Dark Nuclear Physics and Living Matter

M. Pitkanen1, June 24, 2003

1 Department of Physical Sciences, High Energy Physics Division,PL 64, FIN-00014, University of Helsinki, Finland.

matpitka@rock.helsinki.fi, http://www.physics.helsinki.fi/∼matpitka/.

Recent address: Kadermonkatu 16,10900, Hanko, Finland.

Contents

1 Introduction 71.1 How to characterize dark matter? . . . . . . . . . . . . . . . . 81.2 Evidence for long range weak forces and new nuclear physics 91.3 k = 113 dark nuclear physics . . . . . . . . . . . . . . . . . . 11

1.3.1 Are valence quarks always dark? . . . . . . . . . . . . 111.3.2 Could only the hadronic space-time sheet be scaled up

for light nuclei? . . . . . . . . . . . . . . . . . . . . . . 131.4 Water and k = 113 exotic nuclear physics . . . . . . . . . . . 141.5 Implications of the partial darkness of condensed matter . . . 15

2 General ideas about dark matter and condensed matter 152.1 Living matter, dark matter, and conformal confinement . . . 152.2 How dark matter and visible matter interact? . . . . . . . . . 16

2.2.1 Basic implications from the scaling of h . . . . . . . . 162.2.2 Simple model for dark atoms . . . . . . . . . . . . . . 172.2.3 How dark photons transform to ordinary photons? . . 182.2.4 Hierarchy of dark matters and hierarchy of minds . . . 182.2.5 Wave-length hierarchy, coherent metabolism, and proton-

electron mass ratio . . . . . . . . . . . . . . . . . . . . 192.2.6 Dark condensed matter and grey matter . . . . . . . . 212.2.7 A connection with bio-photons . . . . . . . . . . . . . 23

3 A more detailed theoretical view about dark matter 243.1 Characterization of dark matter as large h phase . . . . . . . 253.2 Criterion for the presence of exotic electro-weak bosons and

gluons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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3.3 Do Gaussian Mersennes define a hierarchy of dark electro-weak physics? . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Interpretations of the dark matter hierarchies . . . . . . . . . 313.5 Conformal weights of dark matter . . . . . . . . . . . . . . . 323.6 How dark and visible matter interact? . . . . . . . . . . . . . 33

3.6.1 Constraints on matter dark matter interaction . . . . 343.6.2 Space-time correlates for matter dark matter interaction 34

4 k = 113 dark matter and light weak bosons 354.1 Stability of the large h nucleon condensate . . . . . . . . . . . 354.2 What is the value of integer n? . . . . . . . . . . . . . . . . . 354.3 Large h phase with real conformal weights and ordinary weak

bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Could also weak bosons make a transition to conformally non-

trivial large h phase? . . . . . . . . . . . . . . . . . . . . . . . 374.5 Formation and de-coherence of k = 113 dark nucleon phase . 37

4.5.1 Option I: Dark weak bosons are not present in k = 113phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5.2 Option II: Dark weak bosons are present in k = 113dark phase . . . . . . . . . . . . . . . . . . . . . . . . 38

4.6 Dark k = 113 weak bosons and doubly dark k = 107 valencequarks as a minimal option explaining anomalies . . . . . . . 39

4.7 Exotic chemistries and electromagnetic nuclear darkness . . . 41

5 TGD based model for water based on magic dark nuclei andmagic dark dark super-nuclei 425.1 The 41 anomalies of water . . . . . . . . . . . . . . . . . . . . 425.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Dark and ordinary hydrogen bonds . . . . . . . . . . . . . . . 455.4 Icosahedral clusters and dark dark matter . . . . . . . . . . . 465.5 Comments on 41 anomalies . . . . . . . . . . . . . . . . . . . 47

6 Dark matter, long ranged weak force, condensed matter,and chemistry 496.1 What is the most conservative option explaining chiral selec-

tion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Questions related to ordinary condensed matter and chemistry 516.3 Dark-to visible phase transition as a general mechanism of

bio-control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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6.4 Long ranged weak forces in chemistry and condensed matterphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4.1 Exotic nuclear quarks as sources of long ranged weak

force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4.2 Could long ranged weak forces be key players in bio-

catalysis? . . . . . . . . . . . . . . . . . . . . . . . . . 566.5 Z0 force and van der Waals equation of state for condensed

matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.6 Z0 force and chemical evolution . . . . . . . . . . . . . . . . . 626.7 Parity breaking effects at molecular level . . . . . . . . . . . . 63

