Year 2017

How to demonstrate quantum superposition of classical gravitational fields?

There was rather interesting article in Nature (see this) by Marletto and Vedral about the possibility of demonstrating the quantum nature of gravitational fields by using weak measurement of classical gravitational field affecting it only very weakly. There is also an article in arXiv by the same authors (see this). The approach relies on quantum information theory.

The gravitational field would serve as a measurement interaction and the weak measurements would be applied to gravitational witness serving as probe - the technical term is ancilla. Authors claim that weak measurements giving rise to analog of Zeno effect could be used to test whether the quantum superposition of classical gravitational fields (QSGR) does take place. One can however argue that the extreme weakness of gravitation implies that other interactions and thermal perturbations mask it completely in standard physics framework. Also the decoherence of gravitational quantum states could be argued to make the test impossible.

One must however take these objections with a big grain of salt. After all, we do not have a theory of quantum gravity and all assumptions made about quantum gravity might not be correct. For instance, the vision about reduction to Planck length scale might be wrong. There is also the mystery of dark matter, which might force considerable motivation of the views about dark matter. Furthermore, General Relativity itself has conceptual problems: in particular, the classical conservation laws playing crucial role in quantum field theories are lost. Superstrings were a promising candidate for a quantum theory of gravitation but failed as a physical theory.

In TGD, which was born as an attempt to solve the energy problem of TGD and soon extended to a theory unifying gravitation and standard model interactions and also generalizing string models, the situation might however change. In zero energy ontology (ZEO) the sequence of weak measurements is more or less equivalent to the existence of self identified as generalized Zeno effect! The value of h_eff/h=n characterizes the flux tubes mediating various interactions and can be very large for gravitational flux tubes (proportional to GMm/v₀, where v₀<c has dimensions of velocity, and M and m are masses at the ends of the flux tube) with Mm> v₀m_Pl² (m_Pl denotes Planck mass) at their ends. This means long coherence time characterized in terms of the scale of causal diamond (CD). The lifetime T of self is proportional to h_eff so that for gravitational self T is very long as compared to that for electromagnetic self. Selves could correspond sub-selves of self identifiable as sensory mental images so that sensory perception would correspond to weak measurements and for gravitation the times would be long: we indeed feel the gravitational force all the time. Consciousness and life would provide a basic proof for the QSGR (note that large neutron has mass of order Planck mass!).

See the article How to demonstrate quantum superposition of classical gravitational fields? or the chapter Quantum criticality and dark matter.

Anomalous neutron production from an arc current in gaseous hydrogen

I learned about nuclear physics anomaly new to me (actually the anomaly is 64 years old) from an article of Norman and Dunning-Davies in Research Gate (see this). Neutrons are produced from an arc current in hydrogen gas with a rate exceeding dramatically the rate predicted by the standard model of electroweak interactions, in which the production should occur through e-+p→ n+ν by weak boson exchange. The low electron energies make the process also kinematically impossible. Additional strange finding due to Borghi and Santilli is that the neutron production can in some cases be delayed by several hours. Furthermore, according to Santilli neutron production occurs only for hydrogen but not for heavier nuclei.

In the following I sum up the history of the anomaly following closely the representation of Norman and Dunning-Davies (see this): this article gives references and details and is strongly recommended. This includes the pioneering work of Sternglass in 1951, the experiments of Don Carlo Borghi in the late 1960s, and the rather recent experiments of Ruggiero Santilli (see this).

The pioneering experiment of Sternglass

The initial anomalously large production of neutrons using an current arc in hydrogen gas was performed by Earnest Sternglass in 1951 while completing his Ph.D. thesis at Cornell. He wrote to Einstein about his inexplicable results, which seemed to occur in conditions lacking sufficient energy to synthesize the neutrons that his experiments had indeed somehow apparently created. Although Einstein firmly advised that the results must be published even though they apparently contradicted standard theory, Sternglass refused due to the stultifying preponderance of contrary opinion and so his results were preemptively excluded under orthodox pressure within discipline leaving them unpublished. Edward Trounson, a physicist working at the Naval Ordnance Laboratory repeated the experiment and again gained successful results but they too, were not published.

One cannot avoid the question, what physics would look like today, if Sternglass had published or managed to publish his results. One must however remember that the first indications for cold fusion emerged also surprisingly early but did not receive any attention and that cold fusion researchers were for decades labelled as next to criminals. Maybe the extreme conservatism following the revolution in theoretical physics during the first decades of the previous century would have prevented his work to receive the attention that it would have deserved.

The experiments of Don Carlo Borghi

Italian priest-physicist Don Carlo Borghi in collaboration with experimentalists from the University of Recife, Brazil, claimed in the late 1960s to have achieved the laboratory synthesis of neutrons from protons and electrons. C. Borghi, C. Giori, and A. Dall'Olio published 1993 an article entitled "Experimental evidence of emission of neutrons from cold hydrogen plasma" in Yad. Fiz. 56 and Phys. At. Nucl. 56 (7).

Don Borghi's experiment was conducted via a cylindrical metallic chamber (called "klystron") filled up with a partially ionized hydrogen gas at a fraction of 1 bar pressure, traversed by an electric arc with about 500V and 10mA as well as by microwaves with 10¹⁰ Hz frequency. Note that the energies of electrons would be below .5 keV and non-relativistic. In the cylindrical exterior of the chamber the experimentalists placed various materials suitable to become radioactive when subjected to a neutron flux (such as gold, silver and others). Following exposures of the order of weeks, the experimentalists reported nuclear transmutations due to a claimed neutron flux of the order of 10⁴ cps, apparently confirmed by beta emissions not present in the original material.

Don Borghi's claim remained un-noticed for decades due to its incompatibility with the prevailing view about weak interactions. The process e^-+p→ n+ν is also forbidden by conservation of energy unless the total cm energy of proton and the electron have energy larger than Δ E= m_n-m_p-m_e=0.78 MeV. This requires highly relativistic electrons. Also the cross section for the reaction proceeding by exchange of W boson is extremely small at low energies (about 10^-20 barn: barn=10^-28 m² represents the natural scale for cross section in nuclear physics). Some new physics must be involved if the effect is real. Situation is strongly reminiscent of cold fusion (or low energy nuclear reactions (LENR), which many main stream nuclear physicists still regard as a pseudoscience.

Santilli's experiments

Ruggero Santilli (see this) replicated the experiments of Don Borghi. Both in the experiments of Don Carlo Borghi and those of Santilli, delayed neutron synthesis was sometimes observed. Santilli analyzes several alternative proposals explaining the anomalyn and suggests that new spin zero bound state of electron and proton with rest mass below the sum of proton and electron masses and absorbed by nuclei decaying then radioactively could explain the anomaly. The energy needed to overcome the kinematic barrier could come from the energy liberated by electric arc. The problem of the model is that it has no connection with standard model.

Both in the experiments of Don Carlo Borghi and those of Santilli, delayed neutron synthesis was sometimes observed. According to Santilli: According to Santilli:

" A first series of measurements was initiated with Klystron I on July 28,2006, at 2 p.m. Following flushing of air, the klystron was filled up with commercial grale hydrogen at 25 psi pressure. We first used detector PM1703GN to verify that the background radiations were solely consisting of photon counts of 5-7 μR/h without any neutron count; we delivered a DC electric arc at 27 V and 30 A (namely with power much bigger than that of the arc used in Don Borghi's tests...), at about 0.125" gap for about 3 s; we waited for one hour until the electrodes had cooled down, and then placed detector PM1703GN against the PVC cylinder. This resulted in the detection of photons at the rate of 10 - 15 μR/hr expected from the residual excitation of the tips of the electrodes, but no neutron count at all.

