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 nf_c of cyclotron frequencies of biologically important biological ions (most of them in alpha band) and to the frequencies f_J +/-nf_c, where f_J 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 f_J +/- nf_c are associated with these communications. For f_J = 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 L_w 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 L_n or even better, U(1) Kac Moody generators J_n 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

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 Z_B and Z_F whereas supersymmetric option is defined by the product Z_B\times Z_F. 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 d_I(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 e^H/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.

For the details of the number theoretic model of genetic code see the new chapter Could Genetic Code Be Understood Number Theoretically?.

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 αm_e 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 z^1/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

What is EEG made of?

In the previous posting I discussed the model for the effects of ELF em fields on brain and its implications. One implication is a radically new view about genes implying the notions of supergene consisting of strings of genes in large number of different cell nuclei through which magnetic flux sheet traverses: a page of book containing as its lines genetic text is quite precise metaphor. Hypergenes are also unavoidable and consists of this kind of text from several organisms.

Second implication is a model for EEG. Living organisms in biosphere are analogous to proteins on cell membrane consisting of two lipid layers acting as a gigantic dark Josephson junction going through litosphere and ionosphere (counterparts of lipid layers at the level for fourth level of dark matter hierarchy). Josephson junction consists of a magnetic flux sheet going through the nervous system to ionosphere and connects thus brain with two parts of magnetic body of Earth analogous to cell interior and exterior. Josephson oscillation at 3.72 Hz defines basic "drum beat" and in the presence of a voltage perturbation the pattern of generated coherent state of ELF photons provides a representation for the perturbation of EEG which is also a part of EEG. Obviously, our relationship to magnetic body of Earth would be similar to that of cells to organism.

The usual classification of EEG frequencies by EEG bands is more or less a convention and the definitions of various bands vary in frustratingly wide ranges. In a more ambitious approach bands should be replaced with some substructures identified on basis of their physical origin and function. In the proposed framework this is possible and explains successfully basic aspects of EEG in various states of consciousness. This can be regarded as a dramatic victory for the notion of TGD inspired theory of consciousness and the notion of dark matter. This identification of substructures of course applies only to that part of EEG from which noise is subtracted. The contribution of neural activity is one such source of noise, often regarded as the only contribution.

1. Basic contributions to EEG

There are three fundamental contributions to EEG besides the neuronal noise.

1. Schumann resonances characterize Earth's magnetic field and collective aspects of consciousness.

2. Cyclotron frequencies generated in cyclotron transitions of ions. An attractive guess is that cyclotron frequencies correspond to the control signals from magnetic body so that Josephson junctions and magnetic body would form a closed feedback loop. These frequencies can be classified to those associated with bosonic and fermionic ions respectively.

3. The frequencies generated by Josephson currents as coherent photons. Harmonics of cyclotron frequencies shifted upwards and downwards by Josephson frequency f_J=3.72 Hz. If the amplitude of the perturbation at cyclotron frequency is strong the EEG looks locally like it would consists of amplitudes with frequencies f_+/-=f_J¹+/- f_J during most of the cyclotron period so that the visual inspection of time evolution of EEG can be rather misleading. Since these frequencies are involved with communications to the magnetic body of Earth, the natural guess would be that they correlate with the neural processing.

2. Classification of cyclotron frequencies

Consider now the classification of cyclotron frequencies.

1. Cyclotron frequencies can be classified those associated with atomic and molecular ions. For biologically important atomic ions most frequencies are above 7.5 Hz. For molecular ions frequencies are lower and for DNA sequences the frequencies are in delta band. Thermal stability condition suggest a lower bound of ≈ 1 Hz for significant frequencies of this kind.

2. Atomic ions can be classified into bosonic and fermionic ions. Practically all biologically important bosonic ions have Z=2 and in alpha band: f(⁶Li⁺)=50 Hz and f(Mg²⁺)=25 Hz are the only frequencies above alpha band. Situation is essentially the same for biologically interesting ions too. ⁷L⁺ is exception and corresponds to 42.9 Hz having Josephson satellites at 39.2 Hz and 46.6 Hz. Thus the frequency range 7.5-15 Hz is very strongly represented and expected to be fundamental.

3. The integer n characterizing the harmonic of the cyclotron frequency in question is an additional classificational criterion and could correlate with the character of neural processing.

3. Wake-up EEG

The question is whether this classification is consistent with the conventional decomposition into various bands and whether it allows to gain some real insights EEG. Consider first wake-up EEG.

1. The first implication is that each cyclotron frequency f_c is accompanied by by two satellites f_c+/- f_J. For alpha band these satellites correspond to theta band and beta band identifiable as responses to control signals from magnetic body in alpha band. One can ask whether these bands as a whole correspond to the satellites of alpha band. This identification implies that both bands are present and makes sense for wake-up EEG but not as such for the EEG during first and second period of deep sleep during which theta band is present but higher bands are absent.

2. Sensorimotor rhythm in range (12-16) Hz is associated with physical stillness and body presence. The interpretation is as a low amplitude satellite of alpha rhythm with low amplitude control signals from the magnetic body so that rhythmicity is not lost and frequencies are clearly f_c+f_J.

3. Beta band is above 12 Hz and associated with active, busy or anxious thinking and active concentration and is chaotic and highly asynchronous. The natural interpretation is as large amplitude satellite of alpha band involving the activation of communications to the magnetic body and large control signals with f_J¹>> f_c. Hence the spectra would for a considerable part of period 1/f_c effectively consist of frequencies f_+/-= f_J+/- f_J¹, where f_J¹ varies in frequency range characterized by the amplitude of perturbation. There is no definite resonance frequency since f_J¹ can vary continuously. Globally the situation is different since the spectrum can in principle be decomposed to frequencies f_J+/- nf_c. These two descriptions correspond to time domain and genuine frequency domain.

   For sufficiently high harmonics of f_c the chaoticity disappears and frequencies f_J+/- nf_c become more manifest. The Josephson amplitudes of higher harmonics decrease as 1/nf_c.

   Beta band is predicted to have a mirror image in theta band during cognitive activity. The frequencies in theta band are assigned with cognitive activities and memory recall. Note that also alpha band due to cyclotron frequencies should be present as well as the basic "drum beat" defined by f_J for f_J¹>> fc.

4. Odd higher harmonics of cyclotron frequency are expected to be the most important ones and would have interpretation as control signals from magnetic body. Satellites would correspond to responses to the magnetic body.

   For alpha band the third harmonics of most bosonic ions are in the range 28.2-34.2 Hz and roughly in gamma band above 30 Hz assignable with the control of cognitive activities by magnetic body of Earth's magnetic field.

5. 40 Hz thalamocortical resonance band is very important EEG band. ⁷Li⁺ ion corresponds to 42.9 Hz frequency for which the lower satellite is at 39.2 Hz. The third harmonic for Na⁺ is 39 Hz, which also corresponds to Schumann resonance. Its upper satellite belongs to 40 Hz resonance band. The upper satellite of the third harmonic of Mn²⁺ is 37.9 Hz.

4. Mirror satellites exist!

The existence of the mirror satellites might be regarded as a killer prediction. Amazingly, narrow EEG bands which are mirror images of each other with respect to alpha band have been reported by Nunez. Besides alpha band at 11 Hz, Nunez mentions also narrow sub-bands at 3, 5 and 7 Hz at delta and theta range, as well as the bands at 13, 15 and 17 Hz in beta band.

