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Life as islands of rational/algebraic numbers in the seas of
real and p-adic continua
by Dr. Matti Pitkänen / February 15, 2010
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The possibility to define Entropy differently for rational/algebraic entanglement -- and the fact that number theoretic entanglement Entropy can be negative -- raises the question about which kind of systems can possess this kind of entanglement. I have considered several identifications. But the most elegant interpretation is based on the idea that living matter resides in the intersection of real and p-adic worlds -- somewhat like rational numbers live in the intersection of real and p-adic number fields.
The observation that Shannon entropy allows an infinite number of number theoretic variants for which the entropy can be negative in the case that probabilities are algebraic numbers leads to the idea that Living matter in a well-defined sense corresponds to the intersection of real and p-adic worlds.
This would mean that the mathematical expressions for the space-time surfaces (or at least 3-surfaces or partonic 2-surfaces and their 4-D tangent planes) make sense in both real and p-adic sense for some primes p. The same would apply to the expressions defining quantum states. In particular, entanglement probabilities would be rationals or algebraic numbers so that entanglement can be negentropic and the formation of bound states in the intersection of real and p-adic worlds generates information and is thus favored by NMP.
This picture has also a direct connection with Consciousness.
1. Algebraic entanglement is a prerequisite for the realization of intentions as transformations of p-adic space-time sheets to real space-time sheets representing actions. Essentially, a leakage between p-adic and real worlds is in question and makes sense only in Zero Energy Ontology since various quantum numbers in real and p-adic sectors are not, in general, comparable in positive energy ontology so that Conservation Laws would be broken.
Algebraic entanglement could be also called cognitive. The transformation can occur if the partonic 2-surfaces and their 4-D tangent space-distributions are representable using rational functions with rational coefficients in preferred coordinates for the imbedding space dictated by symmetry considerations. Intentional systems must live in the intersection of real and p-adic worlds. For the minimal option, Life would be also effectively 2-dimensional phenomenon and essentially a boundary phenomenon as number theoretical criticality also suggests.
2. The generation of non-rational (non-algebraic) bound state entanglement between the system and external world means that the system loses consciousness during the state function reduction process following the U-process generating the entanglement. What happens is that the Universe corresponding to a given CD decomposes to 2 un-entangled subsystems which in turn decompose. The process continues until all subsystems have only entropic bound state entanglement or negentropic algebraic entanglement with the External World.
3. If the sub-system generates entropic bound state entanglement in the process, it loses consciousness. Note that the entanglement entropy of the sub-system is a sum over entanglement entropies over all subsystems involved. This hierarchy of subsystems corresponds to the hierarchy if sub-CDs so that the survival without a loss of consciousness depends on what happens at all levels below the highest level for a given self.
In more concrete terms, the ability to stay conscious depends on what happens at cellular level also. The stable evolution of systems having algebraic entanglement is expected to be a process proceeding from short to long length scales as the evolution of Life indeed is.
4. U-process generates a superposition of states in which any sub-system can have both real and algebraic entanglement with the External World. This would suggest that the choice of the type of entanglement is a volitional selection. A possible interpretation is as a choice between Good and Evil. The hedonistic complete freedom resulting as the entanglement entropy is reduced to zero on one hand and the algebraic bound state entanglement implying correlations with the External World and meaning giving up the maximal freedom on the other hand.
The hedonistic option is risky since it can lead to non-algebraic bound state entanglement implying a loss of consciousness. The second option means expansion of Consciousness -- a fusion to the ocean of consciousness as described by spiritual practices.
5. This formulation means a sharpening of the earlier statement "Everything is conscious and consciousness can be only lost" with the additional statement "This happens when non-algebraic bound state entanglement is generated and the system does not remain in the intersection of real and p-adic worlds anymore".
Clearly, the quantum criticality of the TGD Universe seems has very many aspects and Life as a critical phenomenon in the number theoretical sense is only one of them besides the criticality of the space-time dynamics and the criticality with respect to phase transitions changing the value of Planck constant and other more familiar criticalities. How closely these criticalities relate remains an open question.
A good guess is that algebraic entanglement is essential for quantum computation which therefore might correspond to a conscious process. Hence, cognition could be seen as a quantum computation like process (a more appropriate term being quantum problem solving). Living-Dead dichotomy could correspond to rational-irrational or to algebraic-transcendental dichotomy. This at least when Life is interpreted as Intelligent Life. Life would in a well-defined sense correspond to islands of rationality/algebraicity in the seas of real and p-adic continua.
The view about the crucial role of rational and algebraic numbers (as far as Intelligent Life is considered) could have been guessed on very general grounds from the analogy with the orbits of a dynamical system. Rational numbers allow a predictable periodic decimal/pinary expansion and are analogous to one-dimensional periodic orbits. Algebraic numbers are related to rationals by a finite number of algebraic operations and are intermediate between periodic and chaotic orbits allowing an interpretation as an element in an algebraic extension of any p-adic number field.