However, about three hours following the test, detector PM1703GN entered into sonic and vibration alarms, specifically, for neutron detections off the instrument maximum of 99 cps at about 5' distance from the klystron while no anomalous photon emission was measured. The detector was moved outside the laboratory and the neutron counts returned to zero. The detector was then returned to the laboratory and we were surprised to see it entering again into sonic and vibrational alarms at about 5' away from the arc chamber with the neutron count off scale without appreciable detection of photons, at which point the laboratory was evacuated for safety.

After waiting for 30 minutes (double neutron's lifetime), we were surprised to see detector PMl703GN go off scale again in neutron counts at a distance of 10' from the experimental set up, and the laboratory was closed for the day."

TGD based model

The basic problems to be solved are following.

1. What is the role of current arc and other triggering impulses (such as microwave radiation or pressure surge mentioned by Santilli): do they provide energy or do they have some other role?
2. Neutron production is kinematically impossible if weak interactions mediate it. Even if kinematically possible, weak interaction rates are quite too slow. The creation of intermediate states via other than weak interactions would solve both problems. If weak interactions are involved with the creation of the intermediate states, how there rates can be so high?
3. What causes the strange delays in the production in some cases but now always? Why hydrogen gas is preferred?

TGD explanation (see this) could be the same for Tesla's findings, for cold fusion (see this), Pollack effect (see this) and for the anomalous production of neutrons. Even electrolysis would involve in an essential manner Pollack effect and new physics.

Could this model explain the anomalous neuron production and its strange features?

1. Why electric arc, pressure surge, or microwave radiation would be needed? Dark phases are formed at quantum criticality (see this) and give rise to the long range correlations via quantum entanglement made possible by large h_eff=n× h. The presence of electron arc occurring as di-electric breakdown is indeed a critical phenomenon.

   Already Tesla discovered strange phenomena in his studies of arc discharges but his discoveries were forgotten by mainstream. TGD explanation (see this) could be the same for Tesla's findings, for cold fusion (see this), Pollack effect (see this) and for the anomalous production of neutrons. Even electrolysis would involve in an essential manner Pollack effect and new physics.

   Also energy feed might be involved. Quite generally, in TGD inspired quantum biology generation of dark states requires energy feed and the role of metabolic energy is to excite dark states. For instance, dark atoms have smaller binding energy and the energies of cyclotron states increase with h_eff/h. For instance, part of microwave photons could be dark and have much higher energy than otherwise.

   Could the production of dark proton sequences at magnetic flux tubes be all that is needed so that the possible dark variant of the reaction e^-+p→ n+ν would not be needed at all?

2. If also weak bosons appear as dark variants, their Compton length is scaled up accordingly and in scales shorter than the Compton length, they behave effectively as massless particles and weak interactions would become as strong as electromagnetic interactions. This would make possible the decay of dark proton sequences at magnetic flux tubes to beta stable dark isotopes via p→ n+e⁺+ν. Neutrons would be produced in the decays of the dark nuclei to ordinary nuclei liberating nuclear binding energy. Note however that TGD allows also to consider p-adically scaled variants of weak bosons with much smaller mass scale possible important in biology, and one cannot exclude them from consideration.
3. The reaction e^-+p→ n+ν is not necessary in the model. One can however ask, whether there could exist a mechanism making the dark reaction e^-+p→ n+ν kinematically possible. If the scale of dark nuclear binding energy is strongly reduced, also p→ n+e⁺+ν in dark nuclei would become kinematically impossible (in ordinary nuclei nuclear binding energy makes n effectively lighter than p).

   TGD based model for nuclei as strings of nucleons (see this and this) connected by neutral or charged (possibly colored) mesonlike bonds with quark and antiquark at its ends could resolve this problem. One could have exotic nuclei in which proton plus negatively charged bond could effectively behave like neutron. Dark weak interactions would take place for neutral bonds between protons and reduce the charge of the bond from q=0 to q= -1 and transform p to effective n. This was assumed also in the model of dark nuclei and also in the model of ordinary nuclei and predicts large number of exotic states. One can of course ask, whether the nuclear neutrons are actually pairs of proton and negatively charged bond.

4. What about the delays in neutron production occurring in some cases? Why not always? In the situations, when there is a delay in neutron production, the dark nuclei could have rotated around magnetic flux tubes of the magnetic body (MB) of the system before entering to the metal target, one would have a delayed production.
5. Why would hydrogen be preferred? Why for instance, deuteron and heavier isotopes containing neutrons would not form dark proton sequences at magnetic flux tubes. Why would be the probability for the transformation of say D=pn to its dark variant be very small?

   If the binding energy of dark nuclei per nucleon is several orders of magnitude smaller than for ordinary nuclei, the explanation is obvious. The ordinary nuclear binding energy is much higher than the dark binding energy so that only the sequences of dark protons can form dark nuclei. The first guess (see this) was that the binding energy is analogous to Coulomb energy and thus inversely proportional to the size scale of dark nucleus scaling like h/h_eff. One can however ask why D with ordinary size could not serve as sub-unit.

Non-local production of photon pairs as support for h_eff/h=n hypothesis

Again a new anomaly! Photon pairs have been created by a new mechanism. Photons emerge at different points! See this.

Could this give support for the TGD based general model for elementary particle as a string like object (flux tube) with first end (wormhole contact) carrying the quantum numbers - in the case of gauge boson fermion and antifermion at opposite throats of the contact. Second end would carry neutrino-right-handed neutrino pair neutralizing the possible weak isospin. This would give only local decays. Also emissions of photons from charged particle would be local.

Could the bosonic particle be a mixture of two states. For the first state flux tube would have fermion and antifermion at the same end of the fluxtube: only local decays. For the second state fermion and antifermion would reside at the ends of the flux tubes residing at throats associated with different wormhole contacts. This state in state would give rise to non-local two-photon emissions. Mesons of hadron physics would correspond to this kind of states and in old-fashioned hadron physics one speaks about photon-vector meson mixing in the description of the photon-hadron interactions.

If the Planck constant h_eff/h=n of the emitting particle is large, the distance between photon emissions would be long. The non-local days could make the visible both exotic decay and allow to deduce the value of n! This would how require the transformation of emitted dark photon to ordinary (same would happen when dark photons transform to biophotons).

Can one say anything about the length of fux tube? Magnetic flux tube contains fermionic string. The length of this string is of order Compton length and of the order of p-adic length scale.

What about photon itself - could it have non-local fermion-antifermion decays based on the same mechanism? What the length of photonic string is is not clear. Photon is massless, no scales! One identification of length would be as wavelength defining also the p-adic length scale.

To sum up: the nonlocal decays and emissions could lend strong support for both flux tube identification of particles and for hierarchy of Planck constants. It might be possible to even measure the value of n associated with quantum critical state by detecting decays of this kind.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

For details see the chapter Quantum criticality and dark matter.

Hierarchy of Planck constants, space-time surfaces as covering spaces, and adelic physics

From the beginning it was clear that h_eff/h=n corresponds to the number of sheets for a covering space of some kind. First the covering was assigned with the causal diamonds. Later I assigned it with space-time surfaces but the details of the covering remained unclear. The final identification emerged only in the beginning of 2017.