All these frequencies are expressible in the form f_c+/-f_J, f_J=5 Hz, which is one half of the frequency 10 Hz of the memetic code and by 14 per cent higher than 3.7 Hz predicted assuming λ=2¹¹. The value of λ deduced from these frequencies would be λ=1902 and about 7 per cent smaller than λ=2¹¹. This estimate cannot be taken too seriously since it is quite possible that the thickness of Josephson junction is not scaled up completely exactly.

The cyclotron frequencies associated with the bands are 8, 10, and 12 Hz. The cyclotron frequencies of bosonic ions ⁸⁰Se^{2-}, ⁶³Cu²⁺, and ⁵⁵Mn²⁺ for a magnetic field strength B=.526 Gauss are 8.00, 10.11, and 12.00 Hz. The cyclotron frequencies of bosonic ions ⁶⁴Zn²⁺, ⁵⁹Co²⁺, ⁵⁶Fe²⁺ would be 9.90 Hz, 10.52 Hz, and 11.36 Hz and have satellites at frequencies 4.90 Hz, 5.52 Hz, 6.36 Hz and 12.90, 15.52, and 16.36 Hz. All these frequencies belong to the bands reported by Nunez since their widths are 1-2 Hz. Thus the frequencies of all bosonic ions in alpha band and in their satellites belong to the bands reported by Nunez for values of λ and B very near to their nominal values used in calculations!

What is also very remarkable that the 10 Hz magic frequency of the memetic code corresponding to the secondary p-adic length scale L(2,127) associated with Mersenne prime M₁₂₇ characterizing electron appears. It should be also noticed that f_J=5 Hz frequency corresponds to cognitive theta appearing during tasks requiring mathematical skills.

5. Alpha band dominance during relaxed state

In a relaxed state beta band disappears and the spectral power in alpha band increases. This seems to be in conflict with the idea that beta band is a mere satellite. There are two mutually non-inclusive manners to understand this.

1. The first possibility is that cyclotron frequencies in alpha band are not actually present and only Schumann frequency 7.8 Hz and 10 Hz resonance frequency associated with the excitations of electric field in ionospheric cavity behaving like 2-dimensional waves on sphere.

2. Second possibility is that lito-ionospheric Josephson junction is somehow closed so that only the cyclotron contribution of various ions is present. This might be caused by DNA level mechanism which simply prevents the flow of the Josephson currents flowing along magnetic flux sheets through DNA strands. This mechanism would be completely analogous to the closing of ionic channel associated with cell membrane protein.

6. EEG during sleep

The EEG during sleep provides a testing ground for the proposed anatomy of EEG. Sleep consists of 90 + 90 minute periods of NREM and REM sleep. This period is also the period of brain hemisphere dominances during wake up and day dreaming occurs with the same period as REM sleep. During REM sleep the EEG is essentially similar to that during wake-up. These observations inspire the hunch that brain hemisphere dominance dictates whether REM or NREM is in question. This turns out to be a correct guess.

a) EEG during stage 1

During stage 1 theta of deep sleep waves in frequency range 4-8 Hz dominate and amplitudes increase when frequency is reduced. The control signals from magnetic body are expected to be weak so that f_J¹f_J approximation should hold true implying that frequencies f_J+/- f_c should dominate and EEG would look rhythmic rather than chaotic as indeed observed. The amplitudes behave as 1/f_c and thus increase with decreasing f_c. The fact that amplitudes increase with decreasing EEG frequency suggests that the frequencies they correspond to different cyclotron frequencies.

These facts does not conform with the general picture as such. If theta and beta bands are mere satellites of alpha band, both of them should be present during stage 1 sleep but this is not the case. The idea that cyclotron frequencies of heavier ions in B=.5 Gauss (strength of Earth's magnetic field) could replace those appearing during wake-up does not work. Theta band simply does not contain the cyclotron frequencies of biologically important ions for B=.5 Gauss.

The only reasonable way out of difficulty seems to be that the value of the magnetic field associated with active flux sheets is reduced by a factor of 1/2. This would mean that the most important range 7.5-15 Hz of cyclotron frequencies would be scaled down to 3.75-7.5 Hz which indeed corresponds to the theta band. If one excludes Ca²⁺, the range for bosonic ion reduces from 7.5-11.4 to 3.75-5.7 Hz. The satellites correspond to the range .05-8.7 Hz and 7.45-9.4 Hz plus Ca²⁺ satellites at 3.8 Hz and 11.2 Hz. With Ca²⁺ forming a possible exception, the resulting frequency ranges are consistent with empirical facts. Of course, it is quite possible that magnetic body does not generate cyclotron transitions at Ca²⁺ cyclotron frequency.

The resolution of the puzzle might relate to the character of ions at the flux sheets in left and right hemisphere.

1. The quantization of magnetic flux reads as

   Ze×INT BdS= n× hbar

   and for Cooper pairs and bosonic ions with Z=2 (Z refers to absolute value of charge) it gives magnetic field strength which is one half from that for fermionic singly charged ions. Both fermionic ions with Z= 1 and bosonic ions and Cooper pairs with Z=2 are allowed in this case by the single valuedness of wave functions. For Z=2 the quantization condition allows single valued wave functions for Z=2 ions or Cooper pairs only.

2. Assume the quantization condition corresponds to Z=1 for the right hemisphere and Z=1 for the right hemisphere. The presence of fermionic ions implies additional cyclotron frequencies on left hemisphere and the presence of fermionic ions conforms with the old proposal that fermionic Fock states provide a realization of quantal version of Boolean algebra. This conforms with the view that left brain is more reductionistic and performs linear logic operations whereas right brain is more holistic.

3. As a consequence the cyclotron frequency scale in right hemisphere is reduced by a factor of 1/2 and during right hemisphere dominated NREM sleep alpha band would be scaled down to theta band.

4. The prediction is that, apart from the Schumann frequencies and neural noise, left hemisphere EEG spectrum consists of right hemisphere EEG specrum scaled up by a factor of 2 plus the contribution of fermionic ions and the Josephson satellites of these frequencies.

   The assumption that the two quantization conditions correspond to just left and right hemispheres rather some other pair is of course un-necessarily strong and one can imagine also other correspondences.

b) EEG during stage 2

Sleep spindles appearing in the state 2 of deep sleep are sudden increases in EEG amplitude and frequency from theta band to 12-16 Hz. The spindles .5-.1.5 seconds and appear with a period of about minute. In some sources frequency range 7-16 Hz is given as sleeping spindle range. The so called K-complexes are sudden increases in EEG amplitude but no change in frequency associated with spindles.

One interpretation is that sleep spindles correspond to the occasional wake-ups of the left hemisphere. Sleep spindles would thus correspond to the satellites of alpha band identifiable as responses of the corresponding Josephson junctions to occasional strong control signals at cyclotron frequencies in alpha band. K complexes could be interpreted as signals from magnetic body to left hemisphere but inducing no response. It might be that these sudden responses reflect the fact that the left brain is not fully asleep yet.

c) EEG during stages 3 and 4

Most of EEG power during deep sleep stages 3 and 4 is in the in the range .75-4.5 Hz. This implies that control signals at cyclotron transition frequencies of ions from the magnetic body cannot be appreciably present and the control signals at cyclotron frequencies of molecular ions, such as DNA with cyclotron frequencies below 1 Hz, should be responsible for the EEG. The small amplitude of control signal implies 1/f_c behavior and large amplitude as compared to the corresponding amplitudes at higher bands at weak amplitude limit.

