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 n^th strand commute with the generators of braid group which do not act on n^th strand. G would be naturally subgroup of the covering group of rotation group acting in spin degrees of spin 1/2 object. Since U(1) transformations generate only an overall phase to the state, the presence of this factor might not be necessary. A possible candidate for U(1) factor is as a rotation induced by a time-like parallel translation defined by the electromagnetic scalar potential Φ=A_t.

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∫ A_tdt/hbar) coming from the presence of scale potential Φ=A_t 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 of "Genes and Memes".

We know!

TGD was born 4 years before superstring model. Around 1994 I performed first version of p-adic mass calculations of elementary particle masses. Now I am now applying TGD at precise quantitative level to model bio-systems as macroscopic quantum systems. At this moment M-theorist's most notable attempts to build connection to observed reality are speculations about the possibility of blackholes and time machines in LHC.

In today's Cosmic Variance, only five centuries after Newton, Sean Carroll - after stating clearly that we cannot know anything for sure- told us that - after all- "we know" that there is definitely no new physics involved with living cell or brain. To induce the desired associations the title of posting referred to telekinesis but the message was "No New Physics-Period". Lubos echoed Sean Carroll's "we-know"'s (the title referred now to spoon bending).

Neither Sean nor Lubos bothered to mention dark matter- roughly 95 percent of all matter that exists- and its possible relevance for biology since they "we-know" that it has only cosmological relevance. Lubos also emphasized that quantum theory in the realm acceccible to experimentation has reached its final form. In particular, "we-know" that Planck constant has only single value. Message taken.

The grateful audience joined to stormy applauds for these "we-knower"'s but there was also an exception. Someone not-so-well-informed mentioned that DNA double strands have "telepathic abilities". DNA double strands are able to gather together in solution: the same ability of single strand and conjugate has been known for a long time.

In the case of single DNA strands TGD explains this in terms magnetic flux tubes connecting DNA and conjugate DNA strands: also attraction of double strands could be explained by this mechanism. These magnetic flux tubes carrying particles in large Planck constant phase and defining braids could make DNA topological quantum computer. The phase transitions changing hbar and inducing shortening (or lengthening) of the braid strands would allow biomolecules to find each other and play key role in bio-catalysis: consider only DNA replication, transcription of DNA to mRNA, and translation of mRNA to aminoacid sequences.

The reaction of Lubos Motl was what one might have expected: "The full article is here, http://pubs.acs.org/cgi-bin/abst.../jp7112297.html - it is really amazing or, more likely, complete rubbish" (note that the ability of single strands to find each other is known for decades!).

So many theoreticians in this state of "we-knowing" see only what their theory allows them to see. Brings in my mind similar situation more than century ago before quantum mechanics: also then "we-know"'ed that classical mechanics is all that is needed to describe every imaginable observation. History does not seem to be a good teacher but also its students seem to be often rather silly.

Quantum model of nerve pulse V: Summary

Quite recently I learned [1,2,3,4,5] (thanks to Ulla Mattfolk) that nerve pulse propagation seems to be an adiabatic process and thus does not dissipate: the authors propose that 2-D acoustic soliton is in question. Adiabaticity is what one expects if the ionic currents are dark currents (large hbar and low dissipation) or even supra currents. Furthermore, Josephson currents are oscillatory so that no pumping is needed. Combining this input with the model of DNA as topological quantum computer (tqc) [8] leads to a rather precise model for the generation of nerve pulse. The following gives a brief summary of main points of the model in its recent form.

1. The system would consist of two superconductors- microtubule space-time sheet and the space-time sheet in cell exterior- connected by Josephson junctions represented by magnetic flux tubes defining also braiding in the model of tqc. The phase difference between two super-conductors would obey Sine-Gordon equation allowing both standing and propagating solitonic solutions. A sequence of rotating gravitational penduli coupled to each other would be the mechanical analog for the system. Soliton sequences having as a mechanical analog penduli rotating with constant velocity but with a constant phase difference between them would generate moving kHz synchronous oscillation. Periodic boundary conditions at the ends of the axon rather than chemistry determine the propagation velocities of kHz waves and kHz synchrony is an automatic consequence since the times taken by the pulses to travel along the axon are multiples of same time unit. Also moving oscillations in EEG range can be considered and would require larger value of Planck constant in accordance with vision about evolution as gradual increase of Planck constant.

2. During nerve pulse one pendulum would be kicked so that it would start to oscillate instead of rotating and this oscillation pattern would move with the velocity of kHz soliton sequence. The velocity of kHz wave and nerve pulse is fixed by periodic boundary conditions at the ends of the axon implying that the time spent by the nerve pulse in traveling along axon is always a multiple of the same unit: this implies kHz synchrony. The model predicts the value of Planck constant for the magnetic flux tubes associated with Josephson junctions and the predicted force caused by the ionic Josephson currents is of correct order of magnitude for reasonable values of the densities of ions. The model predicts kHz em radiation as Josephson radiation generated by moving soliton sequences. EEG would also correspond to Josephson radiation: it could be generated either by moving or standing soliton sequences (latter are naturally assignable to neuronal cell bodies for which hbar should be correspondingly larger): synchrony is predicted also now.

3. The previous view about microtubules in nerve pulse conduction can be sharpened. Microtubular electric field (always in the same direction) could explain why kHz and EEG waves and nerve pulse propagate always in same direction and might also feed energy to system so that solitonic velocity could be interpreted as drift velocity. This also inspires a generalization of the model of DNA as topological quantum computer [7] since also microtubule-cell membrane systems are good candidates for performers of tqc. Cell replication during which DNA is out of game seems to require this and microtubule-cell membrane tqc would represent higher level tqc distinguishing between multi-cellulars and mono-cellulars.

4. New physics would enter in several manners. Ions should form Bose-Einstein cyclotron condensates. The new nuclear physics predicted by TGD [8] predicts that ordinary fermionic ions (such as K⁺, Na⁺, Cl^-) have bosonic chemical equivalents with slightly differing mass number. Anomalies of nuclear physics and cold fusion provide experimental support for the predicted new nuclear physics. Electronic supra current pulse from microtubules could induce the kick of pendulum inducing nerve pulse and induce a small heating and expansion of the axon. The return flux of ionic Josephson currents would induce convective cooling of the axonal membrane. A small transfer of small positive charge into the inner lipid layer could induce electronic supra current by attractive Coulomb interaction. The exchange of exotic W bosons which are scaled up variants of ordinary W^+/- bosons is a natural manner to achieve this if new nuclear physics is indeed present. There are a lot of support for this new physics: cold fusion and nuclear transmutations in living matter [8] ( these I have discussed in previous postings).

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

References

[1] Soliton model.

[2] T. Heimburg and A. D. Jackson (2005), On soliton propagation in biomembranes and nerves, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), On the action potential as a propagating density pulse and the role of anesthetics, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.

[5] Physicists challenge notion of electric nerve impulses; say sound more likely.

[6] Saltation.

[7] The chapter DNA as Topological Quantum Computer of "Genes and Memes".

[8] The chapter Nuclear String Physics of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

TGD assigns 10 Hz biorhythm to electron as an intrinsic frequency scale

p-Adic coupling constant evolution and origins of p-adic length scale hypothesis have remained for a long time poorly understood. The progress made in the understanding of the S-matrix of the theory (or rather, its generalization M-matrix) (see this) has however changed the situation. The unexpected prediction is that zero energy ontology assigns to elementary particles macroscopic times scales. In particular, the time scale assignable to electron correspond to the fundamental biorhythm of 10 Hz.