The projections of the orbit to various coordinate directions of the algebraic extension represent now periodic orbits. The decimal/pinary expansions of transcendentals are un-predictable being analogous to chaotic orbits.
The special role of rational and algebraic numbers was realized already by Pythagoras. The fact that the ratios for the frequencies of the musical scale are rationals supports the special nature of rational and algebraic numbers. The special nature of the Golden Mean (which involves 51/2) conforms the view that algebraic numbers rather than only rationals are essential for Life.
For details, see for instance the article "TGD-inspired theory of Consciousness" and the chapter "Quantum Theory of Self-Organization" of Bio-Systems as Self-Organizing Systems.
Comments
1. At 3:27 AM, Matti Pitkanen said...
The new territory will be the connection between Number Theory and Physics. The gist of my posting is that the quantum physics of Intention, Cognition, and Life could be what one might call "number theoretical quantum field theories". Basic notion would be rational points and subset of algebraic points as intersection of partonic 2-surfaces. At these points, Intention and Action would meet.
This notion is certainly familiar for mathematicians but totally unknown land for physicists. Maybe this physics is standard stuff of theoretical physics seminars before year 2050;-).
2. At 3:46 AM, Matti Pitkanen said...
Nuclear string model predicts a lot of new nuclear physics -- part of which has already shown itself as anomalies. The essentially new element are color flux tubes with quark and anti-quark at their ends connecting nucleons. They can have charges 1,-1 besides vanishing charges. Charged color bonds imply almost degeneracy of nuclear ground states.
One can have bosonic counterparts of fermionic nuclei with almost same mass and giving rise to almost the same chemistry but playing a key role in the new dark matter chemistry since they can form Bose-Einstein condensates at the magnetic flux tubes.
The new secondary p-adic time scales assigned with elementary particles. In particular the 0.1 second timescale assignable with electron as timescale of corresponding "Causal Diamond" and defining fundamental biorhythm will mean a direct connection between living and dead matter.
By the way, I received from Gabor an article published by Nobelist Szent-Györgyi as early as 1941 with the title "Towards a New Biochemistry" [included below]. The necessity for new Biochemistry has been realized by the greatest visioniaries for 70 years ago. But the mainstream of Big Science is still trying to make progress with textbook wisdom! Mediocricy has replaced democracy in science.
3. At 7:47 PM, Matti Pitkanen said...
Entropy (that is, negative information) is the only thing that Physics today can offer. Shannon entropy is non-negative so that Shannon negentropy as its negative defines negative information measure. In case of entanglement, this is easy to understand since we do not know with certainty whether Schrodinger's cat is dead or alive. The point is to find measure for the entanglement itself as carrier of information.
If entanglement probabilities are rational numbers (even algebraic numbers are allowed), one can define number theoretic (p-adic) variants of Shannon entropy satisfying the basic defining conditions. These entropies can be negative and one can find prime p for which the entropy is maximally negative. Rational (algebraic) entanglement carries information- even more: conscious information. Entangling in this manner does not lead to a loss of consciousness but expands it. Joining to the ocean of consciousness as meditators express it. Suddenly even this talk about "cosmic consciousness" begins to make sense.
What is remarkable that this condition is true if Life is in the intersection of p-adic and real worlds. Life and intelligent Consciousness mean a physical counterpart for number theoretical criticality. p-Adic Intention can transform to real Action.
Also my own great experience -- which took me to the road leading to Consciousness theory and about which I have told a lot at my homepage -- can be finally understood. I have really experienced what it feels when all this background noise in body disappears and the entire body (or is it the magnetic one?) goes in a flowing liquid-like state of well-being.
All evolution is basically spiritual in this framework. Just expansion of Consciousness experienced as well-being and love. The basic ethical value is here. Moral rules can provide means of achieving this depends on situation. The basic rule is that the rule should not cause violence (which means by definition reduction of consciousness). Krishnamurti said all this in different words. He was quite spiritual number theorists. Indians are!
4. At 5:32 AM, Matti Pitkanen said...
I have now begun to understand how far reaching the implications of number theoretic Shannon entropy are. The presence of the negentropic entanglement implies the directedness of the biological processes since the outcome of the state function reduction would be far from random. The behavior of negentropic bonds could be almost deterministic. In the case of time-like entanglement, this would select only particular initial final state pairs so that determinism would emerge also in this manner and could lead to almost deterministic irreversible cellular automaton behavior characteristic for the living matter and very different from the reversible determinism of Classical Physics and very difficult to understand in Quantum context.