Taking into account the fact that magnetic field strength is scaled down by factor of 1/2 this means that mass numbers of the ions in question must satisfied A/Z > 150 for f_c< 1 Hz. For DNA sequences with charge of 2 units per single base-pair one would have A> 300. The atomic weights for base pairs plus phosphate group and deoxyribose sugar are 327, 321, 291, 344 corresponding to A, T, C, G.

7. Transcendental states of consciousness and EEG

Transcendental states of consciousness are characterized by the presence of alpha and theta bands (note that theta band is present also during childhood, youth and even early adolescence but usually disappears at older age). It is found that that theta and alpha bands are preserved also during deep sleep. A possible interpretation is that the presence of alpha band signifies that left brain remains awake in a state of relaxed alertness involving weak signals from magnetic body.

For more details see the section "What is EEG made of?" in the just updated chapter Bio-Systems as Super-Conductors: Part II of "TGD Inspired Theory of Consciousness....".

Dark matter and supersolids

Dark matter solids, in particular those in large hbar phase, should move through ordinary matter without friction. This could explain the puzzling findings about artifacts in completely wrong places. It might be that the strange ability of solid matter objects corresponding to different space-time sheets to move through each other has been already observed in laboratory.

The existence of so called supersolids was predicted long time ago by Russian theoreticians Andreev and Liftschitz, who proposed that ⁴He, which is a bosonic solid at low temperatures and as a noble element has weak bonds between atoms, could make under a high enough pressure a phase transition to a supersolid state which is the analog of super-fluid state. Supersolid can be regarded as a Bose-Einstein condensate behaving like single particle and having vanishing friction with respect to other solids.

What the loss of friction with respect to other solid means is not quite clear. The conservative view is that supersolid can glide along another solid without friction. A more radical view is that supersolid can flow through other solids, even ordinary, without any friction.

There is empirical support for the latter option. The experimental procedure used by M. Chan and his student E. Kim is roughly the following.

1. Compress ⁴He gas into a small glass disk made of fused silica containing atomic-sized pores.

2. Construct a torsional oscillator by hanging this disk to a string spinning with kHz frequency back and forth. Increase the pressure gradually from the solidification pressure 40 atm to 62 atm and monitor the oscillation frequency as the pressure is increased.

What is observed is a sudden increase of oscillation frequency at .2 K as if some of ⁴He had leaked out and the system would have lost some of its inertia. The experimenters have however excluded the leakage. Their explanation is in terms of the emergence of ⁴He supersolid phase which remains at rest since it has no friction: glass simply flows through it. The effect is absent for $^3$He, which is a fermionic solid, and this supports the conclusion of the experimentalists.

The ability to flow through another solid makes the notion of supersolid highly counterintuitive in ordinary single-sheeted space-time. Intuitively one tends to assign the friction with the boundary of the supersolid and solid rather than interior and this would mean that supersolid would only glide without friction along but not flow through another solid.

Quantum classical correspondence is the basic interpretational guideline of TGD and forces to ask what the space-time correlate for the formation of supersolid, or whatever is in question in the experiment, might be. The ability of the two solids to flow through each other can be understood if the formation of the supersolid involves the separation of supersolid phase to its own space-time sheet. This separation becomes possible if the friction between ordinary solid and supersolid vanishes. Obviously this interpretation is consistent with the conservative view and implies the radical view in many-sheeted space-time.

Large hbar supersolids could exist even at room temperature so that technological implications would be quite dramatic. Also the artifacts in wrong places could result by a temporary phase transition to large hbar supersolid phase. The low temperature of the phase transition suggests that ⁴He is not large hbar phase.

For more about interpretation of artifacts in wrong places as solid objects which have suffered temporary transformation to large hbar dark matter see the end of the chapter Quantum Coherent Dark Matter and Bio-Systems as Macroscopic Quantum Systems of "Genes, Memes, Qualia,...".

Further empirical support for TGD inspired cosmology

TGD inspired cosmology predicts that critical cosmology is followed by a hyperbolic cosmology. A natural question is whether the travel of microwave photons through the negative curvature cosmology might induce some signatures in microwave background. This is indeed the case.

The geodesics in negative curvature 3-space diverge exponentially. The divergence of the nearly parallel light-like geodesic lines is due to the negative curvature making 2-dimensional sections of 3-space analogous to saddle surfaces. The scatterings during the travel of light induce geodesic mixing so that light from regions with differing temperature mix. Hence negative curvature tends to smooth out the anisotropies of the temperature distribution.

Negative curvature has also a more dramatic signature. Gurzadyan (Astronomy and Astrophysics, vol. B21, 1992 and arXiv:astro-ph/0312523) has developed very refined arguments involving algorithmic information theory and complexity theory to show that in the hyperbolic cosmology the hot and cold spots of the temperature distribution of the cosmic microwave radiation look elongated. The direction of elongation is random but the shape of the ellipse is characterized by the curvature of 3-space and does not depend on temperature or size of the spot. For a flat or positively curved space this kind of elongation does not occur.

The emergence of a preferred direction in a Lorentz invariant cosmology looks highly counter-intuitive. My humble understanding is that a scattering of photons from a large geometric structure must be involved somehow. The elongation should relate to what happens at the last scattering surface whose position together with the positions of observer and previous scattering surface define a plane whose normal defines the preferred direction, which would presumably correspond to the shorter axis of the ellipse. In TGD framework the transfer of photons from a larger space-time sheet to that of observer might correspond to this scattering process. Scattering surface would correspond to the boundary of the space-time sheet of the observer whereas scattering would correspond to refraction at the boundary.

The analysis of BOOMERanG, COBE and WMAP CMB maps indeed shows that the spots have elliptic shape with ellipticity parameter ≈ 2 whereas the prediction for hyperbolic RW cosmology is 1.4 (arXiv:astro-ph/0402399). This would suggest that some additional effect is involved and TGD inspired bet have been already described.

For TGD inspired cosmology see chapter TGD and Cosmology of "TGD".

Matti Pitkanen

The effects of ELF em fields on brain and new view about genes

The text below extends my response to an email from Maria Gaudos who has written an interesting book "The Theory of Natural System", which can be found at Amazon. My goal is to give an online documentary about the fascinating things taking place in TGD based theory of living matter after the discovery of large hbar hierarchy. I am perfectly aware about the importance of this progress and my sincere hope is that these pieces of text could help theoretical colleagues to experience a liberation from M-theory hype, to accept the reality as it is, and to start to dig gold with me. I of course do not dare to wish that details are comprehensible to a layman. Popular books about TGD are still waiting their writers.

1. Hierarchy of quantum critical high T_c superconductors and EEGs generated by hierarchy of Josephson

What I have been elaborating during last weeks is a model of biosystems based on quantum critical superconductors and fractal hierarchy of EEGs identified as a coherent radiation produced by Josephson currents of Josephson junction hierarchy. First some words about the background. The idea about fractal hierarchy of EEGs emerged when I constructed for some years ago a model for the observations of Peter Gariaev relating to the response of DNA to laser light. The idea about Josephson junction generating EEG is fifteen years old: the first guess was that cell membrane would define a Josephson junction generating EEG as coherent photons assignable to Josephson current. The guess was wrong: the frequency involved in case of cell membrane is around 10¹³ Hz rather than say EEG frequency. I constructed a model for quantum critical high T_c super-conductors only month or two ago and the model predicts that cell membrane thickness and cell size as universal length scales associated with these superconductors (p-adic length scale hypothesis). No wonder I am eager to apply this model to living systems!