Number theoretical universality (NTU) leads to the notion of adelic space-time surface (monadic manifold) involving a discretization in an extension of rationals defining particular level in the hierarchy of adeles defining evolutionary hierarchy. The first formulation was proposed here and more elegant formulation here.

The key constraint is NTU for adelic space-time containing sheets in the real sector and various padic sectors, which are extensions of p-adic number fields induced by an extension of rationals which can contain also powers of a root of e inducing finite-D extension of p-adic numbers (e^p is ordinary p-adic number in Q_p).

One identifies the numbers in the extension of rationals as common for all number fields and demands that imbedding space has a discretization in an extension of rationals in the sense that the preferred coordinates of imbedding space implied by isometries belong to extension of rationals for the points of number theoretic discretization. This implies that the versions of isometries with group parameters in the extension of rationals act as discrete versions of symmetries. The correspondence between real and p-adic variants of the imbedding space is extremely discontinuous for given adelic imbedding space (there is hierarchy of them with levels characterized by extensions of rationals). Space-time surfaces typically contain rather small set of points in the extension (xⁿ+yn²=zⁿ contains no rationals for n>2!). Hence one expects a discretization with a finite cutoff length at space-time level for sufficiently low space-time dimension D=4 could be enough.

After that one assigns in the real sector an open set to each point of discretization and these open sets define a manifold covering. In p-adic sector one can assign 8:th Cartesian power of ordinary p-adic numbers to each point of number theoretic discretization. This gives both discretization and smooth local manifold structure. What is important is that Galois group of the extension acts on these discretizations and one obtains from a given discretization a covering space with the number of sheets equal to a factor of the order of Galois group, typically equal to the order of Galois.

h_eff/h=n was identified from the beginning as the dimension of poly-sheeted covering assignable to space-time surface. The number n of sheets would naturally a factor of the order of Galois group implying that h_eff/h=n is bound to increase during number theoretic evolution so that the algebraic complexity increases. Note that WCW decomposes into sectors corresponding to the extensions of rationals and the dimension of the extension is bound to increase in the long run by localizations to various sectors in self measurements (see this). Dark matter hierarchy represents number theoretical/adelic physics and therefore has now rather rigorous mathematical justification. It is however good to recall that h_eff/h=n hypothesis emerged from an experimental anomaly: radiation at ELF frequencies had quantal effects of vertebrate brain impossible in standard quantum theory since the energies E=hf of photons are ridiculously small as compared to thermal energy.

Indeed, since n is positive integer evolution is analogous to a diffusion in half-line and n unavoidably increases in the long run just as the particle diffuses farther away from origin (by looking what gradually happens near paper basket one understands what this means). The increase of n implies the increase of maximal negentropy and thus of negentropy. Negentropy Maximization Principle (NMP) follows from adelic physics alone and there is no need to postulate it separately. Things get better in the long run although we do not live in the best possible world as Leibniz who first proposed the notion of monad proposed!

For details see the chapter Quantum criticality and dark matter.

Time crystals, macroscopic quantum coherence, and adelic physics

Time crystals were (see this) were proposed by Frank Wilzek in 2012. The idea is that there is a periodic collective motion so that one can see the system as analog of 3-D crystal with time appearing as fourth lattice dimension. One can learn more about real life time crystals here.

The first crystal was created by Moore et al (see this) and involved magnetization. By adding a periodic driving force it was possible to generate spin flips inducing collective spin flip as a kind of domino effect. The surprise was that the period was twice the original period and small changes of the driving frequency did not affect the period. One had something more than forced oscillation - a genuine time crystal. The period of the driving force - Floquet period- was 74-75 μs and the system is measured for N=100 Floquet periods or about 7.4-7.5 milliseconds (1 ms happens to be of same order of magnitude as the duration of nerve pulse). I failed to find a comment about the size of the system. With quantum biological intuition I would guess something like the size of large neuron: about 100 micrometers.

Second law does not favor time crystals. The time in which single particle motions are thermalized is expected to be rather short. In the case of condensed matter systems the time scale would not be much larger than that for a typical rate of typical atomic transition. The rate for 2P → 1S transition of hydrogen atom estimated here gives a general idea. The decay rate is proportional to ω³d², where ω= Δ E/hbar is the frequency difference corresponding to the energy difference between the states, d is dipole moment proportional to α a₀, a₀ Bohr radius and α∼ 1/137 fine structure constant. Average lifetime as inverse of the decay rate would be 1.6 ns and is expected to give a general order of magnitude estimate.

The proposal is that the systems in question emerge in non-equilibrium thermodynamics, which indeed predicts a master-slave hierarchy of time and length scales with masters providing the slowly changing background in which slaves are forced to move. I am not a specialist enough to express any strong opinions about thermodynamical explanation.

What does TGD say about the situation?

1. So called Anderson localization (see this) is believed to accompany time crystal. In TGD framework this translates to the fusion of 3-surfaces corresponding to particles to single large 3-surface consisting of particle 3-surfaces glued together by magnetic flux tubes. On can say that a relative localization of particles occurs and they more or less lose the relative translation degrees of freedom. This effect occurs always when bound states are formed and would happen already for hydrogen atom.

   TGD vision would actually solve a fundamental problem of QED caused by the assumption that proton and electron behave as independent point like particles: QED predicts a lot of non-existing quantum states since Bethe-Salpeter equation assumes degrees of freedom, which do not actually exist. Single particle descriptions (Schrödinger equation and Dirac equation) treating proton and electron effectively as single particle geometrically (rather than independent particles) having reduced mass gives excellent description whereas QED, which was thought to be something more precise, fails. Quite generally, bound states are not properly understood in QFTs. Color confinement problem is second example about this: usually it is believed that the failure is solely due to the fact that color interaction is strong but the real reason might be much deeper.

2. In TGD Universe time crystals would be many-particle systems having collection of 3-surfaces connected by magnetic flux tubes (tensor network in terms of condensed matter complexity theory). Magnetic flux tubes would carry dark matter in TGD sense having h_eff/h=n increasing the quantal scales - both spatial and temporal - so that one could have time crystals in long scales.

   Biology could provide basic examples. For instance, EEG resonance frequency could be associated with time crystals assignable to the magnetic body of brain carrying dark matter with large h_eff/h=n - so large that dark photon energy E=h_efff would correspond to an energy above thermal energy. If bio-photons result from phase transitions h_eff/h=n→ 1, the energy would be in visible-UV energy range. These frequencies would in turn drive the visible matter in brain and force it to oscillate coherently.

3. The time crystals claimed by Monroe and Lurkin to be created in laboratory demand a feed of energy (see this) unlike the time crystals proposed by Wilzek. The finding is consistent with the TGD based model. In TGD the generation of large h_eff phase demands energy. The reason is that the energies of states increase with h_eff. For instance, atomic binding energies decrease as 1/h²_eff. In quantum biology this requires feeding of metabolic energy. Also now interpretation would be analogous to this.
4. Standard physics view would rely in non-equilibrium thermodynamics whereas TGD view about time crystals would rely on dark matter and hierarchy of Planck constants in turn implied by adelic physics suggested to provide a coherent description fusing real physics as physics of matter and various p-adic physics as physics of cognition.