1. M-matrix and coupling constant evolution

The final breakthrough in the understanding of p-adic coupling constant evolution came through the understanding of S-matrix, or actually M-matrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). M-matrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. S-matrix is analogous to the phase of Schrödinger amplitude multiplying positive and real square root of density matrix analogous to modulus of Schrödinger amplitude.

The notion of finite measurement resolution realized in terms of inclusions of von Neumann algebras allows to demonstrate that the irreducible components of M-matrix are unique and possesses huge symmetries in the sense that the hermitian elements of included factor N subset M} defining the measurement resolution act as symmetries of M-matrix, which suggests a connection with integrable quantum field theories.

It is also possible to understand coupling constant evolution as a discretized evolution associated with time scales T_n, which come as octaves of a fundamental time scale: T_n=2ⁱT₀. Number theoretic universality requires that renormalized coupling constants are rational or at most algebraic numbers and this is achieved by this discretization since the logarithms of discretized mass scale appearing in the expressions of renormalized coupling constants reduce to the form log(2ⁱ)=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results.

2. p-Adic coupling constant evolution

Could the time scale hierarchy T_n= 2ⁱT₀ defining hierarchy of measurement resolutions in time variable induce p-adic coupling constant evolution and explain why p-adic length scales correspond to L_p propto p^1/2R, p≈ 2^k, R CP² length scale? This looks attractive but there is a problem. p-Adic length scales come as powers of 2^1/2 rather than 2 and the strongly favored values of k are primes and thus odd so that n=k/2 would be half odd integer. This problem can be solved.

1. The observation that the distance traveled by a Brownian particle during time t satisfies r²= Dt suggests a solution to the problem. p-Adic thermodynamics applies because the partonic 3-surfaces X² are as 2-D dynamical systems random apart from light-likeness of their orbit. For CP² type vacuum extremals the situation reduces to that for a one-dimensional random light-like curve in M⁴. The orbits of Brownian particle would now correspond to light-like geodesics \gamma₃ at X³. The projection of γ₃ to a time=constant section X² subset X³ would define the 2-D path γ² of the Brownian particle. The M⁴ distance r between the end points of γ² would be given r²=Dt. The favored values of t would correspond to T_n=2ⁱT₀ (the full light-like geodesic). p-Adic length scales would result as L²(k)= D T(k)= D2^kT₀ for D=R²/T₀. Since only CP² scale is available as a fundamental scale, one would have T₀= R and D=R and L²(k)= T(k)R.

2. p-Adic primes near powers of 2 would be in preferred position. p-Adic time scale would not relate to the p-adic length scale via T_p= L_p/c as assumed implicitly earlier but via T_p= L_p²/R₀= p^1/2L_p, which corresponds to secondary p-adic length scale. For instance, in the case of electron with p=M₁₂₇ one would have T₁₂₇=.1 second which defines a fundamental biological rhythm. Neutrinos with mass around .1 eV would correspond to L(169)≈ 5 μm (size of a small cell) and T(169)≈ 1.× 10⁴ years. A deep connection between elementary particle physics and biology becomes highly suggestive.

3. In the proposed picture the p-adic prime p≈ 2^k would characterize the thermodynamics of the random motion of light-like geodesics of X³ so that p-adic prime p would indeed be an inherent property of X³.

4. The fundamental role of 2-adicity suggests that the fundamental coupling constant evolution and p-adic mass calculations could be formulated also in terms of 2-adic thermodynamics. With a suitable definition of the canonical identification used to map 2-adic mass squared values to real numbers this is possible, and the differences between 2-adic and p-adic thermodynamics are extremely small for large values of for p≈ 2^k. 2-adic temperature must be chosen to be T²=1/k whereas p-adic temperature is T_p= 1 for fermions. If the canonical identification is defined as

   ∑_n≥0 b_n 2ⁿ→ ∑_m≥12^-km∑_{0≤ n< m} b_km+n2ⁿ.

   It maps all 2-adic integers n<2^k to themselves and the predictions are essentially same as for p-adic thermodynamics. For large values of p≈ 2^k 2-adic real thermodynamics with T_R=1/k gives essentially the same results as the 2-adic one in the lowest order so that the interpretation in terms of effective 2-adic/p-adic topology is possible.

3. p-Adic length scale hypothesis and biology

The basic implication of zero energy ontology is the formula T(k)≈ 2^k/2L(k)/c= L(2,k)/c. This would be the analog of E=hf in quantum mechanics and together hierarchy of Planck constants would imply direct connection between elementary particle physics and macroscopic physics. Especially important this connection would be in macroscopic quantum systems, say for Bose Einstein condensates of Cooper pairs, whose signature the rhythms with T(k) as period would be. The presence of this kind of rhythms might even allow to deduce the existence of Bose-Einstein condensates of hitherto unknown particles.

1. For electron one has T(k)=.1 seconds which defines the fundamental f_e=10 Hz bio-rhythm appearing as a peak frequency in alpha band. This could be seen as a direct evidence for a Bose-Einstein condensate of Cooper pairs of high T_c super-conductivity. That transition to "creative" states of mind involving transition to resonance in alpha band might be seen as evidence for formation of large BE condensates of electron Cooper pairs.

2. TGD based model for atomic nucleus (see this) predicts that nucleons are connected by flux tubes having at their ends light quarks and anti-quarks with masses not too far from electron mass. The corresponding p-adic frequencies f_q= 2^kf_e could serve as a biological signature of exotic quarks connecting nucleons to nuclear strings . k_q=118 suggested by nuclear string model would give f_q= 2¹⁸f_e=26.2 Hz. Schumann resonances are around 7.8, 14.3, 20.8, 27.3 and 33.8 Hz and f_q is not too far from 27.3 Hz Schumann resonance and the cyclotron frequency f_c(¹¹B⁺)=27.3 Hz for B=.2 Gauss explaining the effects of ELF em fields on vertebrate brain.

3. For a given T(k) the harmonics of the fundamental frequency f=1/T(k) are predicted as special time scales. Also resonance like phenomena might present. In the case of cyclotron frequencies they would favor values of magnetic field for which the resonance condition is achieved. The magnetic field which in case of electron gives cyclotron frequency equal to 10 Hz is B_e≈ 3.03 nT. For ion with charge Z and mass number A the magnetic field would be B_I= (A/Z)× (m_p/m_e)×B_e. The B=.2 Gauss magnetic field explaining the findings about effects of ELF em fields on vertebrate brain is near to B_I for ions with f_c alpha band. Hence the value of B could be understood in terms of resonance with electronic B-E condensate.

4. The hierarchy of Planck constants predicts additional time scales T(k). The prediction depends on the strength of the additional assumptions made. One could have scales of form nT(k)/m with m labeling the levels of hierarchy. m=1 would give integers multiples of T(k). Integers n could correspond to ruler and compass integers expressible as products of first powers of Fermat primes and power of 2. There are only four known Fermat primes so that one has n=2ⁱ∏_i F_i, F_i in {3,5,17,257, 2¹⁶+1}. In the first approximation only 3- and 5- and 17-multiples of 2-adic length scales would result besides 2-adic length scales. In more general case products m¹m² and ratios m¹/m² of ruler and compass integers and their inverses 1/m¹m² and m²m¹ are possible.

5. Mersenne primes are expected to define the most important fundamental p-adic time scales. The list of real and Gaussian (complex) Mersennes M_n possibly relevant for biology is given by n=89, 107, 113*, 127, 151*,157*, 163*, 167* ('*' tells that Gaussian Mersenne is in question). See the table.