The model involves combination of the notions many-sheeted DNA and magnetic body with the new idea of dark matter hierarchy involving a hierarchy of Planck constants coming as powers of λ≈2¹¹ multiplying its standard value and making possible macroscopic quantum coherence. This dark matter hierarchy is something new and something absolutely essential since for sufficiently high levels of it (large enough hbar) EEG frequencies correspond to photon energies (E= hbar×ω) above thermal threshold: this is true for higher than third levels of the dark matter hierarchy. The earlier assumption that large space-time sheets are cold can be relaxed. Also the cyclotron energy scales of ions in Earth's magnetic field turn out to be above the thermal threshold about .086 eV for k=4 and higher levesls of the hierachy. EEG frequencies can thus have effects in energy range of metabolic energy quantum.

2. Strange effects of ELF em fields on vertebrate brains

My original goal was to construct a model explaining the strange effects of ELF em fields on living matter found during last decades but I ended up with a new vision about EEG and DNA. This shows how enormously important the connection with quantitative empirical facts is for a theoretician. These effects appear in certain amplitude windows and harmonics of cyclotron frequencies and in temperature window 36-37 C. This anomaly is really something for Sherlock Holmes: in particular, amplitude windows seemed to be hopeless to understand but turned out to correspond to EEG bands in the hierarchy of scaled up variants of EEGs. Both p-adically scaled up EEGs suggested earlier and dark matter variants of EEG with frequency scales coming as powers of 2^-11 are present so that an enormously complex fractal information processing structure is involved and extends to cosmic distances (magnetic bodies involved are of astrophysical size).

3. Many-sheeted DNA, hierarchy of magnetic flux sheets, and dark matter hierarchy

As an unexpected side product I ended up with a vision for how DNA codes for information and what the roles of nucleus and cell membrane might be in biocontrol and information processing. I want to emphasize that this picture is based on numbers and his highly predictive.

The picture that emerges conforms and generalizes the earlier picture about many-sheeted DNA based on the notion of magnetic body. The earlier picture involved notions like magnetic circulation and magnetic homeostasis involving fine tuning of the local strength of Earth's magnetic field. In the new picture highly convoluted magnetic flux sheets of thickness of DNA strand corresponding to various levels of dark matter hierarchy go through the DNA strand and connect biological body to various magnetic bodies. This highly convoluted structure is required to satisfy magnetic flux quantization condition for large hbar. Magnetic flux quantization implies that human DNA allows at most 7 levels ("chakras"!) of dark matter hierarchy and hierarchy of magnetic bodies. k=4 corresponds to a connection to the magnetic body of Earth ("heart chakra"). Largest magnetic body has size measured in hundred light years (k=7, "crown chakra", cosmic consciousness")! Once again it seems that mystics have been able to grasp something about the general structure of consciousness and expressed it poetically so that skeptics would not take it seriously;-).

The amount of DNA does not matter much, it is the number of these hierarchy levels ("chakras"!) which characterizes the level in evolution. The simplest hypothesis is that k:th level codes for functions and structures at k:th level of dark matter hierarchy: the higher the level the more abstract are the coded functions involved. Introns can be understood as dark genes coding mostly for dark matter functions and structures. The emergence of new dark matter level corresponds to a sudden jump in evolution as completely new functions and structure appear. For instance, k=4 corresponds to the emergence of ordinary EEG (vertebrates).

The highly convoluted magnetic flux sheets with transverse section define a string going along DNAs in a large number number of cells. Sincle human cell contains roughly one meter of DNA. The number is about 10¹² or even 10¹⁵ for k=7 level if most of DNA length correspond to magnetic flux sheet. The estimate for total number of human brain neurons is of order 10¹². This sheet combines genes from a huge number of cells to single supergene which is like a page of book. Our bodies would be quite literally text strings and single cell contributes only a sentence or two! This means that the representative power of genome has nothing to do with the amount of genes in single cell. Genes from say 10¹² neurons can define something comparable to encyclopedia (total length for the string is about .1 light years)! It would be interesting to see whether and how the basic hierarchical structures of language correspond to the 7-level dark matter hierarchy. The evolution at the level of individual would be presumably generation of new texts of this kind and their expression: just this what I am doing now!

Even more dramatic generalization of genome can be considered. There is in principle no reason why magnetic flux sheets could not pass through several organisms so that kind of hyper-genes would result. These hyper-genes could make possible the emergence of complex social structures with individuals obeying rules making possible complex behavioral patterns.

What is most fascinating that you can more or less see this chromosomes-attached-to-strings picture in the pictures about cell division, which brings strongly in mind dipole magnetic field dividing into two!

4. Why cyclotron frequencies, why amplitude windows, why temperature window?

I have not yet said anything about the original problem: the understanding of amplitude windows and harmonics of cyclotron frequencies of biological important ions in Earth's magnetic field at which ELF em fields have these strange effects in brain of vertebrates. Also temperature window 36-37 C should be understood. The general picture looks like follows.

1. Cyclotron frequencies at Earth's magnetic field induce coherent cyclotron transitions of Bose-Einstein condensate of bosonic ions (in particular Ca²⁺) at magnetic flux sheets going through DNA. Also the transitions of fermionic ions are important. These transitions relate to resonant frequencies in EEG.

2. First amplitude window corresponds to electric field in the range 1-10 V/m: note that EEG amplitudes are in the range 5-10 V/m. Second amplitude window corresponds to 10^-7 V/m (very small!). In both cases the electric fields are transformed to oscillatory voltages over Josephson junction so that the net voltage is

   V= V₀+ V₁sin(ω_ct),

   where V₀ is the constant background voltage, .08 V for cell membrane and possible its p-adically scaled up variants and dark variants of it with scaled up thickness. For 5-10 V/m k=2 level of dark matter hierarchy turns out to be involved: this explains why the effects are relatively low level effects in brain tissue. For 10^-7 V/m k=4 level of dark matter hierarchy ("heart chakra") is involved: this explains why the effects are high level effects related to say navigation and prey detection in case of marine vertebrates.

3. Josephson current is proportional to sin(2e×INT V(t)dt) with amplitude which does not depend on V(t). Hence the coherent state of photons generated by V has overall intensity which does not depend on strength of the perturbing voltage representing biologically relevant information: this standardization is something extremely nice from engineerish point of view. The coherent photon state in turn communicates this information to the relevant magnetic body which then responds appropriately.

4. If the Josephson frequency f_J¹= 2eV₁/h assignable to the perturbation is considerably larger than cyclotron frequency, the perturbation looks locally like a superposition of frequencies f₊=f_J+f_J¹ and f_-= f_J-f_J¹ for most of the time during the period defined by cyclotron frequency. If f₊ or f_- or both correspond to EEG resonance band, one expects strong biological effects. This picture translates the frequency windows to EEG bands and the detailed study of these frequencies allows to conclude that the hypothesis works. There is no parameter fitting involved except that the integer valued scaling factor must be very near to λ= 2¹¹. There are number theoretical reasons to believe that this holds true exactly. For instance, the dark variants of p-adic length scales coming in powers of λ would coincide with p-adic length scales. Of course, entire fractal spectrum of frequency windows is predicted.

