Tuesday, November 29, 2005

Good ideas are published sooner or later

Risto Raitio told in his blog about a paper titled as Differential Structures - the Geometrization of Quantum Mechanics by Torsten Asselmeyer-Maluga and Helge Rose'.

I looked the introduction of the paper and the basic physical ideas turned out to be very familiar from my own homepage! Kind of dejavu experience, definitely! Although authors do not refer to my work, I am happy that these ideas finally begin to find their way to physics archives. Personally I am not allowed to add anything to archives although American Mathematical Society has a link to my homepage in subject classification tables. This is of course understandable: since M theory is the theory of everything there is no need to publish or even archive anything not consistent with M theory.

Only few weeks ago Lubos Motl told in this article A Hagedorn alternative to inflation? about a proposal of A. Nayeri, R. H. Brandenberger, and C. Vafa for a cosmology with Hagedorn temperature as a limiting temperature titled as Producing a Scale-Invariant Spectrum of Perturbations in a Hagedorn Phase of String Cosmology. Cosmology with Hagedorn temperature as a primordial temperature is now more than ten year old piece of TGD: dejavu again! Also these authors forgot to mention my work or they have not yet learned to use internet but this does not spoil my happy mood of mind. It is nice to see that good ideas find their publishers sooner or later.

Returning to the original topic, the authors introduce 3-dimensional singularities of space-time as particles and propose that fields in the interior of space-time and these singularities correspond basically to wave particle duality. In TGD framework partons are identified as lightlike causal horizons of space-time surface and the interior of space-time surface corresponds to field description. In particular, zero modes of configuration space metric defining classical macroscopic degrees of freedom in TGD based generalization of quantum measurement theory are assignable to the interior of space-time.

What is especially nice that the light-like 3-surfaces representing partons allow generalized conformal invariance by their metric 2-dimensionality. Hence 4-dimensional space-time completely unique in that it allows superconformal invariance. This also leads naturally to the Temperley-Lieb algebras and von Neumann algbras known as hyperfinite factors of type II1 appearing in conformal field theories. My sincere but perhaps unrealistic dream is that someone could some day communicate this discovery to people from Harward and Princeton (not an easy task knowing the uni-directional communication abilities of this arrogantzia) since it would put end to the long lasting state of stagnation in theoretical physics.

These factors (as opposed to factors of type I and II appearing in QM and 4-D quantum field theories) emerge also automatically from the construction of the spinor structure in the infinite-dimensional configuration space of 3-surfaces (the "world of classical worlds"), and in their very basic structure code for instance minimal conformal field theories, braid group representations, quantum groups, etc...

One can say that single concept: the "world of classical worlds" with metric fixed uniquely by the mathematical existence requirement and possessing spinor structure determined by this metric implies the basic mathematical structures characterizing conformal field theories and topological quantum field theories.

Even more, hyperfinite type II1 quantum theory leads the elimination of infinities of quantum field theories, and one ends up to a generalized Feynman diagrammatics based on the generalization of braid diagrams and duality as it was defined in the original string models. The generalization of this duality eliminates the mathematically non-existent path integral and thus corresponding infinities.

The detailed summary of this general picture can be found in the first four parts of TGD at my homepage.

Dark matter hierarchy and quantum control and coordination

The understanding of quantum control and coordination is one of the big challenges of TGD inspired theory of consciousness and of living systems. The rapid evolution of the ideas about dark matter hierarchy has lead to the deepening of the views also in this respect. I do not want to repeat all the ideas here but instead refer to earlier posting about EEG and generalization of genetic code. The following general overview about quantum communication and control emerges in this framework.
  1. A fractal hierarchy of Josephson junctions with Josephson currents generates coherent photon and dark intermediate gauge boson states. These states have identification as a fractal hierarchy of EEGs and its electro-weak generalizations (ZEGs and WEGs). The levels of hierarchy result by repeated scalings of Planck constant by a factor about 211 and zooming up various quantum length and time scales. Corresponding scaling down occurs for frequencies assignable to a photon with a given energy. For instance, at the 4:th level of the dark matter hierarchy EEG frequencies correspond to energies above thermal energy at room temperature.