For background see that chapter New Physics and Qualia of "Quantum Hardware of Living Matter".

Quantum model of nerve pulse IV: Could microtubule-axon system perform topological quantum computation?

The proposed picture is consistent with the model of DNA as a topological quantum computer [7] and with the idea that also microtubuli could be involved with tqc. The model of DNA as tqc in its basic form assumes that DNA is connected to the nuclear membrane and cell membranes associated with the cell body by magnetic flux tubes such that each nucleotide is connected to single lipid. Tqc programs are coded to the temporal braiding patters of lipids. This requires that lipid layer is liquid crystal and thus below the critical temperature. The flux tube connecting DNA to inner lipid layer and that beginning from outer lipid layer can form single flux tube or be split. If they form single flux tube braiding and tqc are not possible. During tqc the braid strands going through cell membrane are split and the dance of lipids induced by water flow defining time like braiding (hydrophilic lipid ends are anchored to the cellular water) induces braiding of the magnetic flux tubes which write the tqc program to memory. Furthermore, the lifetimes of flux tubes in the connected state must be short enough to prevent the generation of a nerve pulse. This is the case if the temperature is sufficiently below the critical temperature. The ionic supra currents are identifiable as the observed quantal non-dissipative currents flowing through the cell membrane when tqc is not on.

Centrioles have their own genetic code realized in terms of RNA and they play key role during gene replication when DNA is out of the game. This encourages to think that microtubuli make possible an independent tqc like process. How microtubule-axon system could then perform tqc? One can consider two options and also their hybrid in the proposed model for nerve pulse.

1. Option I: Magnetic flux tubes connect microtubules to the space-time sheet of cell exterior. In the model of DNA as tqc these flux tubes continue back to the nucleus or another nucleus: the flux tubes must be split at cell membrane during tqc and this splitting induces the required isolation from the external world during tqc. During nerve pulse the situation would be different and the flow of lipids in liquid phase could induce braiding: the isolation would however fail now. Tqc would explain why the axon melts during nerve pulse.

   One can become critical and ask why also the magnetic flux tubes from DNA could not end to the space-time sheet of the cell exterior. The justification for 'No' (besides isolation) could be that also cell soma would possess standing soliton sequence like waves and standing nerve pulses.

   Could one then see this tqc as a special variant of DNA-membrane tqc? The idea about magnetic flux tubes emanating from DNA and flowing along microtubuli interiors and radiating to the axonal membrane looks ugly: in any case, this would not affect tqc and nerve pulse could be seen as a direct gene expression.

2. Option II: For some years ago I considered the possibility of a gel-sol-gel phase transition proceeding along the surface surface of the micro-tubuli, accompanying nerve pulse, perhaps inducing nerve pulse, and coding for long term sensory memories in terms of 13 13-bit sequences defined by the tubulin helices with bit represented as a conformation of microtubule. This hypothesis might be easily shown to be wrong on basis of the available experimental facts already now. Suppose however that this phase transition happens and that the braid strands do not continue from the microtubular surface to the cell nucleus. In this case the braiding could be induced by a gel-sol-gel transition accompanying and perhaps generating the nerve pulse at the microtubular level and inducing the disassembly of the tubulins followed by re-assembly inducing the braiding. Also this braiding would contribute to tqc like process or at least memory storage by braiding.

The following considerations do not depend on the option used.

1. What comes first in mind is that the braiding codes memories, with memories understood in TGD sense using the notion of 4-D brain: that is in terms of communications between brain geometrically now and brain in the geometric past. In standard neuroscience framework braiding of course cannot code for memories since it changes continually. Nerve pulse sequences would code for long term sensory memories in a time scale longer than millisecond and microtubular phase transition could have a fine structure coding for sensory data in time scales shorter than nerve pulse duration. The fact is that we are able to distinguish from each other stimuli whose temporal distance is much shorter than millisecond and this kind of coding could make this possible. Also the direct communication of the auditory (sensory) input using photons propagating along MEs parallel to axon could also explain this.

2. In the model of DNA as tqc nucleotides A,T,C,G are coded into a 4-color of braid strand represented in terms of quarks u,d and their antiquarks. An analogous coding could be present also now. The coding would result if DNA is connected to microtubuli but this option does not look attractive. If each aminoacid can be accompanied by 3-braid with colors of any of DNA codons coding each aminoacid of tubulin would be connected to 3 lipids. As a matter fact, 3-braids can be regarded as fundamental sub-modules of tqc programs since 3-braid is the smallest N-braid which can do non-trivial tqc. Tubulins could be seen as higher level modules consisting of order hundred 500 amino-acids. This corresponds to a DNA strand with length of about .5 μm corresponding to p-adic length scale L(163) which is one of the four magic p-adic length scales (k=151,157,163,167) which correspond to Gaussian Mersennes. This higher level language character of microtubular tqc programs would conform with the fact that only eukaryotes possess them.

3. Cellular cytoskeleton consists of microtubuli. Could microtubular tqc -in either of the proposed forms- take place also at the cell soma level? Could DNA-nuclear membrane system define the primordial tqc and microtubular cytoskeleton-cell membrane system a higher level tqc that emerged together with the advent of the multicellulars? Is only the latter tqc performed at the multicellular level? The notions of super- and hypergenome encourage to think that both tqcs are performed in all length scales. One can imagine that ordinary cell membrane decomposes into regions above and below the critical point (the value of the critical temperature can be controlled. Those below it would be connected to DNA by flux tube bundles flowing inside the microtubular cylinders. Microtubular surfaces would in turn be connected to the regions above the critical point. One should also understand the role of M₁₃=2¹³-1 12-bit higher level "genetic code" assignable naturally to microtubuli. For instance, could the bit of this code tell whether the module defined by the tubulin dimer strand bundle participates tqc or not?

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

References

[1] Soliton model.

[2] T. Heimburg and A. D. Jackson (2005), On soliton propagation in biomembranes and nerves, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), On the action potential as a propagating density pulse and the role of anesthetics, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.

[5] Physicists challenge notion of electric nerve impulses; say sound more likely.

[6] Saltation.

[7] The chapter DNA as Topological Quantum Computer of "Genes and Memes".

Quantum model of nerve pulse III: Relation to Hodgkin-Huxley model

Before the replacement of Hodgkin-Huxley model with a genuinely quantal model can be taken seriously, one must answer many difficult questions which also Hodgkin and Huxley must have faced as they developed their own model. In the following I will go through the basic questions and quantum answers to them.

1. Questions and answers

Q: In the resting state membrane potential is negative and cell has a negative net charge. What stabilizes the cell against the leakage of the negative charge if pumps and channels are not responsible for this?

A: The findings about the strange behavior of cell membrane inspire TGD based answer. Cell membrane space-time sheet is its own quantum world and the flow of ions occurs only in the presence of magnetic flux tubes connecting it to the external world. These currents a however oscillatory Josephson currents if dissipation is absent. Hence there is no need to cut completely the connections to the external world.

Q: How the resting state can result spontaneously if pumps are absent?

A: If ionic currents are Josephson currents, they are automatically oscillating and the return to the original state is guaranteed. The flux tubes carrying the ionic currents will be assumed to connect axonal microtubules to the space-time sheet of the cell interior. Consider first the most obvious objections.

1. Dark ions could not transform to ordinary ones in the exterior of the cell membrane. This might indeed kill the model.