This would provide a general explanation for the ability of the living matter to overcome the Second Law of Thermodynamics basically implied by quantum randomness of the standard Quantum Theory. Non-equilibrium thermodynamics and cellular automaton models could be seen as phenomenological descriptions for the actual breaking of the Second Law in the intersection of real and p-adic worlds.
I reworked to today the model for ATP→ADP+Pi (the fundamental piece of Biology). The first basic elements of the model are the magnetic flux tubes forming an Indra's net. For instance, they would connect ATP to catalyst molecules to catalyst molecules so that reduction of hbar would contract the flux tubes and bring the molecules near to each other.
The second element is liberation of metabolic energy as electron or proton drops to a larger space-time sheet. The third element is the assumption that the O=:s of phosphates act as universal plugs to which flux tubes are connected.
The new element is the assumption that negentropic entanglement is associated with the flux tubes and is between the ends of flux tubes. The innermost flux O= is connected to catalyst molecule so that ATP and O= can find each other when needed by reducing the value of Planck constant.
This identifies high energy phosphate bond as a flux tube between outer O=:s of ATP and carrying negentropic entanglement and positive energy. No binding energy would be needed since Negentropy Maximization Principle would stabilize the system.
All flux tube connections can be regarded as high energy bonds made possible by NMP which allows to have correlated pairs without binding energy. This binding would be a free love affair whereas ordinary binding to the bottom of potential well would be like the jail of the organized marriage. Krishnamurti was a brilliant biologists too;-)!
It really becomes possible to understand ATP to ADP transformation and various related processes. I attached the first draft of pdf file [below] as an example.
5. At 1:42 AM, Matti Pitkanen said...
from Ulla: "How is it then with free will in TGD if everything is deterministic?"'
I did not say that everything is deterministic. Only in the intersection of real and p-adic worlds it is possible to get rid of total randomness of the state function reduction process simply because negentropy maximization does not choose between entangled eigenstates of density matrix with vanishing entropy now but leads to highly unique state with minimum negative entanglement entropy.
Thermalization in ensemble can be seen as a consequence of the random outcome of ordinary state function reduction and this is avoided in the case of negentropic entanglement. Thermal noise disappears just as I directly experienced as a bodily sensation in my "big experience".
Second aspect: Every quantum jump involves as a first step unitary process and this makes means re-creation. Only the initial and final states of even (positive and negative energy parts of Zero Energy state) in this intersection tend to be strongly correlated (unlike in the case of ordinary state function reduction).
In any case, however, I am just beginning to realize how powerful a notion this number theoretic negentropy is.
6. At 11:41 PM, Matti Pitkanen said...
All Zero Energy states are in well-defined sense morphic fields. It is a matter of convention at what level one begins to speak about memory.
For instance, electrons causal diamond corresponds to the time scale 0.1 seconds which defines the fundamental biorhythm. This suggests that the magnetic bodies of electrons are crucial for consciousness and life. The self-organization patterns assignable to self plus the hierarchy of its subselves resulting in quantum jump define the habits, memes. The feed of ordered energy is central in self organization and now it is replaced with the feed of energy assignable to negentropic entanglement. Dissipation serves still as the Darwinian selector.
The interpretation of feelings and emotions like pain is very natural if one assigns the fundamental emotions to the expansion or loss of consciousness at some level of hierarchy of subselves.
Negentropic entanglement would allow to understand emotions. Negentropy would be the basic measure. Unfortunately, the positive emotion resulting from negentropic entanglement has a finite duration in Geometric-Time since CD has finite size.
About the beginning of new era. Science is what determines our values (or rather the lack of them) at this moment. In reductionistic science, there are no values; there is no good or evil; there is no love. There is only a fight for survival or the universe as a dead machine. This kind of world view more than explains why society has become a jungle. Our civilization is really about destroying itself.
The only manner to change the situation is through the fusion of Science and Spirituality. My sincere hope is that these ideas could be taken seriously despite the fact that they generate aggression and disgust in a typical colleague for whom science means a specialization to apply a particular algorithm to lengthen curriculum vitae.
TOWARDS A NEW BIOCHEMISTRY? [1]
by Professor A. Szent-Gygrgyi
University, Szeged, Hungary
The atom consists of a nucleus surrounded by a system of electrons. By sharing one-or-more electrons, atoms can join to form molecules. In such a molecule, as a rule every electron belongs to 1-or-2 atoms. This is our idea of a single small molecule. And this picture has hitherto unconsciously governed our thinking in Biochemistry.
The study of crystals and metals, however, has revealed the existence of a different state of matter. If a great number of atoms is arranged with regularity in close proximity (as for instance, in a crystal lattice), the terms of the single valency electrons may fuse into common bands. The electrons in this band cease to belong to 1-or-2 atoms only and belong to the whole system.