   Number theoretical universality (NTU) leads to the notion of adelic space-time surface (monadic manifold) involving a discretization in an extension of rationals defining particular level in the hierarchy of adeles defining evolutionary hierarchy. h_eff/h=n has been identified from the beginning as the dimension of poly-sheeted covering assignable to space-time surface. The action of the Galois group of extensions indeed gives rise to covering space. The number n of sheets would the order of Galois group implying h_eff/h=n, which is bound to increase during evolution so that the complexity increases.

   Indeed, since n is positive integer evolution is analogous to a diffusion in half-line and n unavoidably increases in the long run just as the particle diffuses farther away from origin (by looking what gradually happens near paper basket one understands what this means). The increase of n implies the increase of maximal negentropy and thus of negentropy. Negentropy Maximization Principle (NMP) follows from adelic physics alone and there is no need to postulate it separately. Things get better in the long run although we do not live in the best possible world as Leibniz who first proposed the notion of monad proposed!

For details see the chapter Quantum criticality and dark matter.

Why metabolism and what happens in bio-catalysis?

TGD view about dark matter gives also a strong grasp to metabolism and bio-catalysis - the key elements of biology.

Why metabolic energy is needed?

The simplest and at the same time most difficult question that innocent student can make about biology class is simple: "Why we must eat?". Or using more physics oriented language: "Why we must get metabolic energy?". The answer of the teacher might be that we do not eat to get energy but to get order. The stuff that we eat contains ordered energy: we eat order. But order in standard physics is lack of entropy, lack of disorder. Student could get nosy and argue that excretion produces the same outcome as eating but is not enough to survive.

We could go to a deeper level and ask why metabolic energy is needed in biochemistry. Suppose we do this in TGD Universe with dark matter identified as phases characterized by h_eff/h=n.

1. Why metabolic energy would be needed? Intuitive answer is that evolution requires it and that evolution corresponds to the increase of n=h_eff/h. To see the answer to the question, notice that the energy scale for the bound states of an atom is proportional to 1/h² and for dark atom to 1/h_eff² ∝ n² (do not confuse this n with the integer n labelling the states of hydrogen atom!).
2. Dark atoms have smaller binding energies and their creation by a phase transition increasing the value of n demands a feed of energy - metabolic energy! If the metabolic energy feed stops, n is gradually reduced. System gets tired, loses consciousness, and eventually dies.

   What is remarkable that the scale of atomic binding energies decreases with n only in dimension D=3. In other dimensions it increases and in D=4 one cannot even speak of bound states! This can be easily found by a study of Schrödinger equation for the analog of hydrogen atom in various dimensions. Life based on metabolism seems to make sense only in spatial dimension D=3. Note however that there are also other quantum states than atomic states with different dependence of energy on h_eff.

Bio-catalysis is key mechanism of biology and its extreme efficacy remains to be understood. Enzymes are proteins and ribozymes RNA sequences acting as biocatalysts.

What does catalysis demand?

1. Catalyst and reactants must find each other. How this could happen is very difficult to understand in standard biochemistry in which living matter is seen as soup of biomolecules. I have already already considered the mechanisms making it possible for the reactants to find each other. For instance, in the translation of mRNA to protein tRNA molecules must find their way to mRNA at ribosome. The proposal is that reconnection allowing U-shaped magnetic flux tubes to reconnect to a pair of flux tube connecting mRNA and tRNA molecule and reduction of the value of h_eff=n× h inducing reduction of the length of magnetic flux tube takes care of this step. This applies also to DNA transcription and DNA replication and bio-chemical reactions in general.
2. Catalyst must provide energy for the reactants (their number is typically two) to overcome the potential wall making the reaction rate very slow for energies around thermal energy. The TGD based model for the hydrino atom having larger binding energy than hydrogen atom claimed by Randell Mills suggests a solution. Some hydrogen atom in catalyst goes from (dark) hydrogen atom state to hydrino state (state with smaller h_eff/h and liberates the excess binding energy kicking the either reactant over the potential wall so that reaction can process. After the reaction the catalyst returns to the normal state and absorbs the binding energy.
3. In the reaction volume catalyst and reactants must be guided to correct places. The simplest model of catalysis relies on lock-and-key mechanism. The generalized Chladni mechanism forcing the reactants to a two-dimensional closed nodal surface is a natural candidate to consider. There are also additional conditions. For instance, the reactants must have correct orientation. For instance, the reactants must have correct orientation and this could be forced by the interaction with the em field of ME involved with Chladni mechanism.
4. One must have also a coherence of chemical reactions meaning that the reaction can occur in a large volume - say in different cell interiors - simultaneously. Here MB would induce the coherence by using MEs. Chladni mechanism might explain this if there is there is interference of forces caused by periodic standing waves themselves represented as pairs of MEs.

Hydrogen atom allows also large h_eff/h=n variants with n>6 with the scale of energy spectrum behaving as (6/n)² if the n=4 holds true for visible matter. The reduction of n as the flux tube contracts would reduce n and liberate binding energy, which could be used to promote the catalysis.

The notion of high energy phosphate bond is somewhat mysterious concept. There are claims that there is no such bond. I have spent considerable amount of time to ponder this problem. Could phosphate contain (dark) hydrogen atom able to go to the a state with a smaller value of h_eff/h and liberate the excess binding energy? Could the phosphorylation of acceptor molecule transfer this dark atom associated with the phosphate of ATP to the acceptor molecule? Could the mysterious high energy phosphate bond correspond to the dark atom state. Metabolic energy would be needed to transform ADP to ATP and would generate dark atom.

Could solar light kick atoms into dark states and in this manner store metabolic energy? Could nutrients carry these dark atoms? Could this energy be liberated as the dark atoms return to ordinary states and be used to drive protons against potential gradient through ATP synthase analogous to a turbine of a power plant transforming ADP to ATP and reproducing the dark atom and thus the "high energy phosphate bond" in ATP? Can one see metabolism as transfer of dark atoms? Could possible negentropic entanglement disappear and emerge again after ADP→ATP.

Here it is essential that the energies of the hydrogen atom depend on hbar_eff=n× h in as hbar_eff^m, m=-2<0. Hydrogen atoms in dimension D have Coulomb potential behaving as 1/r^D-2 from Gauss law and the Schrödinger equation predicts for D≠ 4 that the energies satisfy E_n∝ (h_eff/h)^m, m=2+4/(D-4). For D=4 the formula breaks since in this case the dependence on hbar is not given by power law. m is negative only for D=3 and one has m=-2. There D=3 would be unique dimension in allowing the hydrino-like states making possible bio-catalysis and life in the proposed scenario.

It is also essential that the flux tubes are radial flux tubes in the Coulomb field of charged particle. This makes sense in many-sheeted space-time: electrons would be associated with a pair formed by flux tube and 3-D atom so that only part of electric flux would interact with the electron touching both space-time sheets. This would give the analog of Schrödinger equation in Coulomb potential restricted to the interior of the flux tube. The dimensional analysis for the 1-D Schrödinger equation with Coulomb potential would give also in this case 1/n² dependence. Same applies to states localized to 2-D sheets with charged ion in the center. This kind of states bring in mind Rydberg states of ordinary atom with large value of n.

The condition that the dark binding energy is above the thermal energy gives a condition on the value of h_eff/h=n as n≤ 32. The size scale of the dark largest allowed dark atom would be about 100 nm, 10 times the thickness of the cell membrane.