2. If ionic currents are Josephson currents, they are automatically oscillating and the return to the original state is guaranteed. The objection is that all biologically important ions are not bosons and the model for high T_c super-conductor in its recent form allows only electronic and protonic Cooper pairs at room temperature [8]. TGD based nuclear physics however predicts the possibility of exotic nuclei for which one or more color bonds connecting nucleons to the nuclear string are charged. These exotic nuclei with electronic states identical to those of genuine ions could save the situation. The table below describes how cyclotron frequencies for B=.2 Gauss of the most important ions are modified in the simplest replacements with exotic ions. For instance, the notation Mg⁺⁺_- tells that there is double electronic ionization and electron shell of Argon as usual but that one color bond is negatively charged.

   ²³Na⁺ →¹⁹Ne₊: 13.1 Hz → 15.7 Hz

   ²³Na⁺²⁴→ Mg⁺⁺_-: 13.1 Hz→ 12.5 Hz

   ³⁹K⁺⁴⁰→ A₊: 7.7 Hz→ 7.5 Hz

   ³⁹K⁺→ ⁴⁰Ca⁺⁺_-: 7.7 Hz→ 7.5 Hz

   ³⁵Cl^-→⁴⁰A_-: 8.6 Hz →7.5 Hz

   f_c(K⁺) and f_c(Cl^-) are replaced with the frequency 7.5 Hz and one can do only using the cyclotron frequencies f_c(Ca⁺⁺)/2=7.5 Hz, f_c(Mg⁺⁺)=12.5 Hz, and f_c(Ca⁺⁺)=15 Hz. The nominal values of the lowest Schumann frequencies are 7.8 Hz and 14.3 Hz. All ions with relevance for nerve pulse and EEG could be bosonic ions or bosonic pseudo-ions. I do not know how well the needed ionization mechanisms are understood in the standard framework.

For small oscillations the maximal charge transfer ΔQ generated by an oscillating ionic Josephson current during the cycle is proportional to hbar /f_J propto hbar² and hbar /Ω propto hbar for solitonic situation. ΔQ is very small for the ordinary value of hbar : also the oscillation period is very small. For large values of hbar situation changes and large maximal ion transfers are possible.

An hbar increasing phase transition could be involved with the generation of the nerve pulse. Quantum criticality during nerve pulse generation indeed suggest the presence of flux tubes with varying values of hbar . The lifetimes of the connected flux tubes could be proportional to hbar at criticality. A fractal hierarchy of pulses and EEG like oscillations of the membrane potential corresponding to various values of hbar is suggestive.

Q: Can one make this more quantitative?

A: One can construct a model based on Sine-Gordon wave equation [7] for the phase different Φ between the superconductors connected by Josephson junction sequences defined by magnetic flux tubes and idealizable as a continuous Josephson junction.

1. For a Josephson junction idealizable as a hollow cylinder with radius R and thickness d the expression of the Josephson current reads as

   J= J₀sin(Ze∫ Vdt/hbar)

   J₀ is in case of cell membrane given by

   J₀= (Ze2π dR/Λ²) ×(hbar/m) ,

   where R and d would be now the radius and thickness of the axon, Λ is the magnetic peneration length, and m is the mass of the charge carrier. Although this expression does not hold true as such when Josephson junctions are replaced by magnetic flux tubes connecting microtubuli and axon, one can can safely make some qualitative conclusions. The amplitude of the Josephson current increases with hbar . For electron the value of the amplitude is by a factor x≈ Am_p/m_e≈ 2¹¹A larger than for ion with a mass number A. This gives for electron Cooper pairs a unique role as an initiator of the nerve pulse. Note that the amplitudes of the Josephson currents of electron and ions are quite near to each other if one has hbar (ion)= 2¹¹A×hbar: this might explain why the powers of 2¹¹ for hbar seem to be favored.

2. Electronic Josephson current dominates and makes it ideal for the generation of nerve pulse (kick to gravitational pendulum). This is possible if the net amount of electronic charge is so small that it flows out during the generation of flux tubes. For ions this need not occur even if ion densities are of same order of magnitude. Constant voltage V creates an oscillating current and no catastrophic leakage takes place and the resting state results automatically. The ionic Josephson currents assignable to the magnetic flux tubes connecting microtubules through the cell membrane to the external world could be responsible for the nerve pulse.

3. The mechanical analog for Sine-Gordon system [8] assignable to Josephson junction is rotating pendulum but one must be cautious in applying this analogy. There are two options concerning the modeling of the situation.
   1. Membrane potential represents an external voltage V(t) and one has Φ_i= Z_ie∫ Vdt/hbar, where Φ_i is the phase difference between Bose-Einstein condensates.
   2. System is autonomous and membrane potential V(t)=hbar (dΦ_i/dt)/Z_ie is completely determined by the dynamics of any phase Φ_i. This option is highly predictive and discussed in the sequel.

4. The analogy with gravitational pendulum allows to identify the phase angle Φ as the counterpart of angle Θ characterizing angular position of mathematical pendulum (note that this analogy can be misleading since it implicitly brings in 3-D thinking).
   1. In this picture rotating pendulum corresponds to a soliton sequence containing infinite number of solitons: both stationary and moving soliton sequences are obtained. The sign of Ω=dΦ/dt is fixed and approximately constant for large values of Ω. Resting potential could correspond to this kind of situation and Ω ≈ 2π kHz is suggested by kHz synchrony. A mechanism of this synchrony will be discussed below. For large values of hbar even values of Ω in EEG range could correspond to membrane potential. For large values of Ω one as V≈ hbarΩ_i/Z_ie. If also EEG rhythms correspond to Ω they must correspond to different values of hbar and f propto 1/hbar would hold true. Changes in the dominating EEG rhythm (40 Hz, 10 Hz, 5 Hz,..) could correspond to phase transitions changing hbar to given value for a large number of axons. The maximal charge transfer during single period is proportional to Δ Q propto 1/Ω.
   2. Hyperpolarization/polarization would mean fastening/slowing down of the pendulum rotation and slowing down would make the system unstable. Near criticality against the generation of nerve pulse would mean that pendulum is rotating rather slowly (Ω<< f_J ) so that a small kick can transform rotation to oscillation. The sign of V propto dΦ/dt would change and large amplitude oscillatory motion would result for single period only after which a kick in opposite direction would lead back to the resting state. Membrane potential varies between the resting potential V₀=-75 V and V₁=+40 V during nerve pulse: V₁>|V₀| would have killed the model. Note that V₁=40 V is rather near to the critical potential about V₁=50 V: ideally these potentials should be identical.
   3. The so called breathers -both stationary and moving- correspond to soliton-antisoliton bound state (see the visualization here). Breathers could be identified as large amplitude oscillations around Φ=0 ground state. Physical intuition suggests that breathers are possible also for a ground state corresponding to a rotating pendulum (representing moving or stationary waves). They would correspond to kicking of one pendulum in a sequence of penduli along z-axis rotating in phase at the initial moment. The kick could correspond to a genuine external perturbation generated by a pair electronic supra current pulses of opposite sign giving constant velocity increments ΔΩ initiating and halting the nerve pulse just like they would do in the case of tqc but in opposite time order. If the background corresponds to a propagating EEG wave, also nerve pulse is expected to propagate with same velocity. The propagation direction of EEG wave would also explain why nerve pulses propagate only in single direction.