6.7.1 Mechanism of parity breaking . . . . . . . . . . . . . . 636.7.2 Detailed form of the parity breaking interaction . . . . 64

6.8 Hydrogen bond revisited . . . . . . . . . . . . . . . . . . . . . 66

7 Long ranged weak and color forces, phonons, and sensoryqualia 677.1 Slowly varying periodic external force as the inducer of sound

waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2 About space-time correlates of sound waves . . . . . . . . . . 687.3 A more detailed description of classical sound waves in terms

of Z0 force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.4 Does the physics of sound provide an operational definition

of the dark Z0 force? . . . . . . . . . . . . . . . . . . . . . . . 717.5 Weak plasma waves and the physics of living matter . . . . . 727.6 Sensory qualia and dark forces . . . . . . . . . . . . . . . . . 73

8 Mechanisms of Z0 screening 748.1 General view about dark hierarchy . . . . . . . . . . . . . . . 748.2 Vacuum screening and screening by particles . . . . . . . . . 75

8.2.1 Vacuum screening . . . . . . . . . . . . . . . . . . . . 768.2.2 Screening by weakly charged dark particles . . . . . . 768.2.3 Different variants of Z0 screening by particles . . . . . 77

8.3 A quantum model for the screening of the dark nuclear Z0

charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.3.1 Some relevant observations about dark neutrinos . . . 798.3.2 The model of Z0 screening based on harmonic oscilla-

tor potential does not work . . . . . . . . . . . . . . . 798.3.3 The model for Z0 screening based on constant poten-

tial well . . . . . . . . . . . . . . . . . . . . . . . . . . 808.3.4 Is Bose-Einstein condensate generated spontaneously? 81

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9 Appendix: Dark neutrino atoms 829.1 Dark neutrino atoms in non-relativistic approximation . . . . 839.2 A relativistic model for dark neutrino atom . . . . . . . . . . 84

9.2.1 Self consistent relativistic Schrodinger equation as amodel for Z0 screening . . . . . . . . . . . . . . . . . . 85

9.2.2 Bound states . . . . . . . . . . . . . . . . . . . . . . . 869.2.3 Negative energy resonances . . . . . . . . . . . . . . . 869.2.4 An upper bound on the neutron number of nucleus . . 879.2.5 The behavior of the negative energy solutions near origin 889.2.6 The condition determining the energy eigen values . . 89

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Abstract

The unavoidable presence of classical long ranged weak (and alsocolor) gauge fields in TGD Universe has been a continual source of wor-ries for more than two decades. The basic question has been whetherZ0 charges of elementary particles are screened in electro-weak lengthscale or not. The hypothesis has been that the charges are feeded tolarger space-time sheets in this length scale rather than screened byvacuum charges so that an effective screening results in electro-weaklength scale.

A more promising approach inspired by the TGD based view aboutdark matter assumes that weak charges are indeed screened for ordi-nary matter in electro-weak length scale but that dark electro-weakbosons correspond to much longer symmetry breaking length scale.

The large value of h in dark matter phase implies that Comptonlengths and -times are scaled up. In particular, the sizes of nucleonsand nuclei become of order atom size so that dark nuclear physicswould have direct relevance for condensed matter physics. It becomesimpossible to make a reductionistic separation between nuclear physicsand condensed matter physics and chemistry anymore.

1. The notion of dark matter

In this chapter the earlier ideas about dark matter as an expla-nation of various anomalies are abstracted into a coherent theoreticalframework. General considerations lead to the conclusion that onlyspace-time sheets characterized by the same real or complex p-adicprime and same value of h interact quantum coherently, and that ele-mentary particles are characterized by the p-adic primes characterizingthe space-time sheets at which they feed their gauge charges. Hence thenotion of darkness is only relative and there exists an infinite numberof relatively dark space-time sheets.

The simplest form of dark matter corresponds to real conformalweights but large value of h. In the electromagnetic large h phase cor-responding to k = 113 strong and weak physics are standard but theheight of the Coulomb wall of nuclei is reduced dramatically since thesize of nuclei increases by a factor of order 211. This is enough to ex-plain cold fusion anomalies and selection rules and also the anomalousphysics of water.