For details see the chapter Quantum criticality and dark matter.

NMP and self

The preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note ideas related directly to consciousness and cognition are discussed.

1. Adelic approach strongly suggests the reduction of NMP to number theoretic physics somewhat like second law reduces to probability theory. The dimension of extension rationals characterizing the hierarchy level of physics and defined an observable measured in state function reductions is positive and can only increase in statistical sense. Therefore the maximal value of entanglement negentropy increases as new entangling number theoretic degrees of freedom emerge. h_eff/h=n identifiable as factor of Galois group of extension characterizes the number of these degrees of freedom for given space-time surfaces as number of its sheets.
2. State function reduction is hitherto assumed to correspond always to a measurement of density matrix which can be seen as a reaction of subsystem to its environment. This makes perfect sense at space-time level. Higher level measurements occur however at the level of WCW and correspond to a localization to some sector of WCW determining for instance the quantization axes of various quantum numbers. Even the measurement of h_eff/h=n would measure the dimension of Galois group and force a localization to an extension with Galois group with this dimension. These measurements cannot correspond to measurements of density matrix since different WCW sectors cannot entangle by WCW locality. This finding will be discuss in the following.

The view about Negentropy Maximization Principle (NMP) has co-evolved with the notion of self and I have considered many variants of NMP.

1. The original formulation of NMP was in positive energy ontology and made same predictions as standard quantum measurement theory. The new element was that the density matrix of sub-system defines the fundamental observable and the system goes to its eigenstate in state function reduction. As found, the localizations at to WCW sectors define what might be called self-measurements and identifiable as active volitions rather than reactions.
2. In p-adic physics one can assign with rational and even algebraic entanglement probabilities number theoretical entanglement negentropy (NEN) satisfying the same basic axioms as the ordinary Shannon entropy but having negative values and therefore having interpretation as information. The definition of p-adic negentropy (real valued) reads as S_p= -∑ P_klog(|P_k|_p), where | . |_p denotes p-adic norm. The news is that N_p= -S_p can be positive and is positive for rational entanglement probabilities. Real entanglement entropy S is always non-negative.

   NMP would force the generation of negentropic entanglement (NE) and stabilize it. NE resources of the Universe - one might call them Akashic records- would steadily increase.

3. A decisive step of progress was the realization is that NTU forces all states in adelic physics to have entanglement coefficients in some extension of rationals inducing finite-D extension of p-adic numbers. The same entanglement can be characterized by real entropy S and p-adic negentropies N_p, which can be positive. One can define also total p-adic negentropy: N= ∑_p N_p for all p and total negentropy N_tot=N-S.

   For rational entanglement probabilities it is easy to demonstrate that the generalization of adelic theorem holds true: N_tot=N-S=0. NMP based on N_tot rather than N would not say anything about rational entanglement. For extensions of rationals it is easy to find that N-S>0 is possible if entanglement probabilities are of form X_i/n with |X_i|_p=1 and n integer. Should one identify the total negentropy as difference N_tot=N-S or as N_tot=N?

   Irrespective of answer, large p-adic negentropy seems to force large real entropy: this nicely correlates with the paradoxical finding that living systems tend to be entropic although one would expect just the opposite: this relates in very interesting manner to the work of biologists Jeremy England. The negentropy would be cognitive negentropy and not visible for ordinary physics.

4. The latest step in the evolution of ideas NMP was the question whether NMP follows from number theory alone just as second law follows form probability theory! This irritates theoretician's ego but is victory for theory. The dimension n of extension is positive integer and cannot but grow in statistical sense in evolution! Since one expects that the maximal value of negentropy (define as N-S) must increase with n. Negentropy must increase in long run.

Number theoretical Shannon entropy can serve as a measure for genuine information assignable to a pair of entanglement systems. Entanglement with coefficients in the extension is always negentropic if entanglement negentropy comes from p-adic sectors only. It can be negentropic if negentropy is defined as the difference of p-adic negentropy and real entropy.

The diagonalized density matrix need not belong to the algebraic extension since the probabilities defining its diagonal elements are eigenvalues of the density matrix as roots of N:th order polynomial, which in the generic case requires n-dimensional algebraic extension of rationals. One can argue that since diagonalization is not possible, also state function reduction selecting one of the eigenstates is impossible unless a phase transition increasing the dimension of algebraic extension used occurs simultaneously. This kind of NE could give rise to cognitive entanglement.

There is also a special kind of NE, which can result if one requires that density matrix serves a universal observable in state function reduction. The outcome of reduction must be an eigen space of density matrix, which is projector to this subspace acting as identity matrix inside it. This kind NE allows all unitarily related basis as eigenstate basis (unitary transformations must belong to the algebraic extension). This kind of NE could serve as a correlate for "enlightened" states of consciousness. Schrödingers cat is in this kind of state stably in superposition of dead and alive and state basis obtained by unitary rotation from this basis is equally good. One can say that there are no discriminations in this state, and this is what is claimed about "enlightened" states too.

The vision about number theoretical evolution suggests that NMP forces the generation of NE resources as NE assignable to the "passive boundary of CD for which no changes occur during sequence of state function reductions defining self. It would define the unchanging self as negentropy resources, which could be regarded as kind of Akashic records. During the next "re-incarnation after the first reduction to opposite boundary of CD the NE associated with the reduced state would serve as new Akashic records for the time reversed self. If NMP reduces to the statistical increase of h_eff/h=n the consciousness information contents of the Universe increases in statistical sense. In the best possible world of SNMP it would increase steadily.

Does NMP reduce to number theory?

The heretic question that emerged quite recently is whether NMP is actually needed at all! Is NMP a separate principle or could NMP reduced to mere number theory? Consider first the possibility that NMP is not needed at all as a separate principle.

1. The value of h_eff/h=n should increase in the evolution by the phase transitions increasing the dimension of the extension of rationals. h_eff/h=n has been identified as the number of sheets of some kind of covering space. The Galois group of extension acts on number theoretic discretizations of the monadic surface and the orbit defines a covering space. Suppose n is the number of sheets of this covering and thus the dimension of the Galois group for the extension of rationals or factor of it.
2. It has been already noticed that the "big" state function reductions giving rise to death and reincarnation of self could correspond to a measurement of n=h_eff implied by the measurement of the extension of the rationals defining the adeles. The statistical increase of n follows automatically and implies statistical increase of maximal entanglement negentropy. Entanglement negentropy increases in statistical sense.

   The resulting world would not be the best possible one unlike for a strong form of NMP demanding that negentropy does increaes in "big" state function reductions. n also decrease temporarily and they seem to be needed. In TGD inspired model of bio-catalysis the phase transition reducing the value of n for the magnetic flux tubes connecting reacting bio-molecules allows them to find each other in the molecular soup. This would be crucial for understanding processes like DNA replication and transcription.

3. State function reduction corresponding to the measurement of density matrix could occur to an eigenstate/eigenspace of density matrix only if the corresponding eigenvalue and eigenstate/eigenspace is expressible using numbers in the extension of rationals defining the adele considered. In the generic case these numbers belong to N-dimensional extension of the original extension. This can make the entanglement stable with respect to state the measurements of density matrix.

   A phase transition to an extension of an extension containing these coefficients would be required to make possible reduction. A step in number theoretic evolution would occur. Also an entanglement of measured state pairs with those of measuring system in containing the extension of extension would make possible the reduction. Negentropy could be reduced but higher-D extension would provide potential for more negentropic entanglement and NMP would hold true in the statistical sense.