5. For the ordinary value of hbar , the frequency of the Josephson current corresponds to that assignable to energy .07 eV being around 1.6×10¹³ Hz and quite high. For x==hbar /hbar₀=2⁴⁴ the frequency would be near to cyclotron frequency of about 1 Hz assignable to DNA strands. For x=3× 2^{3× 13} the frequency would be near to the fundamental 10 Hz frequency which is secondary p-adic time scale associated with electron and correspond to the temporal duration of negative energy space-time sheet assignable to electron. For x=3× 2^{3× 11} one would obtain a 640 Hz frequency which corresponds to the time scale of nerve pulse. It seems clear that the original hypothesis that only powers of 2¹¹ define the spectrum of Planck constant is too restrictive. The requirement that cyclotron frequencies and Josephson frequencies are proportional to each other for small oscillations would guarantee resonant behavior for common strength of the magnetic field would give hbar propto A. This would require that each ion species lives at its own flux tubes.

Q: What instabilizes the axon? Why the reduction rather than increase of the magnitude of the membrane potential induces the instability? Why the reduction of the resting potential below the critical value induces nerve pulse?

A: Large enough voltage pulse between microtubules and membrane could generate electronic DC supra current. The introduction of a small amount of positive charge to the inner lipid layer and staying there for some time would generate the voltage pulse between microtubules and lipid layer so that DC electronic supra current would be induced, and induce the reduction Δ V≈ .02 eV of the magnitude of the membrane potential. A similar introduction of negative charge would induce hyperpolarization and the direction of the current would be opposite if it is generated at all. The mechanism generating the small positive charge to the inner lipid layer could be based on the exchange of exotic W bosons between pairs of exotic nuclei at opposite sides of the cell membrane so that the negative charge of the inner lipid layer would be reduced.

Q: Can one understand the observed radial force, the increase of the radius of axons and the reduction of its thickness, and heating followed by cooling?

A: The observed outward force acting on a test system might be due to the ionic Josephson currents to which the test system responds. During the second half of the pulse the sign of the ionic force is predicted to change. The pressure caused by the electronic Josephson current pulse before the connection of flux tubes to single flux tube might relate to the increase of the radius of the axonal membrane and with the reduction of its thickness as well as the slight increase of its temperature as being due to the electrons which heat the lipid layer as they collide with it. The ions return at the second half of the pulse and could transfer the heat away by convection.

1. This hypothesis gives the estimate for the force f per unit area as

   f_2e(t)= (dn(lipid)/da) ×(J(t)/2e)× hbar k

   = (dn(lipid)/da) × U× (hbar² k/2m_e)× sin(ω_J(2t)) ,

   U= (2π A/Λ²) .

   The parameter A corresponds to the parameter dR in the case that Josephson junctions have the thickness of axonal membrane, and is not relevant for order of magnitude estimate. R corresponds to the distance between microtubules and cell exterior space-time sheet to which flux tubes end. dn(lipid)/da is the 2-D density Josephson junctions equal to the density of lipids.

   k≈ 1/R is the wave vector of electron Cooper pair at the magnetic flux tube. The 3-momentum of electron is enormous for the proposed value of hbar , and the only possible interpretation is that the four-momentum of electron is virtual and space-like and corresponds to exchange of space-like virtual photon describing Coulomb interaction with lipid layer.

   Λ² satisfies in the first approximation the formula

   Λ^-2 = (4π e²n_e/m_e)+ ∑_I (4π e²n_I/Am_I)= α_em16π² ×hbar₀[(n_e/m_e)+ ∑_I (n_I/A_Im_I)].

   Note that there is no real dependence on hbar . Above critical voltage electrons could dominate in the expression but during nerve pulse ions should give the dominating contributions. U cannot be too far from unity.

2. From this one can integrate the total momentum of Cooper pairs transferred to the lipid layer before the flux tubes fuse together if one knows the value of time t when this happens. F propto hbar²/m_e² proportionality implies that for the required large value of hbar /hbar₀ ≈ 3× 2³³ the force is by a factor 6× 10²⁰ stronger than for hbar₀.

3. The force caused by ionic Josephson currents on piston is given by

   f(t)= ∑_I (2me/m_I) (2/Z_I) × f_2e (τ)

   τ=(Z_I/2)×(Ω/ω_J)× t .

4. The comparison with the observed force gives estimate for the value of magnetic penetration length and thus density of electrons at the flux tube.

   According to [3] in one particular experiment the force on piston of area S= .01 cm² at the maximum of voltage (t= 2π/Ω) is F= 2 nN. This gives a killer test for the model. One obtains an estimate for the parameter U=(Λ²/2π A) as

   U=Λ²/2πA= (dn(lipid)/da) × (hbar² k/m_pcF)× ∑_I (2/A_IZ_I) .

   The value of U should not deviate too much from unity. One can use the estimates

   hbar/hbar₀=3×2³³ , k=2π/R.

   Note that the experimental arrangement forces to use this value of k. The actual value in normal situation would be smaller and depends on the distance of the boundary of the cell exterior space-time sheet on microtubules. Using the values d=10 nm and R=5 μm this gives

   U≈ ∑_I (2/A_IZ_I)× X ,

   X= 9× 2⁶⁶× (hbar₀² 2π/m_pcFR)×(S/S(lipid)).

   The factor X=.9267 is of order unity! Taking into account that hbar/hbar₀ is enormously large number it is difficult to believe that the result could be mere accident. Hence U does not deviate too much from unity and there are good hopes that the model works.

   For n_I= x_I/a³, a=10^-10 m, and A= dR one obtains a direct estimate which combined with above estimate gives two conditions which should be consistent with each other:

   U= 76.1×∑_I(x_I/A_I) ,

   U= .93×∑_I(2/A_IZ_I) .

   These estimates are consistent for x_I≈ 10^-2, which makes sense.

Q: Where the primary wave propagates: along axon or along microtubules?

A: This question need not make sense if microtubules and axon are connected by magnetic flux tubes to form single quantum coherent system. That axonal microtubules have constant electric field which is always in same direction could explain why the background soliton sequences and nerve pulses propagate always in the same direction and suggests that the primary wave propagates along microtubules. On the other hand, if W exchange between cell exterior and exterior reduces the negative charge of the inner lipid layer then axon could be seen as initiator. This could induce conformational or gel-sol phase transition propagating along microtubule and inducing the pair of voltage pulses in turn inducing the fusion of flux tubes at cell membrane which in turn would induce criticality of the axonal membrane. For this option axonal soliton would be a shadow of the microtubular soliton rather than completely independent dynamical process.

Q: How nerve pulse velocities are determined?

A: At first glance it seems v could be determined by boundary conditions guaranteing synchronization of neuronal activity rather than by dissipation as in Hodkin-Huxley model. As a matter fact, dissipation turns out to affect also v just because it is determined by boundary conditions!

1. Hodkin-Huxley model would suggest that nerve pulse velocity v is dictated by frictional effects as an analog of a drift velocity. The rough order of magnitude estimates for the velocities of conformational waves along micro-tubuli are consistent with the velocities of nerve pulses. The proportionality v propto d of nerve pulse velocity to nerve axonal radius might be understood as resulting on the dependence on the length of flux tubes connecting axon and microtubules and mediating a frictional feedback interaction from axon. Feedback would be naturally reduced as d increases. Feedback interaction could explain also the sensitivity of the thermal parameters of the axonal membrane to the proteins in its vicinity. If the frictional feedback is due to the environmental noise at the axon amplified at quantum criticality this is what one expects. Quite generally, quantum criticality would explain the high sensitivity of the thermal parameters on noise. Saltation cannot be responsible for the higher conduction velocity in myelin sheathed portions of axon. The insulation would reduce the environmental noise at the level of axons and thus reduce the frictional feedback from axon to the microtubules.

