understood in terms of classical dynamics which defines space-time surface as a generalized Bohr orbit.

Tqc would be analogous to the generation of a dynamical hologram or "conscious hologram" (see this). EEG rhythm would correspond to reference wave and the contributions of spikes to EEG would correspond to the incoming wave interfering with it. Two remarks are in order.

D.5 How data is feeded into submodules of tqc?

Scale hierarchy obviously gives tqc a fractal modular structure and the question is how data is feeded to submodules at shorter length scales. There are are certainly interactions between different levels of scale hierarchy. The general ideas about master-slave hierarchy assigned with self-organization support the hypothesis that these interactions are directed from longer to shorter scales and have interpretation as a specialization of input data to tqc sub-modules represented by smaller space-time sheets of hierarchy. The call of submodule would occur when the tqc of the calling module halts and the result of computation is expressed as a 4-D pattern. The lower level module would start only after the halting of tqc (with respect to subjective time) and the durations of resulting tqcs would come as Tn= 2-nT0 that geometric series of tqcs would become possible. There would be entire family of tqcs at lower level corresponding to different values of input parameters from calling module.

D.6 The role of dissipation and energy feed

Dissipation plays key role in the theory of self-organizing systems. Its role is to serve as a Darwinian selector. Without an external energy feed the outcome is a situation in which all organized motions disappear. In presence of energy feed highly unique self-organization patterns depending very weakly on initial conditions emerge.

In case of tqc one function of dissipation would be to drive the braidings to static standard configurations, prevent over-braiding, and perhaps even effectively eliminate fluctuations in non-topological degrees of freedom. Note that magnetic fields are important for 1-gates. Magnetic flux conservation however saves magnetic fields from dissipation.

External energy feed is needed in order to generate new braidings. For the proposed model of cellular tqc the flow of intracellular water induces the braiding and requires energy feed. Also now dissipation would drive this flow to standard patterns coding for tqc programs. Metabolic energy would be also needed in order to control whether lipids can move or not by generating cis type unsaturated bonds.

For the model of DNA as topological quantum computer see the chapter DNA as Topological Quantum Computer.



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