Elementary particles can also have complex conformal weights suchthat the conformal weight of the entire system is real. The hypothesisthat given particle has a fixed complex conformal weight characterizedby the complex zero of Riemann Zeta allows natural hierarchy for thesephases and implies no apparent breakings of Fermi statistics. Weakbosons with complex conformal weights are assumed to correspond toGaussian Mersennes instead of real Mersennes and Gaussian Mersennes

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corresponding to k=113 (nuclear length scale),151,157,163,167 (biolog-ically important length scales possibly related to the coiling hierarchyof DNA) define the most interesting examples allowing large paritybreaking effects and non-standard weak nuclear physics.

For each, in general complex, conformal weight an entire hierar-chy of dark matters with increasing values of h is predicted (”darkdark” matter structures consisting of conformally confined dark mat-ter structures, etc...). This hierarchy has an interpretation in termsof the hierarchy of infinite primes constructible by a repeated secondquantization of a supersymmetric arithmetic quantum field theory.

2. What dark nucleons are?

The basic hypothesis is that nucleons can make a phase transitionto dark phase in which the electromagnetic size of both quarks andnucleons is measured in Angstroms. Proton condensates with large hmight be crucial for understanding the properties of water and perhapseven the properties of ordinary condensed matter.

A further phase transition k = 89 → 113 of weak bosons to Gaus-sian Mersenne phase with complex conformal weights would have dra-matic effects on the weak decay rates of resulting nuclei. If also h islarge, one could understand the large parity breaking effects in livingmatter.

TGD based model for nuclei as nuclear strings relies on the notionsof dark valence quark with large h and light exotic quark for whichweak space-time sheet is dark and corresponds to k = 113 and p-adiclength scale of order atomic size. Exotic quark and anti-quark appearat the ends of color bonds connecting nucleons to form a nuclear string.Color bonds can be also charged.

The simplest model for dark nucleons turns out to be the one inwhich valence quarks, which are dark in QCD sense but not in nuclearphysics sense, are transformed to doubly dark quarks and correspond top-adic length scale L(151) ' 10 nm whereas exotic quarks have alwaysdark weak space-time sheet with atomic size. This leads to a model ofpartially dark condensed matter based on the assumption that nucleican form super-nuclei, which are string like structures containing or-dinary nuclei as highly knotted and linked portions separated by colorbonds having length of order interatomic distance. If internuclear colorbonds are em charged, they are also weakly charged, and the repulsiveweak force between exotic quark and anti-quark, which must be actu-ally strong by criticality condition, compensates the internuclear colorforce in equilibrium.

3. Anomalous properties of water and dark nuclear physics

The transparency of water to visible light inspired originally theidea that water is partially dark matter. Second crucial empirical input

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was the H1.5O chemical formula supported by neutron and electronscattering in attosecond time scale, which happens to correspond to theelectromagnetic size of dark proton. This leads to the hypothesis thatone fourth of protons combine to form neutron strings with positivelycharged color bonds between neutrons. Double darkness of the valencequarks explains why the neutrons are dark with respect to nuclearstrong interactions and thus not seen in neutron scattering, whereas thetransformation to neutrons explains why they are not seen in electronscattering.

The crucial property of water is the presence of molecular clusters.Tedrahedral clusters allow an interpretation in terms of magic Z=8protonic dark nuclei. The icosahedral clusters consisting of 20 tedra-hedral clusters in turn have interpretation as magic dark dark nuclei:the presence of the dark dark matter explains large portion of theanomalies associated with water and explains the unique role of waterin biology. In living matter also higher levels of dark matter hierar-chy are predicted to be present. The observed nuclear transmutationsuggest that also light weak bosons are present.

4. Implications of the partial darkness of condensed matter

The model for partially dark condensed matter deriving from nu-clear physics allows to understand the low compressibility of the con-densed matter as being due to the repulsive weak force between exoticquarks, explains large parity breaking effects in living matter, and sug-gests a profound modification of the notion of chemical bond havingmost important implications for bio-chemistry and understanding ofbio-chemical evolution.