2. The introduction of friction is however problematic in the recent situation. In absence of boundary conditions Sine-Gordon equation predicts for the propagating soliton sequences a continuous velocity spectrum and friction should affect Ω and V but it is not clear whether it can affect v.
   1. In this framework the boundary boundary conditions at the ends of the axon or some subunit of axon would dictate the values of v: γΩ L/v=n2π corresponds to periodic boundary conditions (note that γ=(1-(v/c)²)^1/2≈ 1 holds true). v=ΩL/n2π implies that friction indeed affects also v.
   2. This relationship states that the time taken by the nerve pulse propagate through the axon is always T= L/v =n2π/Ω: this would synchronize neurons and Ω≈ 2π kHz is suggested by the well-known 1 kHz synchrony difficult to understand in the standard framework where v would be determined by chemistry rather than geometry. Myelin shielding could in this picture guarantee that coherent wave propagation is possible over the entire axon so that boundary conditions can be applied.
   3. This would give v≈ ΩL/n2π≤ ΩL/2π. Ω= 2π kHz and n=1 would give for L in the range 1 cm -10 cm v in the range 10 m/s-100 m/s corresponding roughly to the observed range of values. For short axons velocity would be lower: for L=10 μm one would have v= .01 m/s. For longer axons the value of n could be higher or the axon would decompose into structural units for which periodic boundary conditions are satisfied. The sections between Ranvier nodes have length measured in millimeters as are also the lengths of axonal microtubules and 1 mm would correspond to a velocity of 1 m/s. The actual velocity for the myelinated sections varies between 18-100 m/s so that basic structural units should be longer.
   4. The proportionality of v to the radius of axon would follow from the proportionality of the length of the axon or its basic sub-unit (not longer than ≈ 10 cm) to its radius: the simplest geometric explanation for this would be in terms of scaling invariance of the axonal geometry consistent with fractality of TGD Universe. In the standard framework this proportionality would be explained by the minimization of dissipative losses in the case of long axons: one cannot exclude some variant of this explanation also now since friction indeed reduces v.
   5. There is an electric field associated with microtubules (always in same direction). Could this electric field play the role of external force feeding energy and momentum to the moving soliton sequence to compensate dissipation so that v would have interpretation as a drift velocity?

Q: Can one understand EEG in this framework?

A: Just like kHz waves also EEG generating waves could correspond to propagating soliton sequences. Since V is not affected, the value of hbar must be much larger and one must have hbar propto f, where f defines the EEG rhythm. It is known that EEG amplitudes associated with EEG rhythms behave roughly like 1/f. This can be understood. By Maxwell's equation the divergence of electromagnetic field tensor is proportional to 4-current implying the amplitude of EEG identified as Josephson radiation is proportional J₀/Ω and therefore to hbar. The propagation velocity v= ΩL/2πn of EEG generating waves is rather slow as compared to kHz waves: for f=10 Hz one would have 10 cm long axon v=1 m /s. Synchronization results automatically from periodic boundary conditions at the ends of the axons.

Nerve pulses during EEG rhythms would have much slower velocity of propagation and the duration of nerve pulse would be much longer. The maximal charge transfer would be proportional to 1/hbar. It would thus seem that EEG and nerve pulse activity should exclude each other for a given axon. Ω is however smaller so that the generation of nerve pulse is easier unless also ion densities are lower so that J₀ (analogous to gravitational acceleration g in pendulum analogy) is reduced. Perhaps this takes place. The consistency with the propagation velocity of microtubular conformational (or even gel-sol-gel) waves might pose additional constraints on v and thus on frequencies Ω for which nerve pulses are possible. That ordinary EEG is not associated with ordinary cells might be due to the fact that hbar is much smaller: the fractal analog of EEG generating waves could be present but these EEG waves would correspond to faster oscillations in accordance with the view about evolution as an increase of hbar.

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

References

[1] Soliton model.

[2] T. Heimburg and A. D. Jackson (2005), On soliton propagation in biomembranes and nerves, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), On the action potential as a propagating density pulse and the role of anesthetics, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.

[5] Physicists challenge notion of electric nerve impulses; say sound more likely.

[6] Saltation.

[7] Sine-Gordon

[8] The chapter Bio-Systems as Super-Conductors: part I of "The Quantum Hardware of Living Matter".

Quantum model of nerve pulse II: Basic inputs of TGD based model of nerve pulse

The model of nerve pulse whose inputs are summarized below can be motivated by the observed adiabaticity of the nerve pulse and by the strange findings about ionic currents associated with the cell membrane and by the model of Danish researchers for the nerve pulse [1,2,3,4]. The model involves also a fusion of various ideas of earlier models. In particular, Josephson currents and solitons are in a key role in the model but with the necessary flexibility brought in by the hierarchy of Planck constants.

The basic inputs of the model are following.

1. The presence of acoustic soliton or density pulse proposed by Danish researchers [3] looks plausible but a a more fundamental quantum control mechanism inducing the acoustic soliton cannot be excluded. Among other things this should explain why acoustic solitons propagate always in the same direction. In particular, one can consider a soliton like excitation (say breather for Sine-Gordon equation) associated with the electronic or ionic Josephson currents running along magnetic flux tubes. The strange effects associated with the ionic currents through the cell membrane suggest quite generally that at least weak ionic currents through normal cell membrane are non-dissipative quantal currents. The adiabaticity of the nerve pulse suggests that also strong ionic currents are quantal.

2. Strong ionic currents generating nerve pulse through axonal membrane are absent in the resting state. The naive explanation is simple: the life time of the magnetic flux tubes connecting the axonal interior to the exterior is short or the flux tubes are altogether absent. The observation that Josephson currents in constant voltage are automatically periodic suggests a less naive explanation allowing the flux tubes to be present all the time. The presence of ionic Josephson currents predicts a small amplitude oscillation of membrane potential for which 1 kHz synchronous oscillation is a natural identification. Josephson oscillation correspond naturally to propagating soliton sequences for Sine-Gordon equation [7]. The dynamics of the simplest modes is equivalent to the rotational motion of gravitational pendulum: the oscillation of membrane potential corresponds to the variation of dΦ/dt propto V. Note that if axon is above the melting temperature, the lipid layer is in gel phase and fluid motion is impossible. The surface density of lipids is dramatically reduced at criticality so that lipid layers behave like fluids [3]. This means that tqc is not possible by the braiding of lipids.

3. Nerve pulse is generated when the magnitude of the negative membrane potential is reduced below the critical value. Generation of the nerve pulse is like a kick to a rotating gravitational pendulum changing the sign of Ω= dΦ/dt so that rotational motion is transformed to oscillatory motion lasting for about the period of rotation. An opposite but slightly stronger kick must reduce the situation to the original one but with a slightly higher value of Ω. These kicks could correspond to voltage pulse between microtubules and inner lipid layer of cell membrane induced by the addition of small positive (negative) charge on lipid layer. This pulse would induce electronic DC Josephson current inducing the kick and thus reducing V. The exchange of scaled variants of W bosons (assignable to W MEs) could mediate the transfer of charge through the cell membrane and reduce the membrane potential below the critical value but one can consider also other mechanisms.