  2. The most important frequencies of EEG correspond to multiplets nfc of cyclotron frequencies of biologically important biological ions (most of them in alpha band) and to the frequencies fJ +/-nfc, where fJ is the Josephson frequency of a zoomed up Josephson junction obtained from cell membrane and having the same resting potential determining the Josephson energy and frequency.

  3. Cyclotron frequencies relate to the control of the biological body by the magnetic body and can be assigned with the magnetic flux sheets going through DNA since it is genome where protein synthesis is initiated and is thus the optimal intermediate step in the cellular control. The magnetic flux sheets organize genomes for sequences of large number of nuclei to what can be regarded as text lines at the pages of book having magnetic flux sheets as pages. The same mechanism at the level of several organisms gives rise to hypergenomes coding for culture: the evolution of hypergenome explains the difference between us and our cousins. Introns, the 97 percent of "junk" in our DNA as materialistic biologists believe, are excellent candidates for the genes involved with hypergenes. Besides chemical expression also 126 bit memetic code expressed in terms of electromagnetic field patterns is highly suggestive. The basic durations of the memetic codons are T=.1 seconds and its zoomed up variants.

  4. One of the basic functions of cell membranes is to perceive the chemical environment using various kinds of receptors as sensors. Neurons have specialized to receive symbolic representations of the sensory input at primary sensory organs. A good guess is that in this case magnetic flux quanta are hollow cylindrical structures serving as templates for axons and possibly other similar structures and define the communication lines connecting cell membranes to the magnetic body.

    The frequencies fJ +/- nfc are associated with these communications. For fJ = 5 H consistent with the value of the basic scaling of hbar, they correspond to beta and theta band obtained as satellites of alpha band. Also harmonics of alpha band and its satellites are present. This is true for the left hemisphere. The basic facts about EEG during sleep force to conclude that for the right hemisphere magnetic flux is quantized using units of charge Z=2 and this implies that the scale of frequencies is scaled down by a factor of 1/2 so that right alpha band would be around 5 Hz, beta at 7.5 Hz, and theta at 2.5 Hz for singly ionized exotic bosonic ions. The narrow resonances at 3,5,7 Hz and 13,15,17 Hz are predicted correctly. During deepest sleep only DNA cyclotron frequencies around 1 Hz are present.

  5. This picture would explain why the temperature of brain must be in the narrow range 36-37 K to guarantee optimal functionality of the organism. If interior superconductivity is lost, magnetic body receives sensory data but is paralyzed since its desires cannot be fulfilled. If boundary superconductivity is lost, magnetic body can make cell do all kinds of things but is blind.

  6. In the length scales below the weak length scale Lw also weak bosons behave as massless particles and the exchange of virtual W bosons makes possible a nonlocal charge transfer. Dark quark-antiquark pairs associated with the color bonds of the atomic nuclei can become charged via the emission of dark W boson and thus produce exotic ions. The same can happen at the higher levels of the dark matter hierarchy. This provides a nonlocal quantal mechanism inducing or changing electromagnetic polarization in turn inducing ordinary charge flows and thus making possible quantum control.

    For instance, the generation of nerve pulse could rely on the reduction of the resting potential below the critical value by this kind of mechanism in turn inducing ordinary charge flow between cell interior and exterior. The mechanism might apply even in the scale of the magnetic body and make possible the control of central nervous system. Also remote mental interactions, in particular telekinesis, might rely on this mechanism. The test would be to look whether psychics can mentally affect charged capacitors very near to dielectric breakdown.