4. If one has higher-D eigen space of density matrix, p-adic negentropy is largest for the entire subspace and the sum of real and p-adic negentropies vanishes for all of them. For negentropy identified as total p-adic negentropy SNMP would select the entire sub-space and NMP would indeed say something explicit about negentropy.

Hitherto I have postulated NMP as a separate principle. Strong form of NMP (SNMP) states that Negentropy does not decrease in "big" state function reductions corresponding to death and re-incarnations of self.

One can however argue that SNMP is not realistic. SNMP would force the Universe to be the best possible one, and this does not seem to be the case. Also ethically responsible free will would be very restricted since self would be forced always to do the best deed that is increase maximally the negentropy serving as information resources of the Universe. Giving up separate NMP altogether would allow to have also "Good" and "Evil".

This forces to consider what I christened weak form of NMP (WNMP). Instead of maximal dimension corresponding to N-dimensional projector self can choose also lower-dimensional sub-spaces and 1-D sub-space corresponds to the vanishing entanglement and negentropy assumed in standard quantum measurement theory. As a matter fact, this can also lead to larger negentropy gain since negentropy depends strongly on what is the large power of p in the dimension of the resulting eigen sub-space of density matrix. This could apply also to the purely number theoretical reduction of NMP.

WNMP suggests how to understand the notions of Good and Evil. Various choices in the state function reduction would correspond to Boolean algebra, which suggests an interpretation in terms of what might be called emotional intelligence . Also it turns out that one can understand how p-adic length scale hypothesis - actually its generalization - emerges from WNMP.

1. One can start from ordinary quantum entanglement. It corresponds to a superposition of pairs of states. Second state corresponds to the internal state of the self and second state to a state of external world or biological body of self. In negentropic quantum entanglement each is replaced with a pair of sub-spaces of state spaces of self and external world. The dimension of the sub-space depends on which pair is in question. In state function reduction one of these pairs is selected and deed is done. How to make some of these deeds good and some bad? Recall that WNMP allows only the possibility to generate NNE but does not force it. WNMP would be like God allowing the possibility to do good but not forcing good deeds.

   Self can choose any sub-space of the subspace defined by k≤ N-dimensional projector and 1-D subspace corresponds to the standard quantum measurement. For k=1 the state function reduction leads to vanishing negentropy, and separation of self and the target of the action. Negentropy does not increase in this action and self is isolated from the target: kind of price for sin.

   For the maximal dimension of this sub-space the negentropy gain is maximal. This deed would be good and by the proposed criterion NE corresponds to conscious experience with positive emotional coloring. Interestingly, there are 2^k-1 possible choices, which is almost the dimension of Boolean algebra consisting of k independent bits. The excluded option corresponds to 0-dimensional sub-space - empty set in set theoretic realization of Boolean algebra. This could relate directly to fermionic oscillator operators defining basis of Boolean algebra - here Fock vacuum would be the excluded state. The deed in this sense would be a choice of how loving the attention towards system of external world is.

2. A map of different choices of k-dimensional sub-spaces to k-fermion states is suggestive. The realization of logic in terms of emotions of different degrees of positivity would be mapped to many-fermion states - perhaps zero energy states with vanishing total fermion number. State function reductions to k-dimensional spaces would be mapped to k-fermion states: quantum jumps to quantum states!

   The problem brings in mind quantum classical correspondence in quantum measurement theory. The direction of the pointer of the measurement apparatus (in very metaphorical sense) corresponds to the outcome of state function reduction, which is now 1-D subspace. For ordinary measurement the pointer has k positions. Now it must have 2^k-1 positions. To the discrete space of k pointer positions one must assign fermionic Clifford algebra of second quantized fermionic oscillator operators. The hierarchy of Planck constants and dark matter suggests the realization. Replace the pointer with its space-time k-sheeted covering and consider zero energy energy states made of pairs of k-fermion states at the sheets of the n-sheeted covering? Dark matter would be therefore necessary for cognition. The role of fermions would be to "mark" the k space-time sheets in the covering.

For details see the chapter Negentropy Maximization Principle or the article Re-examination of the basic notions of TGD inspired theory of consciousness.

WCW and the notion of intentional free will

The preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note ideas related directly to consciousness and cognition are discussed.

1. Adelic approach strongly suggests the reduction of NMP to number theoretic physics somewhat like second law reduces to probability theory. The dimension of extension rationals characterizing the hierarchy level of physics and defined an observable measured in state function reductions is positive and can only increase in statistical sense. Therefore the maximal value of entanglement negentropy increases as new entangling number theoretic degrees of freedom emerge. h_eff/h=n identifiable as factor of Galois group of extension characterizes the number of these degrees of freedom for given space-time surfaces as number of its sheets.
2. State function reduction is hitherto assumed to correspond always to a measurement of density matrix which can be seen as a reaction of subsystem to its environment. This makes perfect sense at space-time level. Higher level measurements occur however at the level of WCW and correspond to a localization to some sector of WCW determining for instance the quantization axes of various quantum numbers. Even the measurement of h_eff/h=n would measure the dimension of Galois group and force a localization to an extension with Galois group with this dimension. These measurements cannot correspond to measurements of density matrix since different WCW sectors cannot entangle by WCW locality. This finding will be discuss in the following.

The original definition of self was as a subsystem able to remain unentangled under state function reductions associated with subsequent quantum jumps. The density matrix was assumed to define the universal observable. Note that a density matrix, which is power series of a product of matrices representing commuting observables has in the generic case eigenstates, which are simultaneous eigenstates of all observables. Second aspect of self was assumed to be the integration of subsequent quantum jumps to coherent whole giving rise to the experienced flow of time.

The precise identification of self allowing to understand both of these aspects turned out to be difficult problem. I became aware the solution of the problem in terms of ZEO (ZEO) only rather recently (2014).

1. Self corresponds to a sequence of quantum jumps integrating to single unit as in the original proposal, but these quantum jumps correspond to state function reductions to a fixed boundary of causal diamond CD leaving the corresponding parts of zero energy states invariant - "small" state function reductions. The parts of zero energy states at second boundary of CD change and even the position of the tip of the opposite boundary changes: one actually has wave function over positions of second boundary (CD sizes roughly) and this wave function changes. In positive energy ontology these repeated state function reductions would have no effect on the state (Zeno effect) but in TGD framework there occurs a change for the second boundary and gives rise to the experienced flow of time and its arrow and self: self is generalized Zeno effect.
2. The first quantum jump to the opposite boundary corresponds to the act of "free will" or birth of re-incarnated self. Hence the act of "free will" changes the arrow of psychological time at some level of hierarchy of CDs. The first reduction to the opposite boundary of CD means "death" of self and "re-incarnation" of time-reversed self at opposite boundary at which the the temporal distance between the tips of CD increases in opposite direction. The sequence of selves and time reversed selves is analogous to a cosmic expansion for CD. The repeated birth and death of mental images could correspond to this sequence at the level of sub-selves.
3. This allows to understand the relationship between subjective and geometric time and how the arrow of and flow of clock time (psychological time) emerge. The average distance between the tips of CD increases on the average as along as state function functions occur repeatedly at the fixed boundary: situation is analogous to that in diffusion. The localization of contents of conscious experience to boundary of CD gives rise to the illusion that universe is 3-dimensional. The possibility of memories made possibly by hierarchy of CDs demonstrates that this is not the case. Self is simply the sequence of state function reductions at same boundary of CD remaining fixed and the lifetime of self is the total growth of the average temporal distance between the tips of CD.