5. Why the effect appears only in the temperature window 36-37 C? Temperature window can be understood in terms of quantum criticality of high T_c superconductivity (see here and here). The BCS type supra currents flowing in the interior of the nucleus and boundary supra currents flowing at the nuclear membrane have different kinds of Cooper pairs and these super-conductivities co-exist only in a narrow temperature range. The body temperature of vertebrates, in particular endotherms, is quite essential for the proper functioning of brains as anyone knows from personal experience. Hence quantum criticality must be essential for high level biocontrol involving EEG and magnetic bodies. This means there must be co-operation of nuclear membrane and nucleus.

6. How can one understand the fact that both cyclotron frequency and amplitude window are needed for a non-trivial effect to result? It seems that an interpretation in terms of AND gate provides the only way to understand this. Both interior super conductor and boundary superconductor defined by cell nucleus or its dark matter scaled up variant (at least as far magnetic flux sheet structure is considered) must react simultaneously in order that a non-trivial response results. The following scenario is the simplest one I have been able to imagine.
   1. The perturbing electric field E₁ generates cyclotron transitions in the cell nucleus with the effect that it opens up the communication line to magnetic body and is ready to receive control messages inducing the coding of say aminoacid sequences needed to realize a response to the perturbation representing relevant information such as sensory input. If frequency is wrong, cell nucleus stays off-line.

   2. Magnetic body in turn receives the EEG signals from very many nuclei if the amplitude is in some window representing resonant EEG band and thus coding genuine information. Magnetic body responsible for intentional actions responds by a signal activating a super-gene defined by the genes in nuclei involved and generates in this manner a coherent biological response.

   3. Biological response in given nucleus results only if it has opened the communication line. Thus only cyclotron frequencies and amplitudes within amplitude windows represent biologically relevant information. This provides an effective manner to eliminate the effects of the electromagnetic noise. The dominance of noise has been indeed used as one the basic argument by skeptics whose world view does not allow the effects of ELF em fields on living matter and who see EEG as a mere side effect of nerve pulse activity.

For more details see the just updated chapter Bio-Systems as Super-Conductors: Part II of "TGD Inspired Theory of Consciousness...". I hope that I manage to update the chapters related to EEG, many-sheeted DNA and genetic code before the end of the year. Many things to do and the passion to get this done almost explodes me!

Matti Pitkanen

The invariance of angular momentum under the scaling of hbar

As explained in the previous posting, number theoretic vision in principle allows to predict the spectrum of quantized values of Planck constant in TGD framework. Basically the scaling of hbar reduces to the overall scaling of the Minkowski metric of an algebraic extension of rational M⁴ factor of M⁴×CP₂.

The assumption that four-momentum is invariant in the scaling of M⁴ metric by λ combined with Poincare invariance implies that also angular momentum is invariant under the scaling by λ. Hence the analog of wave vector defined as L_z/hbar would scale down by a factor 1/λ giving L_z/hbar=m/λ. The one-valuedness of the wave function however requires L_z/hbar=m.

A possible resolution of the paradox is based on following observations.

1. Number theoretic vision suggests strongly that λ is integer. Effective 2-dimensionality means that 2-dimensional partonic surfaces are basic structures to be considered. By conformal invariance scalings and rotations are generated by L₀ and iL₀. Hence it would not be surprising if the radial scaling by λ^k would be accompanied by a similar angular scaling. If M⁴ projection of the partonic 2-surface has dimension D> 0, this scaling would at space-time level mean that partonic 2-surface becomes analogous to λ^k-sheeted Riemann surface defining λ^k-fold covering of E² identified as Euclidian plane of M⁴. Using M⁴ coordinates for the space-time sheet as local coordinates, one finds that induced spinor fields have fractional angular momentum eigenvalues L_z=m/λ^k.

2. In this picture the approach to quantum criticality would correspond to the emergence of classical chaos at space-time level by a step-wise process in which the step hbar--> λ×hbar can be regarded as generalization of the period doubling bifurcation. As hbar increases by a factor λ^k, the space-time sheet representing an orbit of particle closing after one turn transforms to an orbit closing only after λ^k turns. Note that the volume of space-time sheet remains finite only if the orbit closes after finite number of turns. The step k--> k+1 would correspond to a local fractal operation making each sheet of the λ^k sheeted surface λ-sheeted so that λ^k+1 sheeted surface would result. Instead of period doubling one would have period λ-folding with the value of λ depending on p-adic prime p≈ 2^k.

3. The model of Nottale for planetary orbits requires besides the harmonics of λ₀≈ 2¹¹ also some of its sub-harmonics, at least third and fifth one. This conflicts with the assumption that λ is integer unless λ₀ is divisible by 3 and 5. This would be the case for λ₀= 2175=3× 5²× 29 consistent with the mean value λ= 2174 deduced with one per cent accuracy from the model of Nottale for planetary orbits. Only integer factors of λ₀(p≈ 2^k) would be allowed as sub-harmonics for given p for this option and integer valuedness of λ₀ would also pose strong conditions on the values of p if the proposed formula for λ₀ is accepted.

4. If one allows the quantization of angular momentum using integer multiple of hbar₀ as a unit, λ need not be an integer. In this case one could have λ=(r/s)×λ₀, where λ₀≈ 2¹¹ is integer. The orbit represented by the space-time sheet would close after (rλ₀)^k turns and angular momentum would be quantized as L_z=ms^k×hbar₀. Even in this case the proposed formula for λ(p) would give strong constraints on the values of p.

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

Some comments about large hbar, dark matter, and living matter

I am busily updating the two books about TGD inspired quantum biology and theory of consciousness. It still takes a couple months to replace the old interpretation of long ranged weak forces with the new one. I feel guilty of having been wrong so many years, and to gain a peace of mind I want to confess all and summarize briefly some re-interpretations implied by the new view about dark matter.

1. Space-time sheets need not be thermally isolated

The earlier view about the role of many-sheeted space-time was based on the assumption that different space-time sheets are effectively thermally isolated, or at least that the rate of energy transfer between them is very slow. This allows to assume that the temperature at large space-time sheets is very low. Expressing it concisely: these space-time sheets would be cold, dry, and silent rather than hot, wet, and noisy. Quantum control by large space-time sheets would make possible the miraculous behavior of living matter.

In the new view it is not necessary to assume thermal isolation. Energies are invariant under phase transitions increasing hbar and thus the size of the space-time sheet. This means that long enough sequence of phase transitions increasing hbar by a factor λ ≈ 2¹¹ can give rise to a situation in which the space-time sheets are so large that given frequency scale corresponds to an energy above thermal energy. The assumption of thermal stability obviously gives interesting constraints on earlier quantum models of living matter.

2. The quantal effects of ELF em fields on living matter

It has been known since seventies that em fields at E(xtremely) L(ow) F(requencies) have effects on living matter and magnetic quantum transitions in Earth's magnetic field .5× 10^-4 Tesla are in question. This is impossible in the framework of standard quantum physics since magnetic energies are ridiculously small for typical cyclotron frequencies, say 6.2 ×10^-14 eV for 15 Hz cyclotron frequency of Ca⁺⁺ ion in Earth's magnetic field.

The solution of the puzzle comes from dynamical hbar. One manner to perform flux quantization is to keep magnetic field invariant and increase the area of flux quantum. In this process cyclotron energy scales as hbar whereas cyclotron frequency remains invariant so that it is possible to achieve arbitrary high cyclotron energies for a fixed cyclotron frequency.

Consider as an example Earth's magnetic field, which corresponds to p-adic length scale L(k) with k=13²=169 for the normal value of hbar. In this case cyclotron frequencies are in EEG range 1-100 Hz for atoms with mass number A>2. For instance, Ca corresponds to 15 Hz frequency.