1 Introduction

The unavoidable presence of classical long ranged weak (and also color)gauge fields in TGD Universe has been a continual source of worries formore than two decades. The basic question has been whether Z0 charges ofelementary particles are screened in electro-weak length scale or not. Thehypothesis has been that the charges are feeded to larger space-time sheets inthis length scale rather than screened by vacuum charges so that an effectivescreening results in electro-weak length scale.

A more promising approach inspired by the TGD based view about darkmatter assumes that weak charges are indeed screened for ordinary matterin electro-weak length scale but that dark electro-weak bosons correspondto much longer symmetry breaking length scale.

The large value of h in dark matter phase implies that Compton lengths

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and -times are scaled up. In particular, the sizes of nucleons and nuclei be-come of order atom size so that dark nuclear physics would have direct rele-vance for condensed matter physics. It becomes impossible to make a reduc-tionistic separation between nuclear physics and condensed matter physicsand chemistry anymore. This view forces a profound re-consideration of theearlier ideas in nuclear and condensed physics context. It however seemsthat most of the earlier ideas related to the classical Z0 force and inspiredby anomaly considerations survive in a modified form.

1.1 How to characterize dark matter?

The identification of the precise criterion characterizing dark matter phase isfar from obvious. TGD actually suggests an infinite number of phases whichare dark relative to each other in the sense that the particles of differentphases cannot appear in the same vertex and a phase transition changingthe particles to each other analogous to de-coherence is necessary.

Dark matter certainly corresponds to a large h and this characterizationis in principle all that is needed. This characterization relies on intuition thatonly space-time sheets with same size interact quantum coherently. Thisintuition generalizes to the hypothesis that only space-time sheets with samep-adic prime and same value of h have direct quantum coherent interactionsand that incoherent interaction involve a phase transition changing the valueof p-adic prime and h. Infinite number of relatively dark phases is predicted.

Furthermore, each particle is characterized by a collection of p-adicprimes. There are two possible interpretations and both of them allowto resolve objections against p-adic hierarchies of color and electro-weakphysics.

a) These primes characterize the space-time sheets at which it feeds itsgauge fluxes and particles can interact only via their common space-timesheets and are otherwise dark with respect to each other.

b) Number theoretical vision supports the notion of multi-p p-adicity andthe idea that elementary particles correspond to infinite primes, integers, orperhaps even rationals [A3, B5]. To infinite primes, integers, and rationalsit is possible to associate a finite rational q = m/n by a homomorphism. qdefines an effective q-adic topology of space-time sheet consistent with p-adic topologies defined by the primes dividing m and n (1/p-adic topology ishomeomorphic to p-adic topology). The largest prime dividing m determinesthe mass scale of the space-time sheet in p-adic thermodynamics. m and nare exchanged by super-symmetry and the primes dividing m (n) correspondto space-time sheets with positive (negative) time orientation. Two space-

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time sheets characterized by rationals having common prime factors can beconnected by a #B contact and can interact by the exchange of particlescharacterized by divisors of m or n.

The nice feature of this option is that single multi-p p-adic space-timesheet rather than a collection of them characterizes elementary particle.Concerning the description of interaction vertices as generalization of ver-tices of Feynman graphs (vertices as branchings of 3-surfaces) this optionis decisively simpler than option a) and is consistent with earlier numbertheoretic argument allowing to evaluate gravitational coupling strength. Itis also easier to to understand why the largest prime in the collection deter-mines the mass scale of elementary particle.

A particular kind of dark matter corresponds to conformally confinedmatter with particles having complex conformal weights such that the netconformal weight is real. In this case h need not be large. If particles ofgiven phase have a fixed conformal weight corresponding to non-trivial zeroof Zeta or its conjugate, there is no effective violation of Fermi statistics.The space-time correlate for the complexity of conformal weights would beGaussian primeness but also other extensions of p-adic numbers can beconsidered and zeros of Zeta could map to Gaussian Mersennes in the caseof gauge bosons if they quite generally correspond to Mersennes.

The chiral selection in living matter suggest large parity breaking andpresence of light dark weak bosons with complex conformal weights corre-sponding to increasing values for the zeros of Zeta. The Gaussian Mersennes(1 + i)k − 1 for k = 113, 151, 157163, 167 correspond to nuclear length scaleand four biologically important length scales in the range 10 nm-25 µm,which seem to relate directly to the coiling hierarchy of DNA double strands.