  7. Massless extremals (MEs, topological light rays) serve as correlates for dark bosons. Besides neutral massless extremals TGD predicts also charged massless extremals obtained from their neutral counterparts by a mere color rotation (color and weak quantum numbers are not totally independent in TGD framework). The interpretation of the charged MEs has remained open hitherto. In the recent framework charged massless extremals could be seen as correlates for nonlocal quantum control by affecting charge equilibria whereas neutral MEs would serve as correlates for coordination and communication. Color charged MEs could also induce color charge polarization and flows of color charges and thus generate visual color qualia by the capacitor mechanism.

For the details of the model of EEG see the new chapter Dark Matter Hierarchy and Hierarchy of EEGs? of "Genes, Memes, Qualia, and ...."

Yes! Genetic code can be understood number theoretically!

For the last three weeks I have been working (or rather fighting) with a number theoretical model of the genetic code stimulated by an accidental observation: the number of primes smaller than 64, the number of DNA codons, is 18 and together with 0 and 1 this makes 20, the number of aminoacids!

This led to an intense period of work involving a lot of modular arithmetics and painful MATLAB computations. Things are not made easier by the fact that I have to develop program modules in home PC and run them in University computer. It is tragic that the Physics Department of Helsinki University is so poor that it cannot provide me with the number theoretically advanced Mathematica or even MATLAB working in my personal PC. To say nothing about some financial help. Hence I am forced to work as an unemployed with a minimal unemployment money. The far sighted and wise decision makers of Helsinki University must cry for pain and shame when they cannot do nothing to help me. The people working in the universities of rich countries such as India probably cannot realize how difficult the situation of scientists in the underdeveloped countries like Finland is.

I have already told about the basic ideas behind the number theoretical model of the genetic code. The idea is to maximize the negentropy defined as a number theoretic variant of Shannon entropy by replacing the arguments of logarithms with their p-adic norms. The point is that these entropies can have also negative values as a function of the prime defining the p-adic norm. Negentropy maximization makes it possible to assign a unique prime p(n) to a given integer n representing DNA triplet.

The task is to determine the map mapping DNA codons, which are naturally labelled by 3 4-digits in base 4, to the set of integers n in the range 0-63 and to deduce the map by assigning to the partitions (n,r) of n to r summands Boltzmann weights f(r)= exp(-H(r)/T) and by maximizing the negentropy. One can consider bosonic, fermionic, and supersymmetric thermodynamics. All possible partitions correspond to bosonic case, the partitions containing given integer at most once correspond to the fermionic case, and supersymmetric case corresponds to the product of bosonic and fermionic partition functions.

The numerical experimentation led to the conclusion that simplest Hamiltonians do not work. Quantum criticality and fractality of TGD Universe however inspire the idea that the criticality is an inherent property of Hamiltonian rather than only thermodynamical state. Hence Hamiltonian can depend only weakly on the character of the partition so that all partitions contribute with almost equal weights to the partition function. The natural assumption is that the Hamiltonian depends only on the number r of summands in the partition. The super-symmetric variant of this kind of Hamiltonians yield the most realistic candidates for the genetic code and one might hope that a number theoretically small perturbation not changing the divisors p < 61 of partition function but affecting the probabilities could give correct degeneracies.

Unfortunately, numerical experimentation suggests that this might not be the case and that simple analytic form of Hamiltonian is too much to hope for. A simple argument however shows that exp(-H/T)=f(r) could be in quantum critical case be deduced from the genetic code by fixing the 62 values of f(r) so that the desired 62 correspondences n → p(n) result. The idea about almost universality of the genetic code would be replaced with the idea that quantum criticality allows to engineer a genetic code maximizing the total negentropy associated with DNA triplet-aminoacid pairs. In principle this would allow to predict a unique genetic code as the absolute negentropy maximum but this is outside of my computational resources since the crucial assumption 1< f(n)< n still leaves 63! possibilities to consider.

A natural guess is that the map codon → n of codons to integers is given as a small deformation of the map induced by the map of DNA codons to integers induced by the identification of nucleotides with 4-digits 0,1,2, 3 (this identification depends on whether first, second, or third nucleotide is in question). This map predicts an approximate p(n)=p(n+1) symmetry directly visible over finite ranges in the columns of code table and has also a convincing number theoretical justification in terms of a procedure allowing to construct f(n) by a trial and error procedure. This map is also consistent with exact A-G symmetry and almost exact T-C symmetry with respect to the last nucleotide of the codon.