For k=4 level of dark matter hierarchy cyclotron energies for ions with charge Z are above the maximum photon energy .086 eV for blackbody radiation at room temperature T=300 K for A< 233Z. All singly charged stable nuclei obey this stability condition. This state of affairs makes possible Bose-Einstein condensation of bosonic ions in cyclotron energy states. The number of these ions is very limited.

There are many observations making bells ringing. The cyclotron frequencies of bosonic ions correspond to important EEG frequencies and important biological rhythms. Ca²⁺ cyclotron energy corresponds to twice the metabolic energy quantum .5 eV which suggests that the exceptional role of Ca²⁺ ions in living matter is due to the fact that quantum coherent cyclotron transitions induce coherent metabolism in the length scale of entire body. The lowest EEG frequencies correspond to the maximum energy of photons of black body radiation at room temperature. Even the basic building blocks of DNA, which are charged (this is due to the phosphate group PO₄^2-), and thus also DNA sequences, satisfy the thermal stability condition but rather marginally, which could explain the sensitivity of the functioning of living matter to body temperature. Spin interaction energies have the ordinary value and thus spin is thermalized unless thermal isolation is assumed.

3. What distinguishes between the magnetic bodies associated with Earth and living organisms??

The notion of magnetic body is central to the TGD inspired theory of living matter. Every system possesses magnetic body and there are strong reasons to believe that the magnetic body associated with human body is of order Earth size. Therefore the question arises what distinguishes between the magnetic bodies of Earth and human body. The quantization of magnetic flux suggests an answer to this question. There are several manners to achieve quantization of magnetic flux with dynamical hbar.

1. One possibility is that the area S of flux quantum scales as hbar². In this case flux quantization implies that B scales as 1/hbar, cyclotron frequency as 1/hbar whereas cyclotron energy is invariant under the scaling of hbar.

2. Second possibility is that the value of magnetic field remains invariant and S scales as hbar. This is especially natural when flux quanta are magnetic flux walls. In this case cyclotron frequencies remain invariant but cyclotron energy scales as hbar. This quantization looks very natural in case of magnetic fields associated with living matter.

For the latter option one can consider both tubular and sheet like flux quanta.

1. For tubular flux quanta k=4 level of dark matter hierarchy the radii of tubular flux quanta would be about 2²²×L(169)=L(169+44=213)=20 m. This length scale would define the size scale for the quantum coherence regions of the Bose-Einstein condensates of bosonic ions. This scale is enough to guarantee that the behavior of ions in Earth's magnetic field is consistent with the model based on single-sheeted space-time. The idea that Earth's magnetic field consists of flux tubes of this thickness is attractive.

2. One can consider also flux sheets which have thickness L(169)=5 μm (cell nucleus size is in the range 5-10 μm) and the total transversal length L(169+4 ×22)=L(257)=8 × 10⁴ km, which corresponds to about 12 Earth radii. Strongly folded flux sheets of this thickness might be associated with living matter. The photon frequency corresponding to the wavelength L(257) is 3.75 Hz and characterizes theta band. This size scale could be assigned with the human magnetic body in the following manner.

   Suppose that the magnetic flux flows in head to tail direction so that the magnetic flux arrives to the human body through a layer of cortical neurons. Assume that the flux quantum is branched as it arrives cortex so that branches flow through the uppermost layer of neurons and that DNA of each neuronal nucleus in the uppermost neuron layer defines a transversal section for a branch of this kind. The total length of DNA in single human cell is about one meter. If the number of cortical neurons in single layer is about 8× 10⁷ which corresponds roughly to an average neuron radius of 10^-5 meters then the total length of DNA corresponds to the total width of flux quantum.

4. Dark Z⁰ magnetic fields and cognition

One can repeat the same arguments assuming that cells are high T_c super conductors in the sense discussed in the previous postings. TGD assigns the mysterious absorption lines at 27, 41, 50, 68 meV associated with high T_c superconductors with the transitions of "wormholy" Cooper pairs carrying boundary supra currents associated with tube like structures with radius of .2 μm. The boundary of this structure itself has 2-layered structure of total thickness L(151)= 10 nm of cell membrane ith layers having thickness L(149)=5 nm of lipid layer of cell membrane. These scales are characteristic axonal length scales but universal predictions of the model of high T_c superconductivity. These energies correspond expressed in Kelvin correspond 270, 410, 500, 680 Kelvins. By dividing with a factor 2.88 one obtains the temperatures for which 3-D black body radiation has maximum intensity for these frequencies. The energies are below thermal energies but not too much. For effectively 1 and 2-D systems the situation improves since there is no maximum and exponential damping of spectrum sets on at much lower energy.

Similar argument applies in the scale L_w=.2 μm for Z⁰ magnetic transitions with scale about 10⁴ eV much above the thermal energy scale. The hierarchy of length scales is now L_w=.2 μm, .4 mm, .8 m,.... L_w=.4 mm, possibly characterizing mm sized cortical modules, corresponds roughly to a frequency scale 40/A Hz, A atomic weight. The thermal stability supports the earlier idea that Z⁰ force, dark neutrino superconductivity, ννbar wormhole contacts, and ZEG relate to cognition which must be thermally insulated whereas electromagnetic interactions would relate to sensory perception which could be highly sensitive even to temperature differences.

5. Strange behavior of cellular water and quantal ionic currents through cell membrane

The fact that cellular water does not leak out of cell in a centrifugal force suggests that some fraction of water inside cell is in different phase. One explanation is that the nuclei of water inside cell are in doubly dark phase whereas electrons are in singly dark phase (having Compton length of 5 nm and perhaps directly "visible" using day technology!) as indeed predicted by the model of high Tc superconductivity.

This conceptual framework could explain various findings challenging the notions of ionic pumps and channels discussed in more detail in the chapter Biosystems as Superconductors: Part II of "TGD Inspired Theory of Consciousness...." where also references can be found.

1. The selectivity of the cell membrane implies that channels cannot be simple sieves and there must be complex information processing involved.
2. The needed number of pumps specialized to particular ions is astronomical and the first question is where to put all these channels and pumps. On the other hand, if the cell constructs the pump or channel specialized to a given molecule only when needed, how does it know what the pump looks like if it has never seen the molecule? The needed metabolic energy to achieve all the pumping and channelling is huge. Strangely enough, pumping does not stop when cell metabolism stops.
3. One can also wonder why the ionic currents through cell membrane look quantal and are same through cell membrane and silicon rubber membrane.

These observations suggest non-dissipative ionic currents and quantum self-organization. The TGD based explanation would be in terms of high T_c electronic and possibly also ionic superconductivity associated with cell membrane made possible by the large hbar phase for nuclei and electrons in the interior of cell. It however seems that thermal stability conditions allow only protonic Cooper pairs in the model of ionic Cooper pairs based on direct generalization of the model of high T_c electronic super conductivity.

6. How noble gases can act as anesthetes?

Chemically inert noble gases are known to act as anesthetes. Somehow these atoms affect neuronal membrane, probably reducing the nerve pulse activity. A possible explanation is in terms of anomalous weak isospin due to the charged color bonds inside nuclei of noble gas generated in the cellular environment. This bonds carry also em charge so that noble gas atom would behave like ion with nuclear charge Z+1 or Z-1. Also the long ranged color force and dark weak force with range L_w=.2 μm associated with noble gas nuclei in dark phase could be part of the solution of the mystery.