For given prime, TGD predicts actually an entire hierarchy of dark mat-ters corresponding to varying values of h such that the many particle statesat previous level become particles at the next level. The hierarchy for largevalues of h would provide a concrete physical identification for the hierarchyof infinite primes identifiable in terms of a repeated second quantization ofan arithmetic super-symmetric QFT [A3]. The finite primes about infiniteprime is in a well defined sense a composite would correspond to the particlesin the state forming a unit of dark matter.

1.2 Evidence for long range weak forces and new nuclearphysics

There is a lot of experimental evidence for long range electro-weak forces,dark matter, and exotic nuclear physics giving valuable guidelines in the

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attempts to build a coherent theoretical scenario.a) Cold fusion [8] is a phenomenon involving new nuclear physics and the

known selection rules give strong constraints when one tries to understandthe character of dark nuclear matter. In particular, the model requires thatonly nuclear protons are in dark phase [D1].

b) Large parity breaking effects in living matter indicate the presence oflong ranged weak forces, and the reported nuclear transmutations in livingmatter [29, 30] suggest that new nuclear physics plays a role also now.

c) The physics of water involves a large number of anomalies and lifedepends in an essential manner on them. As many as 41 anomalies arediscussed in the excellent web page ”Water Structure and Behavior” of M.Chaplin [1]. The transparency of water is still a mystery for science [7].The fact that the physics of heavy water differs much more from that ofordinary water as one might expect on basis of different masses of watermolecules suggests that dark nuclear physics is involved. The finding thatone hydrogen atom per two water molecules remain effectively invisible inneutron and electron interactions in attosecond time scale [1, 2] suggeststhat water is partially dark. These findings have been questioned in [3]and thought to be erroneous in [4]. If the findings are real, dark matterphase made of super-nuclei consisting of dark protons could explain themas also the clustering of water molecules predicting magic numbers of watermolecules in clusters.

If so, dark nuclear physics could be an essential part of condensed matterphysics and biochemistry. For instance, the condensate of dark protonsmight be essential for understanding the properties of bio-molecules andeven the physical origin of van der Waals radius of atom in van der Waalsequation of state.

d) The scaling law of homeopathy [D5] suggests that the scalings asso-ciated with the transitions to dark matter correspond to scalings by powersn/v0, n = 3, and that a hierarchy of dark matters is involved (dark matter,dark dark matter, etc...)

e) Exotic chemistries [11] in which clusters of atoms of given given typemimic the chemistry of another element. These systems behave as if nucleiwould form a jellium (constant charge density) defining a harmonic oscilla-tor potential for electrons. Magic numbers correspond to full electron shellsanalogous to noble gas elements. It is difficult to understand why the con-stant charge density approximation works so well. If nuclear protons are inlarge h with n = 3 state the electromagnetic sizes of nuclei would be about2.4 Angstroms and the approximation would be natural.

f) The anomalies reported by free energy researchers such as over unity

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energy production in devices involving repeated formation and dissociationof H2 molecules based on the original discovery of Nobelist Irwing Langmuir[18] (see for instance [19]) suggest that part of H atoms might end up to darkmatter phase liberating additional energy. The ”mono-atomic” elements ofHudson suggest also dark nuclear physics [6]. There is even evidence formacroscopic transitions to dark phase [16, 17, 15].

g) Tritium beta decay anomaly [24, 25, 26, 27] suggests exotic nuclearphysics related to weak interactions and that dark anti-neutrino density atthe orbit of Earth around Sung oscillating with one year period is involved.This kind of remnant of dark matter would be consistent with the model forthe formation of planets from dark matter. The evidence for the variationof the rates of nuclear and chemical processes correlating with astrophysicalperiods [20] could be understood in terms of weak fields created by darkmatter and affect by astrophysical phenomena.

1.3 k = 113 dark nuclear physics

k = 113 characterizes electromagnetic size of u and d quarks, of nucleons,and nuclei. k = 107 characterizes the QCD size of hadrons. The basicquestion is what large h phase means for hadrons and for confined quarks.There are two possible answers to this question.

a) Both hadronic k = 107 space-time sheet and space-time sheets ofvalence quarks and possibly also of sea quarks are in the large h phase.

b) Only hadronic space-time sheets suffer this phase transition for lightnuclei. If the criticality condition Z2αem ≥ 1 implying Z ≥ 12 or some ofits variants discussed in [B6] is satisfied, the phase transition increasing hof quarks occurs.