1. There are quantum measurements inducing localization in the moduli space of CDs with passive boundary and states at it fixed. In particular, a localization in the moduli characterizing the Lorentz transform of the upper tip of CD would be measured. The measured moduli characterize also the analog of symplectic form in M⁴ strongly suggested by twistor lift of TGD - that is the rest system (time axis) and spin quantization axes. Of course, also other kinds of reductions are possible.
2. Also a localization to an extension of rationals defining the adeles should occur. Could the value of n=h_eff/h be observable? The value of n for given space-time surface at the active boundary of CD could be identified as the order of the smallest Galois group containing all Galois groups assignable to 3-surfaces at the boundary. The superposition of space-time surface would not be eigenstate of n at active boundary unless localization occurs. It is not obvious whether this is consistent with a fixe value of n at passive boundary.

   The measured value of n could be larger or smaller than the value of n at the passive boundary of CD but in statistical sense n would increase by the analogy with diffusion on half line defined by non-negative integers. The distance from the origin unavoidably increases in statistical sense. This would imply evolution as increase of maximal value of negentropy and generation of quantum coherence in increasingly longer scales.

3. A further abstract choice corresponds to the the replacement of the roles of active and passive boundary of CD changing the arrow of clock time and correspond to a death of self and re-incarnation as time-reversed self.

1. Do all measurements involve entanglement between the moduli or extensions of two CDs reduced in the measurement of the density matrix? Non-diagonal entanglement would allow final states states, which are not eigenstates of moduli or of n: this looks strange. This could also lead to an infinite regress since it seems that one must assume endless hierarchy of entangled CDs so that the reduction sequence would proceed from top to bottom. It looks natural to regard single CD as a sub-Universe.

   For instance, if a selection of quantization axis of color hypercharge and isospin (localization in the twistor space of CP₂) is involved, one would have an outcome corresponding to a quantum superposition of measurements with different color quantization axis!

   Going philosophical, one can also argue, that the measurement of density matrix is only a reaction to environment and does not allow intentional free will.

2. Can one assume that a mere localization in the moduli space or for the extension of rationals (producing an eigenstate of n) takes place for a fixed CD - a kind of self measurement possible for even unentangled system? If there is entanglement in these degrees of freedom between two systems (say CDs), it would be reduced in these self measurements but the outcome would not be an eigenstate of density matrix. An interpretation as a realization of intention would be approriate.
3. If one allows both options, the interpretation would be that state function reduction as a measurement of density matrix is only a reaction to environment and self-measurement represents a realization of intention.
4. Self measurements would occur at higher level say as a selection of quantization axis, localization in the moduli space of CD, or selection of extension of rationals. A possible general rule is that measurements at space-time level are reactions as measurements of density matrix whereas a selection of a sector of WCW would be an intentional action. This because formally the quantum states at the level of WCW are as modes of classical WCW spinor field single particle states. Entanglement between different sectors of WCW is not possible.
5. If the selections of sectors of WCW at active boundary of CD commute with observables, whose eigenstates appear at passive boundary (briefly passive observables) meaning that time reversal commutes with them - they can occur repeatedly during the reduction sequence and self as a generalized Zeno effect makes sense.

   If the selections of WCW sectors at active boundary do not commute with passive observables then volition as a choice of sector of WCW must change the arrow of time. Libet's findings show that conscious choice induces neural activity for a fraction of second before the conscious choice. This would imply the correspondences "big" measurement changing the arrow of time - self-measurement at the level of WCW - intentional action and "small" measurement - measurement at space-time level - reaction.

   Self as a generalized Zeno effect makes sense only if there are active commuting with passive observables. If the passive observables form a maximal set, the new active observables commuting with them must emerge. The increase of the size of extension of rationals might generate them by expanding the state space so that self would survive only as long at it evolves. Self would die and re-incarnate when it could not generate any new observables communicating with those assignable to active boundary to be measured. From personal experience I can say that ageing is basically the loss of the ability to make new choices. When all possible choices are made, all observables are measured or self-measured, it is time to start again.

   Otherwise there would be only single unitary time evolution followed by a reduction to opposite boundary. This makes sense only if the sequence of "big" reductions for sub-selves can give rise to the time flow experienced by self: the birth and death of mental images would give rise to flow of time of self.

For details see the chapter Negentropy Maximization Principle or the article Re-examination of the basic notions of TGD inspired theory of consciousness.

Anomalies of water as evidence for dark matter in TGD sense

The motivation for this brief comment came from a popular article telling that a new phase of water has been discovered in the temperature range 50-60 ^oC (see this ). Also Gerald Pollack (see this ) has introduced what he calls the fourth phase of water. For instance, in this phase water consists of hexagonal layers with effective H_1.5O stoichiometry and the phase has high negative charge. This phase plays a key role in TGD based quantum biology. These two fourth phases of water could relate to each other if there exist a deeper mechanism explaining both these phases and various anomalies of water.

Martin Chaplin (see this ) has an extensive web page about various properties of water. The physics of water is full of anomalous features and therefore the page is a treasure trove for anyone ready to give up the reductionistic dogma. The site discusses the structure, thermodynamics, and chemistry of water. Even academically dangerous topics such as water memory and homeopathy are discussed.

One learns from this site that the physics of water involves numerous anomalies (see this ). The structural, dynamic and thermodynamic anomalies form a nested in density-temperature plane. For liquid water at atmospheric pressure of 1 bar the anomalies appear in the temperature interval 0-100 ^oC.

Hydrogen bonding creating a cohesion between water molecules distinguishes water from other substances. Hydrogen bonds induce the clustering of water molecules in liquid water. Hydrogen bonding is also highly relevant for the phase diagram of H₂O coding for various thermodynamical properties of water (see this ). In biochemistry hydrogen bonding is involved with hydration. Bio-molecules - say amino-acids - are classified to hydrophobic, hydrophilic, and amphiphilic ones and this characterization determines to a high extent the behavior of the molecule in liquid water environment. Protein folding represents one example of this.

Anomalies are often thought to reduce to hydrogen bonding. Whether this is the case, is not obvious to me and this is why I find water so fascinating substance.

TGD indeed suggests that water decomposes into ordinary water and dark water consisting of phases with effective Planck constant h_eff=n× h residing at magnetic flux tubes. Hydrogen bonds would be associated with short and rigid flux tubes but for larger values of n the flux tubes would be longer by factor n and have string tension behaving as 1/n so that they would softer and could be loopy. The portional of water molecules connected by flux tubes carrying dark matter could be identified as dark water and the rest would be ordinary water. This model allows to understand various anomalies. The anomalies are largest at the physiological temperature 37 C, which conforms with the vision about the role of dark matter and dark water in living matter since the fraction of dark water would be highest at this temperature. The anomalies discussed are density anomalies, anomalies of specific heat and compressibility, and Mpemba effect. I have discussed these anomalies already for decade ago. The recent view about dark matter allows however much more detailed modelling.

For details see the chapter Dark Nuclear Physics and Condensed Matter or the article The anomalies of water as evidence for the existence of dark matter in TGD sense.