Matti Pitkänen

Large hbar limit and macroscopic limit of quantum theory

The necessity of large hbar phases has been actually highly suggestive since the first days of quantum mechanics. The classical looking behavior of macroscopic quantum systems remains still a poorly understood problem and large hbar phases provide a natural solution of the problem.

In TGD framework quantum coherence regions correspond to space-time sheets. Since their sizes are arbitrarily large the conclusion is that macroscopic and macrotemporal quantum coherence are possible in all scales. Standard quantum theory definitely fails to predict this and the conclusion is that large hbar phases for which quantum length and time scales are proportional to hbar and long are needed.

Somewhat paradoxically, large hbar phases explain the effective classical behavior in long length and time scales. Quantum perturbation theory is an expansion in terms of gauge coupling strengths inversely proportional to hbar and thus at the limit of large hbar classical approximation becomes exact! Also the Coulombic contribution to the binding energies of atoms vanishes at this limit. The fact that we experience world as a classical only tells that large hbar phase is essential for our sensory perception. Of course, this is not the whole story and the full explanation requires a detailed anatomy of quantum jump.

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

Matti Pitkänen

Can TGD predict the spectrum of Planck constants?

The idea that hbar is dynamical and can have arbitrarily large values is about one and half year old as a write this. A lot of progress has occurred during the last year but I have not yet been able to seriously pose the question whether and how TGD could predict the values of the Planck constant. In the chapter Was von Neumann Right After All? a proposal for how TGD predicts the value spectrum of hbar as one aspect of quantum criticality is discussed and number theoretical arguments are used to make a guess about the spectrum of hbar.

1. The outline of the argument

In a very concise form the argument goes as follows.

1. Generalization of the basic facts about the dependence of Dirac operator on hbar leads to the realization that the freedom to choose the value of hbar corresponds to the freedom to choose the overall scaling λ= hbar/hbar₀ of M⁴ metric associated with various copies of M⁴ obtained identified as various algebraic extensions of rational M⁴ glued together along common set of rationals consistent with the isometric identification. This generalization of number concept is one of the basic ideas of number theoretic vision and allows to fuse real and p-adic physics to single coherent whole by requiring that all of these physics are obtained by an algebraic continuation from rational physics.
2. The modified Dirac operator for space-time surfaces depends on λ via the induced metric determining the fermionic oscillator operator basis. Dirac determinant determines Kähler function. λ cannot be chosen freely. The dependence of λ on the algebraic extension for a given p-adic prime p is fixed by the quantum criticality condition stating that the critical Kähler coupling strength g_K² is same for various algebraic extensions associated with given p-adic prime p.
3. Number theoretic ideas allow to make good guesses concerning the dependence of λ on the algebraic extension.

2. Hierarchy of spinors structures in the world of classical worlds defined by algebraic extensions of rationals

What the rational physics and its algebraic extensions mean in practice is following.

1. The modified Dirac operator D associated with the induced spinor structure at space-time surfaces which depends on λ via the scale of M⁴ metric. The eigen values associated with the eigen spinors of D label the fermionic oscillator operators in turn determining the configuration space spinor Clifford algebra and spinor structure. Rational physics means restriction to a subset of rational eigenvalues and various extensions restriction to eigenvalues in that extension so that one obtains a family of configuration space spinor structures. There is no need for discretization as one might naively think: just a restriction to eigenvalues in a given extension.
2. The inclusion of rationals to a given extension induces an inclusion of type II₁ factors analogous to Jones inclusion characterized by the index M:N telling the dimension of the extended configuration space ("world of classical worlds") Clifford algebra as a rational Clifford algebra module. This generalized index M:N is what contains information about hbar for that extension.
3. Kähler function for a given space-time sheet in given extension can be defined as a Dirac determinant involving only the eigenvalues in that extension. It is quite possible that this restriction automatically guarantees the finiteness of the determinant. The Dirac determinant is exponent of Kähler function with certain value of Kähler coupling g_K². The condition determining the dependence of λ on algebraic extension is that the value of λ belonging to the extension is such that g_K² is same as for rational case.
4. The beauty of this picture is that since M⁴ metric is inversely proportional to λ the induced metric contains λ Kähler action and function and thus configuration space geometry depends highly nonlinearly on λ (hbar). This means that classical theory codes all non-perturbative effects although it is quite possible that loop corrections to configuration space functional integral vanish. Classical theory allows also to obtain information about quantum corrections: for instance, quantization of the magnetic flux classically implied by the basic variational principle uses λ as a unit and thus it might be possible to deduce λ in this manner for a given algebraic extension.

3. λ as homomorphism from space of algebraic extensions of rationals to reals

1. The generating units of extension can be classified to phases, the number of which is at most one, and set E of generating units for which complex norm differs from unity. A natural assumption is that λ is product of term λ_a depending on the phase via Beraha number defined by the phase and of term λ_b depending on E only through the number of units in E. λ_a can be identified either as the inverse of the fractal spinor dimension d or the corresponding space dimension given by 2-based logarithm of d. The latter option is more plausible physically since it is consistent with the assumption that M:N is integer power of M:N for ordinary Jones inclusion meaning that a tensor power of M:N= B_n fractal-dimensional Clifford algebra as rational Clifford algebra module is obtained. This option is also consistent the earlier physics inspired guess.
2. The general form for the dependence of λ_b on the extension can be deduced by requiring that it is a multiplicative homomorphism from the family of algebraic extensions to real numbers. What this means is that if one combines basis of linearly independent extensions E₁ and E₂ with d₁ and d₂ linearly independent generating "imaginary" units with complex norm different from unity, the value of λ_b for the combined extension is product.

   This gives a large factor which comes as a power of N= (1/v₀), 1/v₀≈ 2¹¹. The Clifford algebra associated with the algebraic extension is N^k:th tensor power of minimal M:N= B_n dimensional module or tensor product of these. The powers of 1/v₀ define a hierarchy of values of hbar and thus a hierarchy of macroscopically quantum coherent systems with increasing size. Dark matter hierarchy corresponds to the hierarchy of algebraic extensions of rationals and ordinary matter to rational physics. This should translate more or less easily to the p-adic context so that each p-adic number field defines this kind of hierarchy of extensions.

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

Further progress in the understanding of high Tc superconductivity

I have made further progress in the model for high T_c superconductors and it seems that high T is becoming one of the victories of TGD and involves all key aspects of TGD such as many-sheeted space-time, quantum criticality of TGD Universe, p-adic length scale hypothesis, dark matter as ordinary matter in large hbar phase with precise prediction for the possible values of hbar, etc.. Even the predicted homological magnetic monopoles made possible by CP₂ geometry are crucial for the model of Cooper pairs.

At qualitative level the model explains various strange features of high T_c superconductors such as the existence of pseudogap: this is due to the fact that two kinds of super-conductivities corresponding to BCS type large hbar supra currents at interior and boundary supra currents carried by wormholy Cooper pairs. Model also explains why 2-dimensionality and restriction of currents to "rivers" is essential for superconductivity. At quantitative level the model predicts correctly the four poorly understood photon absorption lines and allows to understand the critical doping ratio. The current carrying structures have structure similar to that of axon including the double layered structure of cell membrane and also the size scales are predicted to be same so that the idea that axons are high T_c superconductors is highly suggestive.

The following gives a rough description for the model for high T_c superconductors.