1.3.1 Are valence quarks always dark?

The first option is that both k = 107 space-time sheet of hadrons andk = 113 space-time sheets of valence quarks have large h and in darkmatter phase as far as QCD is considered. This would mean that theirQCD size (k = 107) is of order electron Compton length. These surpris-ingly long length scales have a natural interpretation as the height of themagnetic/color-magnetic body of nucleon.

The basic criterion for the transition to dark matter phase is that pertur-bation theory for gauge interacting system ceases to converge. A more prac-tical criterion in terms of two particle gauge interactions reads as Q1Q2α '1.

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The criterion suggests that only protons inside nucleus can make a tran-sition to dark matter phase meaning that real p-adic prime p ' 2113 isreplaced by Gaussian Mersenne MG

113. Neutrons would remain in ordinaryphase and proton’s electromagnetic size L(113) would increase by a fac-tor n/v0 ' 211n, where n is integer, for n = 3 this would give size of 2.4Angstroms, order of magnitude for typical van der Waals radius.

The assumption that only p and D make the transition is suggestedby the fact that only tritium but not 3He is produced in cold fusion andthe special properties of water. This assumption is however not consistentwith the assumption that the transition occurs at quark level. Fortunately,also the assumption that the phase transition occurs for both neutrons andprotons allows to understand the selection rules of cold fusion: the point isthat Coulomb repulsion makes the rate for the fusion of p and p resultingin the phase transition of dark p slow. If conformal weights remain complexand large h phase transition occurs for k = 113 sheet only, the lifetimes ofnuclei are not changed and nuclear physics is not affected as far as classicallowest order in h predictions are considered. The basic effects come fromthe dramatic lowering of Coulomb wall by the increase of the nuclear size.

The phase transition increasing only h must be distinguished from aphase transition making conformal weights complex. In this phase tran-sition the real prime corresponding to k = 113 would become GaussianMersenne. This would bring in conformally non-trivial weak bosons withk = 113 with mass scaled down by a factor 2−12. The lifetime of neutronswould become very short unless the mass difference is below electron massand this condition would serve as a criterion for the stability of the resultingexotic nuclei.

If both of these phase transitions occur k = 113 weak bosons would havea Compton length of order atomic size scale. This could allow to understandthe large parity breaking effects in living matter.

The simplest model for dark nucleons turns out to be the one in which va-lence quarks, which are dark in QCD sense but not in nuclear physics sense,are transformed to doubly dark quarks and correspond to p-adic length scaleL(151) ' 10 nm whereas exotic quarks have always dark weak space-timesheet with atomic size. This leads to a model of partially dark condensedmatter based on the assumption that nuclei can form super-nuclei, whichare string like structures containing ordinary nuclei as highly knotted andlinked portions separated by color bonds having length of order interatomicdistance. If internuclear color bonds are em charged, they are also weaklycharged, and the repulsive weak force between exotic quark and anti-quark,which must be actually strong by criticality condition, compensates the in-

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ternuclear color force in equilibrium.

1.3.2 Could only the hadronic space-time sheet be scaled up forlight nuclei?

The model discussed in this chapter is based on guess work and leaves a lotof room for different scenarios. One of them emerged only after a couple ofmonths finishing the work with this chapter.

1. Is only the h associated with hadronic space-time sheet large?

The surprising and poorly understood conclusion from the p-adic masscalculations was that the p-adic primes characterizing light quarks u,d,ssatisfy kq < 107, where k = 107 characterizes hadronic space-time sheet[B3].

a) The interpretation of k = 107 space-time sheet as a hadronic space-time sheet implies that quarks topologically condense at this space-timesheet so that k = 107 cannot belong to the collection of primes characterizingquark.

b) Since hadron is expected to be larger than quark, quark space-timesheets should satisfy kq < 107 unless h is large for the hadronic space-timesheet so that one has keff = 107 + 22 = 129. This would predict twokinds of hadrons. Low energy hadrons consists of u, d, and s quarks withkq < 107 so that hadronic space-time sheet must correspond to keff = 129and large value of h. One can speak of confined phase. This allows alsok = 127 light variants of quarks appearing in the model of atomic nucleus.The hadrons consisting of c,t,b and the p-adically scaled up variants of u,d,shaving kq > 107, h has its ordinary value in accordance with the idea aboutasymptotic freedom and the view that the states in question correspond toshort-lived resonances.