One can deduce both codon-integer and aminoacid-prime correspondences. At least two Boltzmann weight distributions f(n) are consistent with the genetic code and Negentropy Maximization Principle constrained by the degeneracies of the genetic code. Only bosonic thermodynamics works contrary to the expectations raised by the earlier analytic models.

What is so non-trivial is that the natural map assigning to a given codon an integer gives almost correctly the map of codons to integers n in turn allowing to understand genetic code as a correspondence maximizing the individual negentropies of codon-aminoacid pairs as well as their sum. This motivates the attempts to find the physical interpretation of the number theoretical thermodynamics. The interpretation in terms of a broken conformal invariance is highly suggestive since bosonic partitions can be assigned with the states of a fixed conformal weight n constructed by using ordered sequences of conformal generators Ln or even better, U(1) Kac Moody generators Jn so that basically a breaking of Kac Moody symmetry would be in question. What is this system: for instance, could it be associated with the lightlike boundaries of magnetic flux quanta which are key actors in TGD based model of topological quantum computation?

For the details of the number theoretic model of genetic code see the new chapter Could Genetic Code Be Understood Number Theoretically? of Genes,Memes, Qualia, and ...".

Matti Pitkänen

Monday, November 21, 2005

Could Genetic Code Be Understood Number Theoretically?

The discussions with Sampo Vesterinen about topics only remotely related to genetic code led by a pure accident to an observation which stimulated feverish two week's period of work with a number theoretical model of genetic code. The number of DNA triplets is 64. This inspires the idea that DNA sequence could be interpreted as an expansion of an integer using 64 as the base. Hence given DNA triplet would represent some integer in n=1,...,6 (sequences of I Ching symbols give a beautiful representation of numbers in 64 base)). The discussions stimulated an observation putting bells ringing. The number of primes smaller than 64 is 18. Together with 0, and 1 this makes 20: the number of aminoacids!

1. Questions

The finding just described stimulates a whole series of questions.

Do aminoacids correspond to integers in the set consisting of primes ≤ 61 and {0,1}. Does aminoacid sequence have an interpretation as a representation as a sequence of integers consisting of 0, 1 and products of primes p=2,...,61? Does the aminoacid representing 0 have an interpretation as kind of period separating from each other structural units analogous to genes representing integers in the sequence so that we would quite literally consists of sequences of integers? Do 0 and 1 have some special biological properties, say the property of being biologically inert both at the level of DNA and aminoacids?

Does genetic code mediate a map from integers 0,...,63 to set S such that 0 and 1 are mapped to 0 and 1? If so then three integers 2≤= n &le 63 must correspond to stopping sign codons rather than primes. What stopping sign codon property means at the level of integers? How the map from integers 2,...,61 to the primes p=2,...,61 is determined?

2. The chain of arguments leading to a number theoretical model for the genetic code

The following chain of arguments induced to large part by concrete numerical experimentation leads to a model providing a partial answer to many of these questions.

  1. The partitions of any positive integer n can be interpreted in terms of number theoretical many boson states. The partitions for which a given integer appears at most once have interpretation in terms of fermion states. These states could be identified as bosonic and fermionic states of Super Virasoro representation with given conformal weight n.

  2. The generalization of Shannon entropy by replacing logarithms of probabilities with the logarithms of p-adic norms of probabilities allows to have systems with negative entropy and thus positive negentropy. The natural requirement is that n corresponds to such prime p≤ 61 that the negentropy assigned to n is maximal in some number theoretic thermodynamics. The resulting correspondence n → p(n) would naturally determine the genetic code.

  3. One can assign to the bosonic and fermionic partitions a number theoretic thermodynamics defined by a Hamiltonian. Purely bosonic and fermionic thermodynamics are defined by corresponding partition functions ZB and ZF whereas supersymmetric option is defined by the product ZB\times ZF. Supersymmetric option turns out to be the most realistic one.