1. Neuronal axon as a geometric model for current carrying "rivers"

Neuronal axons, which are bounded by cell membranes of thickness L(151) consisting of two lipid layers of thickness L(149) are high T_c superconductors (this was not the starting point but something which popped out naturally). The interior of this structure is in large hbar nuclear phase, which is partially dark. Since the thickness of the tube should be smaller than the quantum size of the dark nuclei, a lower limit for the the radius r of the corresponding nuclear space-time sheets is obtained by scaling up the weak length scale L_w(113)=2^(11-89)/2L_w(89) defined by W boson Compton length by a factor 2²² to doubly dark weak length scale L_w=2²²L_w(113)=.2 μm.

These flux tubes with radius r>L_w define "rivers" along which conduction electrons and various kinds of Cooper pairs flow. Scaled up electrons have size L(k_eff=149) corresponding to 5 nm, the thickness of the lipid layer of cell membrane. The observed quantum fluctuating stripes of length 1-10 nm might relate very closely to scaled up electrons with Compton length 5 nm, perhaps actually representing zoomed up electrons! The critical doping fraction p_cr=.14 can be understood by assuming that holes gather to the L(151) thick boundary of tube with radius r>L_w, and that the density of holes is at most one hole per Cu atom, which is indeed very natural. This gives the estimate r=3L_w/2=.3 μm for neuronal axons r≈ 5L_w/2= .5 μm holds true.

2. Cuprate layers are in partially dark phase

It is far from clear what it means that cuprate layers are in a partially dark phase. One can imagine several models but the following one seems to be the most plausible one.

1. The model explaining the anomalies of H₂O assumes 1/4:th of protons are in dark phase. The guess would be that some fraction of Cu atoms perhaps related to the doping fraction is in dark phase with large hbar. The large charge of Cu(29,63) nuclei would by the general criterion force a transition to large hbar phase. This kind of mechanism might be quite general manner to obtain dark nuclear matter.
2. The dark phase would differ dramatically from the ordinary one. Cu atom size would be scaled by a factor 2² and ground state energy would be scaled down by a factor 2^-22 to 3.2× 10^-6 eV. The large density of dark nuclei and electrons does not allow scaled up Cu atoms. The overlap of nuclei would make possible formation of Cooper pairs or larger structures, some kind of super fluidity is expected to result. Also the dropped electrons would in large hbar phase and form Cooper pairs and a good guess is that this occurs by a generalization of the conventional BCS mechanism.

3. At least two kinds of superconductivities are involved

Two kinds of superconductivities are involved and dark Cu nuclei define superconducting superfluid phase. These superconductivities correspond to interior supra currents and surface supra currents having quite different character. The pseudogap and upper critical temperature at which system becomes conductive would naturally relate to the presence of these two superconductivities.

1. In the interior there is large hbar BCS type superconductivity based on phonon exchange in interior with T_c scaled up by a factor 2² whereas critical magnetic field is scaled down by a factor 2^-11/2 so that it has same order of magnitude as the Earth's magnetic field. Super-conductivity is made possible by the overlap of scaled up conduction electrons which in normal phase do not have overlap so that system would behave as insulator. It is tempting to associate the higher critical temperature with this superconductivity. The magnetic perturbations produced by antiferromagnet would tend to destroy this superconductivity above lower critical temperature since the value of the critical magnetic field is so low and because the surface supra currents hindering the penetration of magnetic fields would not be present. What would remain is conductivity which would depend on temperature via simple scaling law.
2. At the L(151) "cell membrane" like space-time sheet there is exotic superconductivity based on wormholy Cooper pairs whose model has been already discussed briefly. Stability conditions require that electrons of wormholy Cooper pairs have ordinary value of hbar but have dropped to L(149) thick space-time sheets. Critical magnetic field is of order Tesla since hbar is ordinary and this critical magnetic field determines the effective critical magnetic field of high T_c super conductor below the lower critical temperature since surface supra currents prevent the penetration of the external magnetic field to superconductor.

4. Model for wormholy Cooper pair

Electrons are condensed at k=149 space-time sheets of size 5 nm and feed the electric gauge flux to k=151 space-time sheet through wormhole contacts carrying quantum numbers of u and dbar quarks. There is a monopole flux through u type wormhole contact returning back through dbar type wormhole contact. This is the first time that homological magnetic monopoles predicted by CP₂ topology appear in an explicit TGD inspired physical model. Interestingly, the first 23 year old model of color confinement was based on the identification of color hyper charge as homological charge. In the recent conceptual framework the the space-time correlate for color hyper charge Y of quark could be homological magnetic charge Q_m= 3Y so that color confinement for quarks would have purely homological interpretation at space-time level.

These electronic flux tubes are connected by color bonds and the quarks at wormhole throats for (udbar)² type color singlet such that udbar does not reduce to color singlet so that color confinement binds the Cooper pair and electrons are "free-travellers". These exotic Cooper pairs are not energy minima if electrons are in large hbar phase, which forces BCS type large hbar superconductivity in interior.

Both the assumption that electrons condensed at k=149 space-time sheets result from scaled up large hbar electrons and minimization of energy imply the the scales L(149) and L(151) for the space-time sheets involved so that there is remarkable internal consistency. The model explains the spins of the wormholy Cooper pairs and their angular momenta. The dark BSC type Cooper pairs are expected to have S=0 and L=0.

5. Which elements could be high T_c superconductors?

More generally, elements having one electron in s state plus full electronic shells are good candidates for doped high T_c superconductors. If the atom in question is also a boson the formation of atomic Bose-Einstein condensates at Cooper pair space-time sheets is favored. Superfluid would be in question. Thus elements with odd value of A and Z possessing full shells plus single s wave valence electron are of special interest. The six stable elements satisfying these conditions are ⁵Li, ³⁹K, ⁶³Cu, ⁸⁵Rb, ¹³³Cs, and ¹⁹⁷Au. Partially dark Au for which dark nuclei form a superfluid could correspond to what Hudson calls White Gold: Hudson claims that also other transition metals have similar properties in their "monoatomic" phase. Most of the properties claimed by Hudson are understandable if these elements are high T_c superconductors with nuclei in a partially dark phase. Regrettably professional physicists are unable to consider the possibility that these claims might have some germ of truth despite the fact that a lot non-scientific looking claims are involved.

6. Are living systems high T_c superconductors?

The idea about cells and axons as superconductors has been one of the main driving forces in development of the vision about many-sheeted space-time. Despite this the realization that the supra currents in high T_c superconductors flow along structure similar to axon and having same crucial length scales came as a surprise. Axonal radius which is typically of order r=.5 μm. lambda;=2¹¹ would predict r>.2 μm. The model can be easily generalized so that it allows also the interpretation of microtubuli and endoplasma membranes as high T_c superconductors.

Interestingly, Cu is one of the biologically most important trace elements. For instance, copper is found in a variety of enzymes, including the copper centers of cytochrome c-oxidase, the Cu-Zn containing enzyme superoxide dismutase, and copper is the central metal in the oxygen carrying pigment hemocyanin. The blood of the horseshoe crab, Limulus polyphemus uses copper rather than iron for oxygen transport. Hence there are excellent reasons to ask whether living matter might be able to build high T_c superconductors based on copper oxide.

For more details see the chapters Superconductivity in Many-Sheeted Space-Time of p-Adic TGD and Bio-Systems as Superconductors: part I of "TGD Inspired Theory of Consciousness...". For dark neutrino super-conductivity and its application in the quantum model of hearing and cognition see the chapter Bio-Systems as Superconductors: part II of "TGD Inspired Theory of Consciousness..." and the chapter Quantum Model for Hearing of "Genes, Memes, Qualia,......"