This picture is very elegant but would mean that it would be light hadronrather than quark which should have large h and scaled up Compton length.This does not affect appreciably the model of atomic nucleus [B6] since thecrucial length scales L(127) and L(129) are still present.

2. Under what conditions quarks correspond to large h phase?

What creates worries is that the scaling up of k = 113 quark space-time sheets of quarks forms an essential ingredient of condensed matterapplications assuming also that these scaled up space-time sheets coupleto scaled up k = 113 variants of weak bosons. Thus one must ask underwhat conditions k = 113 quarks, and more generally, all quarks, can make

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a transition to a dark phase accompanied by a simultaneous transition ofhadron to a doubly dark phase.

The criterion for the transition to a large h phase at the level of valencequarks would require that the criticality criterion is satisfied at k = 111space-time sheet and would be expressible as Z2αem = 1 or some variant ofthis condition discussed above.

The scaled up k = 127 quark would correspond to k = 149, the thicknessof the lipid layer of cell membrane. The scaled up hadron would correspondto k = 151, the thickness of cell membrane. This would mean that alreadythe magnetic bodies of hadrons would have size of cell membrane thickness sothat the formation of macroscopic quantum phases would be a necessity sincethe average distance between hadrons is much smaller than their Comptonlength.

1.4 Water and k = 113 exotic nuclear physics

The transparency of water to visible light inspired originally the idea thatwater is partially dark matter. Second crucial empirical input was the H1.5Ochemical formula supported by neutron and electron scattering in attosecondtime scale, which happens to correspond to the electromagnetic size of darkproton. This leads to the hypothesis that one fourth of protons combine toform neutron strings with positively charged color bonds between neutrons.Double darkness of the valence quarks explains why the neutrons are darkwith respect to nuclear strong interactions and thus not seen in neutronscattering, whereas the transformation to neutrons explains why they arenot seen in electron scattering.

There is evidence for two kinds of hydrogen bonds [32, 31]: a possibleidentification is in terms of ordinary and dark proton. Tedrahedral wa-ter clusters consisting of 14 water molecules would contain 8 dark protonswhich corresponds to a magic number for dark nucleus consisting of pro-tons. Icosahedral water clusters in turn consist of 20 tedrahedral clustersand the interpretation would be as magic ”dark” dark nucleus associatedwith k = 151 dark dark electro-weak bosons. The appearance of the darkdark hierarchy level could make water completely exceptional and make itunique from the point of view of living matter in which also higher hierar-chy levels would be present and correspond quite concretely to the coilinghierarchy of DNA at the level of ordinary matter.

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1.5 Implications of the partial darkness of condensed matter

The model for partially dark condensed matter deriving from nuclear physicsallows to understand the low compressibility of the condensed matter asbeing due to the repulsive weak force between exotic quarks, explains largeparity breaking effects in living matter, and suggests a profound modificationof the notion of chemical bond having most important implications for bio-chemistry and understanding of bio-chemical evolution.

2 General ideas about dark matter and condensedmatter

2.1 Living matter, dark matter, and conformal confinement

Dark matter is identified as a macroscopic quantum phase with large h forwhich particles have complex conformal weights and by conformal confine-ment behaves like single coherent whole. Dark matter controls living matterand this explains the weird looking findings about Bohr rules for planetaryorbits.

Also living matter would be matter with a large value of h and hencedark, and form conformally confined blobs behaving like single units withextremely quantal properties, including free will and intentional action intime scales familiar to us! Dark matter and conformal confinement wouldbe responsible for the mysterious vital force.

Any system for which some interaction becomes so strong that pertur-bation theory does not work, gives rise to this kind of system in a phasetransition in which h increases to not lose perturbativity gives rise to thiskind of ”super-quantal” matter. In this sense emergence corresponds tostrong coupling. One must however remember that emergence is actuallymuch more and involves the notion of quantum jump. Dark matter madepossible by dynamical h is necessary for macroscopic and macro-temporalquantum coherence and is thus prerequisite for emergence.