  4. The simplest option is that Hamiltonian depends only on the number r of the integers in the partition. The dynamics would be in a well defined sense local and would not depend on the sizes of summands at all. The thermodynamical states would be degenerate with degeneracy factors given by total numbers dI(n,r) of partitions of type I=B,F. The invariants known as rank and crank define alternative candidates for basic building blocks of Hamiltonian.

  5. Ordinary exponential thermodynamics based on, say eH/T= q0^{r-1}, q0 a rational number, produces typically unrealistic genetic codes for which most integers are mapped to small primes p≤ 11 and many primes are not coded at all. The idea that realistic code could result at some critical temperature fails also.

  6. Quantum criticality and fractality of TGD Universe inspire the idea that the criticality is an inherent property of Hamiltonian rather than only thermodynamical state. Hence Hamiltonian can depend only weakly on the character of the partition so that all partitions contribute with almost equal weights to the partition function. Fractality is achieved if Boltzmann factors are given by e-H/T=(r+r0)n0 so that H(r)=log(r+r0) serves as Hamiltonian and n0 corresponds to the inverse temperature. The super-symmetric variant of this Hamiltonian yields the most realistic candidates for the genetic code and there are good hopes that a number theoretically small perturbation not changing the divisors p≤ 61 of partition function but affecting the probabilities could give correct degeneracies.

  7. Numerical experimentation suggests however that this might not be the case and that simple analytic form of Hamiltonian is too much to hope for. A simple argument however shows that e-H/T=f(r) could be in quantum critical case be deduced from the genetic code by fixing the 62 values of f(r) so that the desired 62 correspondences n→ p(n) result. The idea about almost universality of the genetic code would be replaced with the idea that quantum criticality allows to engineer a genetic code maximizing the total information associated with DNA triplets.


Saturday, November 05, 2005

Hydrinos and TGD

The work of Randell Mills related to hydrinos has created discussion in blobs. This morning Czech Lubos Motl applied his genuinely American shoot-first approach to problem solving routinely applied also by his intellectual hero George W. Bush;-).

One comment came from "Andre", who realized the importance of distinguishing between experimental work and theoretical work. The theoretical side in the work of Mills indeed looks as horrible as the work of M-theorists from experimental viewpoint. The tragedy is that people having the maths do not usually have physics and vice versa.

Mills claims of having observed inverted hydrogen spectrum with the principal quantum number n replaced with its inverse 1/n. This is something totally different from what one obtains from Dirac equation as a non-square integrable solution (rest energy becomes about αme for non-square integrable solutions which are integrable if interpreted as solutions of Klein-Gordon!). I do not understand why these solutions are associated with Mills claimed experimental findings.

For a couple of years ago I played with the idea that many-sheeted space-time, 4-surface in M4×CP2, might allow to have a rational valued principal quantum number n as a generalization of fractional spin and ended up with a proposal that transition to chaos might have analog at level of Bohr orbits.

The first idea was that for bound states the orbit of particle is kind of higher level particle, tubular space-time sheet, inside which particle moves. Stepwise transition to chaos could mean classically that in single step closed orbit is replaced by one which closes only after N turns. N=2 corresponds to period doubling.

Fractional spin m/N could mean that orbit closes only after N turns. At space-time level the orbit of particle is replaced with flux tube like structure and fractionization would mean that flux tube like space-time sheet transforms to one closing only after N turns when looked from M4 view point. The tube would look like N-fold covering of M4 locally, analogous to Riemann surface associated with z1/N.

For principal quantum number this would mean that the period in radial direction becomes N-fold in single step to chaos. Not so easy to imagine. Principal quantum number would become a rational of form n/N in this process: you would have "radial anyon". Mills would have observed few lowest values of N with n=1 (ground state of the fractal hydrogen).

For details look here.

Matti Pitkanen