4. The conservative option would be that ordinary ionic currents take care of the rest and Hogkin-Huxley model applies. This was assumed in the earliest model in which soliton sequence for Josephson current was assumed to induce nerve pulse sequence: in the recent model this assumption does not make sense. The findings of Danish researchers do not however support the conservative option [3]. Nerve pulse could be due to dark ionic (possibly supra -) currents with large hbar with a low dissipation rate. Their flow would be made possible by the presence of magnetic flux tubes connecting cell interior and exterior.

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

References

[1] Soliton model.

[2] T. Heimburg and A. D. Jackson (2005), On soliton propagation in biomembranes and nerves, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), On the action potential as a propagating density pulse and the role of anesthetics, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.

[5] Physicists challenge notion of electric nerve impulses; say sound more likely.

[6] Saltation.

[7] Sine-Gordon

[8] The chapter DNA as Topological Quantum Computer of "Genes and Memes".

Quantum model of nerve pulse I: Soliton model of nerve pulse

In the first part of series I will briefly summarize soliton model of nerve pulse proposed by Danish researchers [1,2,3,4].

1. The temperature of the axon is slightly above the critical temperature T^c for the phase transition leading from crystal like state of the lipid layers to a liquid crystal state. Near criticality the elastic constants and heat capacity of the membrane vary strongly and have maxima at criticality so that also sound velocity varies strongly near criticality. Also the relaxation times are long. There is also dispersion present meaning that the frequency of sound wave depends nonlinearly on wave vector. Non-linearity and dispersion are prerequisites for the presence of solitons which by definition do not dissipate energy.

2. Variations of temperature, volume, area, and thickness and also other mechanical effects are known to accompany nerve pulse propagation. It is also known that the heat density and temperature of the cell membrane increases slightly first and is then reduced. This suggests adiabaticity in average sense. These findings motivate the assumption that nerve pulse actually corresponds to acoustic soliton [2,3].

3. Soliton model reproduces correctly the velocity of nerve pulse inside myelin sheaths but it is not clear to me how well the much lower conduction velocity in non-myelin sheathed regions is reproduced. It is not clear how the lower values of the conduction velocity and its proportionality to the axonal radius in non-myelinated regions can be understood. Intuitively it however seems clear that the lower velocity is due to the feedback from the interaction of ions with the region exterior to cell membrane. In the case of myelin sheaths the conduction of nerve pulse is usually believed to take place via saltation [6]: the depolarization induced at Ranvier node is believed to be enough to take the membrane potential below critical value in the next node so that nerve pulse hops between the nodes. Insulation would improve the insulation and make this process possible. The reversible heat transfer process is however known to be present also in the myelinated portions of axon so that there must be a pulse propagating also in these regions [3]. It is not clear how the myelin sheet can increase the velocity in the soliton model but the reduction of the feedback inducing friction suggests itself.

4. Soliton property predicts adiabaticity. Ordinary ionic currents however dissipate so that adiabaticity assumption is questionable in standard physics context. The model does not predict the growth of entropy followed by its reduction. This behavior is consistent with adiabaticity in a time resolution of order millisecond.

5. The estimate for the capacitor energy density during the nerve pulse is considerably smaller than the energy density is many times magnitude smaller than that of the acoustic wave. This might allow to demonstrate that Hodgkin-Huxley model is not a complete description of the situation.

6. Authors notice [2,3] that the shapes curves representing solitonic energy density and the capacitor energy density as a function of time are essentially identical. Same applies to the experimentally deduced heat change release curve and capacitor energy density for garfish axon. Also heat release and the deviation of the membrane potential from its resting value are in exact phase. These similarities could reflect a control signal responsible for the nerve pulse originating somewhere else, perhaps at microtubuli. This could explain why secondary nerve pulse is not generated immediately after the first one although the temperature is slightly lower after the pulse than before it. This could of course be also due to the exhaustion of the metabolic resources.

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

References

[1] Soliton model.

[2] T. Heimburg and A. D. Jackson (2005), On soliton propagation in biomembranes and nerves, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), On the action potential as a propagating density pulse and the role of anesthetics, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), Function of Nerves-Action of Anesthetics, Gamma 143, An elementary Introduction.

[5] Physicists challenge notion of electric nerve impulses; say sound more likely.

[6] Saltation.

How quantum classical correspondence is realized at parton level?

Quantum classical correspondence must assign to a given quantum state the most probable space-time sheet depending on its quantum numbers. The space-time sheet X⁴(X³) defined by the Kähler function depends however only on the partonic 3-surface X³, and one must be able to assign to a given quantum state the most probable X³ - call it X³_max - depending on its quantum numbers.

X⁴(X³_max) should carry the gauge fields created by classical gauge charges associated with the Cartan algebra of the gauge group (color isospin and hypercharge and electromagnetic and Z⁰ charge) as well as classical gravitational fields created by the partons. This picture is very similar to that of quantum field theories relying on path integral except that the path integral is restricted to 3-surfaces X³ with exponent of Kähler function bringing in genuine convergence and that 4-D dynamics is deterministic apart from the delicacies due to the 4-D spin glass type vacuum degeneracy of Kähler action.

Stationary phase approximation selects X³_max if the quantum state contains a phase factor depending not only on X³ but also on the quantum numbers of the state. A good guess is that the needed phase factor corresponds to either Chern-Simons type action or a boundary term of YM action associated with a particle carrying gauge charges of the quantum state. This action would be defined for the induced gauge fields. YM action seems to be excluded since it is singular for light-like 3-surfaces associated with the light-like wormhole throats (not only (det(g₃)^1/2 but also det(g₄)^1/2 vanishes).

The challenge is to show that this is enough to guarantee that X⁴(X³_max) carries correct gauge charges. Kind of electric-magnetic duality should relate the normal components F_ni of the gauge fields in X⁴(X³_max) to the gauge fields F_ij induced at X³. An alternative interpretation is in terms of quantum gravitational holography. The difference between Chern-Simons action characterizing quantum state and the fundamental Chern-Simons type factor associated with the Kähler form would be that the latter emerges as the phase of the Dirac determinant.

One is forced to introduce gauge couplings and also electro-weak symmetry breaking via the phase factor. This is in apparent conflict with the idea that all couplings are predictable. The essential uniqueness of M-matrix in the case of HFFs of type II₁ (at least) however means that their values as a function of measurement resolution time scale are fixed by internal consistency. Also quantum criticality leads to the same conclusion. Obviously a kind of bootstrap approach suggests itself.

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

How p-adic coupling constant evolution and p-adic length scale hypothesis emerge from quantum TGD proper?

What p-adic coupling constant evolution really means has remained for a long time more or less open. The progress made in the understanding of the S-matrix of theory has however changed the situation dramatically.

1. M-matrix and coupling constant evolution

The final breakthrough in the understanding of p-adic coupling constant evolution came through the understanding of S-matrix, or actually M-matrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). M-matrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. S-matrix is analogous to the phase of Schrödinger amplitude multiplying positive and real square root of density matrix analogous to modulus of Schrödinger amplitude.

The notion of finite measurement resolution realized in terms of inclusions of von Neumann algebras allows to demonstrate that the irreducible components of M-matrix are unique and possesses huge symmetries in the sense that the hermitian elements of included factor N subset M defining the measurement resolution act as symmetries of M-matrix, which suggests a connection with integrable quantum field theories.

It is also possible to understand coupling constant evolution as a discretized evolution associated with time scales T_n, which come as octaves of a fundamental time scale: T_n=2ⁿT₀. Number theoretic universality requires that renormalized coupling constants are rational or at most algebraic numbers and this is achieved by this discretization since the logarithms of discretized mass scale appearing in the expressions of renormalized coupling constants reduce to the form log(2ⁿ)=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results.