About number theoretic aspects of NMP

There is something in NMP that I still do not understand: every time I begin to explain what NMP is I have this unpleasant gut feeling. I have the habit of making a fresh start everytime rather than pretending that everything is crystal clear. I have indeed considered very many variants of NMP. In the following I will consider two variants of NMP. Second variant reduces to a pure number theory in adelic framework inspired by number theoretic vision. It is certainly the simplest one since it says nothing explicit about negentropy. Second variant says essentially the as "strong form of NMP", when the reduction occurs to an eigen-space of density matrix.

I will not consider zero energy ontology (ZEO) related aspects and the aspects related to the hierarchy of subsystems and selves since I dare regard these as "engineering" aspects.

What NMP should say?

What NMP should state?

1. NMP takes in some sense the role of God and the basic question is whether we live in the best possible world or not. Theologists asks why God allows sin. I ask whether NMP demand increase of negentropy always or does it allow also reduction of negentropy? Why? Could NMP lead to increase of negentropy only in statistical sense - evolution? Could it only give potential for gaining a larger negentropy?

   These questions have turned to be highly non-trivial. My personal experience is that we do not live in the best possible world and this experience plus simplicity motivates the proposal to be discussed.

2. Is NMP a separate principle or could NMP be reduced to mere number theory? For the latter option state function would occur to an eigenstate/eigenspace of density matrix only if the corresponding eigenvalue and eigenstate/eigenspace are expressible using numberes in the extension of rationals defining the adele considered. A phase transition to an extension of an extension containing these coefficients would be required to make possible reduction. A step in number theoretic evolution would occur. Also an entanglement of measured state pairs with those of measuring system in containing the extension of extension would make possible the reduction. Negentropy would be reduced but higher-D extension would provide potential for more negentropic entanglement. I will consider this option in the following.
3. If one has higher-D eigenspace of density matrix, p-adic negentropy is largest for the entire subspace and the sum of real and p-adic negentropies vanishes for all of them. For negentropy identified as total p-adic negentropy strong from of NMP would select the entire sub-space and NMP would indeed say something explicit about negentropy.

The notion of entanglement negentropy

1. Number theoretic universality demands that density matrix and entanglement coefficients are numbers in an algebraic extension of rationals extended by adding root of e. The induced p-adic extensions are finite-D and one obtains adele assigned to the extension of rationals. Real physics is replaced by adelic physics.
2. The same entanglement in coefficients in extension of rationals can be seen as numbers is both real and various p-adic sectors. In real sector one can define real entropy and in various p-adic sectors p-adic negentropies (real valued).
3. Question: should one define total entanglement negentropy as
   1. sum of p-adic negentropies or
   2. as difference for the sum of p-adic negentropies and real etropy. For rational entanglement probabilities real entropy equals to the sum of p-adic negentropies and total negentropy would vanish. For extensions this negentropy would be positive under natural additional conditions as shown earlier.
   Both options can be considered.

State function reduction as universal measurement interaction between any two systems

1. The basic vision is that state function reductions occur all the for all kinds of matter and involves a measurement of density matrix ρ characterizing entanglement of the system with environment leading to a sub-space for which states have same eigenvalue of density matrix. What this measurement really is is not at all clear.
2. The measurement of the density matrix means diagonalization of the density matrix and selection of an eigenstate or eigenspace. Diagonalization is possible without going outside the extension only if the entanglement probabilities and the coefficients of states belong to the original extension defining the adele. This need not be the case!

   More precisely, the eigenvalues of the density matrix as roots of N:th order polynomial with coefficients in extension in general belong to N-D extension of extension. Same about the coefficients of eigenstates in the original basis. Consider as example the eigen values and eigenstates of rational valued N× N entanglement matrix, which are roots of a polynomial of degree N and in general algebraic number.

   Question: Is state function reduction number theoretically forbidden in the generic case? Could entanglement be stable purely number theoretically? Could NMP reduce to just this number theoretic principle saying nothing explicit about negentropy? Could phase transition increasing the dimension of extension but keeping the entanglement coefficients unaffected make reduction possible. Could entanglement with an external system in higher-D extension -intelligent observer - make reduction possible?

3. There is a further delicacy involved. The eigen-space of density matrix can be N-dimensional if the density matrix has N-fold degenerate eigenvalue with all N entanglement probabilities identical. For unitary entanglement matrix the density matrix is indeed N×N unit matrix. This kind of NE is stable also algebraically if the coefficients of eigenstates do not belong to the extension. If they do not belong to it then the question is whether NMP allows a reduction to subspace of and eigen space or whether only entire subspace is allowed.

   For total negentropy identified as the sum of real and p-adic negentropies for any eigenspace would vanish and would not distinguish between sub-spaces. Identification of negentropy as as p-adic negentropy would distinguish between sub-spaces and´NMP in strong form would not allow reduction to sub-spaces. Number theoretic NMP would thus also say something about negentropy.

   I have also consider the possibility of weak NMP. Any subspace could be selected and negentropy would be reduced. The worst thing to do in this case would be a selection of 1-D subspace: entanglement would be totally lost and system would be totally isolated from the rest of the world. I have proposed that this possibility corresponds to the fact that we do not seem to live in the best possible world.

NMP as a purely number theoretic constraint?

Let us consider the possibility that NMP reduces to the number theoretic condition tending to stabilize generic entanglement.

1. Density matrix characterizing entanglement with the environment is a universal observable. Reduction can occur to an eigenspace of the density matrix. For rational entanglement probabilities the total negentropy would vanish so that NMP formulated in terms of negentropy cannot say anything about the situation. This suggests that NMP quite generally does not directly refer to negentropy.
2. The condition that eigenstates and eigenvalues are in the extension of rationals defining the adelic physics poses a restriction. The reduction could occur only if these numbers are in the original extension. Also rational entanglement would be stable in the generic case and a phase transition to higher algebraic extension is required for state function reduction to occur. Standard quantum measurement theory would be obtained when the coefficients of eigenstates and entanglement probabilities are in the original extension.
3. If this is not the case, a phase transition to an extension of extension containing the N-D extension of it could save the situation. This would be a step in number theoretic evolution. Reduction would lead to a reduction of negentropy but would give potential for gaining a larger entanglement negentropy. Evolution would proceed through catastrophes giving potential for more negentropic entanglement! This seems to be the case!

   Alternatively, the state pairs of the system + complement could be entangled with observer in an extension of rationals containg the needed N-D extension of extension and state function possible for observer would induce reduction in the original system. This would mean fusion with a self at higher level of evolutionary hierarchy - kind of enlightment. This would give an active role to the intelligent observer (intelligence characterized by the dimension of extension of rationals). Intelligent observer would reduce the negentropy and thus NMP would not hold true universally.

   Since higher-D extension allows higher negentropy and in the generic case NE is stable, one might hope that NMP holds true statistically (for rationals total negentropy as sum or real and total p-adic negentropies vanishes).

   The Universe would evolve rather than being a paradize: the number theoretic NMP would allow temporary reduction of negentropy but provide a potential for larger negentropy and the increase of negentropy in statistical sense is highly suggestive. To me this option looks like simplest and most realistic one.

4. If negentropy is identified as total p-adic negentropy rather than sum of real and p-adic negentropies, strong form of NMP says something explicit about negentropy: the reduction would take place to the entire subspace having the largest p-adic negentropy.

For background see the chapter Negentropy Maximization Principle. or the article About number theoretic aspects of NMP.

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