2. p-Adic coupling constant evolution

One can wonder how this picture relates to the earlier hypothesis that p-adic length coupling constant evolution is coded to the hypothesized log(p) normalization of the eigenvalues of the modified Dirac operator D. There are objections against this normalization. log(p) factors are not number theoretically favored and one could consider also other dependencies on p. Since the eigenvalue spectrum of D corresponds to the values of Higgs expectation at points of partonic 2-surface defining number theoretic braids, Higgs expectation would have log(p) multiplicative dependence on p-adic length scale, which does not look attractive.

Is there really any need to assume this kind of normalization? Could the coupling constant evolution in powers of 2 implying time scale hierarchy T_n= 2ⁿT₀ induce p-adic coupling constant evolution and explain why p-adic length scales correspond to L_p propto p^1/2R, p≈ 2^k, R CP₂ length scale? This looks attractive but there is a problem. p-Adic length scales come as powers of 2^1/2 rather than 2 and the strongly favored values of k are primes and thus odd so that n=k/2 would be half odd integer. This problem can be solved.

1. The observation that the distance traveled by a Brownian particle during time t satisfies r²= Dt suggests a solution to the problem. p-Adic thermodynamics applies because the partonic 3-surfaces X² are as 2-D dynamical systems random apart from light-likeness of their orbit. For CP₂ type vacuum extremals the situation reduces to that for a one-dimensional random light-like curve in M⁴. The orbits of Brownian particle would now correspond to light-like geodesics γ₃ at X³. The projection of γ₃ to a time=constant section X² subset X³ would define the 2-D path γ₂ of the Brownian particle. The M⁴ distance r between the end points of γ₂ would be given r²=Dt. The favored values of t would correspond to T_n=2ⁿT₀ (the full light-like geodesic). p-Adic length scales would result as L²(k)= D T(k)= D2^kT₀ for D=R²/T₀. Since only CP₂ scale is available as a fundamental scale, one would have T₀= R and D=R and L²(k)= T(k)R.

2. p-Adic primes near powers of 2 would be in preferred position. p-Adic time scale would not relate to the p-adic length scale via T_p= L_p/c as assumed implicitly earlier but via T_p= L_p²/R₀= p^1/2L_p, which corresponds to secondary p-adic length scale. For instance, in the case of electron with p=M₁₂₇ one would have T₁₂₇=.1 second which defines a fundamental biological rhythm. Neutrinos with mass around .1 eV would correspond to L(169)≈ 5 μm (size of a small cell) and T(169)≈ 10⁴ years. A deep connection between elementary particle physics and biology becomes highly suggestive.

3. In the proposed picture the p-adic prime p≈ 2^k would characterize the thermodynamics of the random motion of light-like geodesics of X³ so that p-adic prime p would indeed be an inherent property of X³.

4. The fundamental role of 2-adicity suggests that the fundamental coupling constant evolution and p-adic mass calculations could be formulated also in terms of 2-adic thermodynamics. With a suitable definition of the canonical identification used to map 2-adic mass squared values to real numbers this is possible, and the differences between 2-adic and p-adic thermodynamics are extremely small for large values of for p≈ 2^k. 2-adic temperature must be chosen to be T₂=1/k whereas p-adic temperature is T_p= 1 for fermions. If the canonical identification is defined as

   ∑_{n≥ 0} b_n 2ⁿ→ ∑_{m ≥1} 2^-m+1∑_{0≤ n< k} b_n+(k-1)m2ⁿ ,

   it maps all 2-adic integers n<2^k to themselves and the predictions are essentially same as for p-adic thermodynamics. For large values of p≈ 2^k 2-adic real thermodynamics with T_R=1/k gives essentially the same results as the 2-adic one in the lowest order so that the interpretation in terms of effective 2-adic/p-adic topology is possible.

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

The notion of prime Hilbert space 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 H_p 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 H_p with its n+1:th tensor power. One can "add fermions" to this state by deleting some prime factors H_p 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 particles in which certain many-particle states at the previous level of hierarchy become particles 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 ζ_p 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 ζ_p. 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 TGD as a Generalized Number Theory III: Infinite Primes of "TGD as a Generalized Number Theory".

Unparticles and p-adic physics

The notion of unparticle introduced by the particle physics Nobelist Howard T. Georgi (see this) is the meme of the last year in particle physics. Unparticles differ from ordinary particles in that they have continuum of mass values. Unparticles result in a conformally invariant quantum field theory theory unless one assumes that the fundamental free fields carry only massless excitations. Conformal invariance is understood here as conformal symmetries of the 4-D Minkowski space. Note that the theories possessing conformal symmetry in the stringy sense predict infinite towers of particles with discrete masses and only the lowest lowest excitations are massless.

For unparticles the measurement of momentum does not fix the energy nor does the measurement of energy fix the magnitude of 3-momentum. An objection against the notion of unparticle is that continuous mass spectrum for free particle means that the Fourier expansion of the field is arbitrary in 4-D sense and the field behaves non-deterministically.

The particles of TGD Universe have certain resemblance to unparticles.

1. In TGD framework scaling invariance is replaced by its discretized variant so that the mass scale as a continuous variable is replaced by a discrete variable whose values are proportional to the powers of square roots of primes.

2. Massless fermions suffer massivation by p-adic thermodynamics for the scaling generator of the generalization of stringy superconformal algebra - something completely new - whereas gauge bosons gain their mass dominantly via their coupling to Higgs, which in the simplest scenario does not contribute to the fermion massless at all.

3. The p-adic mass scale of the particle represents a genuinely new degree of freedom, which becomes pseudo-continuous at the limit of very large mass scales so that something analogous to the unparticle like behavior might emerge.

4. There is a connection with the non-determinism since the failure of the strict determinism is the basic characteristic of TGD Universe and p-adic thermodynamics describes the implications of the light-like randomness of partonic 3-surfaces representing particles/particle orbits. In the case of CP₂ type extremals serving as a model for elementary particles it corresponds to the light-like randomness of M⁴ projection of the extremal appearing above p-adic length scale.

Not all primes are equally favored.

1. Ordinary particles correspond to p-adic primes near integer powers of 2. Mersenne primes and their complex cousins Gaussian Mersennes are especially favored as are also primes near prime powers of 2. The interpretation is that the counterpart of Darwinian selection has picked up the favored p-adic primes from the pseudo-continuum during the cosmic evolution. Also the particles with unfavored p-adic mass scale could be created with some small but non-vanishing probability in particle reactions, and would be analogous to the creation of unparticles. p-Adic unparticles are however expected to decay rapidly to normal particles and could be seen as on mass-shell analogs of virtual particles.

2. The prediction is that both quarks, leptons, Higgs and massive gauge bosons can appear with several favored p-adic mass scales. In the case of neutrinos there is a direct evidence for several p-adic mass scales. TGD based model for hadron massess predicts that even in the case of low lying hadrons quarks can possess several favoured mass scales. Experimentally quark masses have been localized only in certain ranges. The observed bumpiness of top mass distribution might have explanation in terms of several mass scales for U type quarks. The most probable values of Higgs mass deduced from leptonic and hadronic measurements of Weinberg angle seem to differ by a factor of 8 and a possible explanation is that Higgs can appear at two different p-adic mass scales.

For more details see the chapters in the first part of the book p-Adic Length Scale Hypothesis and Dark Matter Hierarchy.