Physically large h means a larger unit for quantum numbers and thisrequires that single particle states form larger particle like units. This kindof collective states with weak mutual interactions are of course very naturalin strongly interacting systems. At the level of quantum jumps quantumjumps integrate effectively to single quantum jump and longer moments ofconsciousness result. Conformal confinement guarantees all this. Entirehierarchy of size scales for conformally confined blobs is predicted corre-sponding to values of h related to Beraha numbers [A2, C1, A4] but there

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would be only single value corresponding to very large h for given values ofsystem parameters (gravitational masses, charges,...). The larger the valueof h the longer the characteristic time scale of consciousness and of a typicallife cycle.

In RHIC color glass condensate resembles incompressible liquid. Liquidsmight be liquids because they contain some dark matter at magnetic/Z0

magnetic flux tubes (darkness follows from the large value of h). Incom-pressibility of liquid could correspond to maximal density of flux tubes andto the fact that magnetic fields have no sources. In accordance with theprevious ideas already water would be living and conscious system in someprimitive sense.

The notion of field body in turn means that dark matter at the mag-netic flux tubes would serve as an intentional agent using biological bodyas a motor instrument and sensory receptor. Dark matter would be themiraculous substance that living systems are fighting for, and perhaps themost important substance in metabolic cycle.

2.2 How dark matter and visible matter interact?

The hypothesis that the value of h is dynamical, quantized and becomeslarge at the verge of a transition to a non-perturbative phase in the ordinarysense of the word has fascinating implications. In particular, dark matter,would correspond to a large value of h and could be responsible for theproperties of the living matter. In order to test the idea experimentally, amore concrete model for the interaction of ordinary matter and dark mattermust be developed and here of course experimental input and the consistencywith the earlier quantum model of living matter is of considerable help.

2.2.1 Basic implications from the scaling of h

It is relatively easy to deduce the basic implications of the scaling of h.a) If the rate for the process is non-vanishing classically, it is not affected

in the lowest order. For instance, scattering cross sections for say electron-electron scattering and e+e− annihilation are not affected in the lowest ordersince the increase of Compton length compensates for the reduction of αem.Photon-photon scattering cross section, which vanishes classically and isproportional to α4

emh2/E2, scales down as 1/h2.b) Higher order corrections coming as powers of the gauge coupling

strength α are reduced since α = g2/4πh is reduced. Since one has hs/h =αQ1Q2/v0, αQ1Q2 is effectively replaced with a universal coupling strength

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v0 ' 4.6 × 10−4. In the case of QCD the paradoxical sounding implicationis that αs becomes very small in non-perturbative phase.

2.2.2 Simple model for dark atoms

The simplest model for dark nuclei is as blobs of Ncr dark nuclei surroundedby dark electrons at distance which is scaled by a factor k2, k = hs/h fromthat for ordinary atom. Thus the dark nucleus looks point-like from thepoint of view of the electron cloud. As far as electrons are considered thereare two options.

i) Dark electrons could behave as independent particles with the inter-action strength with the dark nucleus given by NZα. The critical valueNcr of N is determined from the condition of criticality for single electrondark nucleus interaction: [X] = 1, X = NcrZα. Here [x] denotes the largestinteger smaller than x. Criticality implies hs/h = X/v0 ∼ 1/v0. Note thatthe condition 1 ≤ X < 2 implying 1/v0 ≤ hs/h < 2/v0 holds true.

ii) Dark electrons are in the same state apart from the values of super-canonical conformal weights making possible to satisfy fermionic statistics.They behave like a single super-electron with mass Ncrme and em chargeNcre. In this case the criticality condition reads [X] = 1, X = N2

crZα. Thecriticality implies hs/h = X/v0 ∼ 1/v0.

b) The binding energy scale E ∝ Z2α2emme of atoms scales as 1/h2

so that a partially dark matter for which protons have large value of hdoes not interact appreciably with the visible light. Scaled down spectrumof atomic binding energies would be the experimental signature of darkatoms. The resulting binding energy spectrum is independent of the atomin the approximation Ncr = 1/Zα. The binding energy scale defined by theionization energy E0 = Z2α2me/4 as given by Bohr model is replaced withE0 = v2

0me/4X2 ' 26.5/X2 meV. Different values of X allow to distinguishbetween different atoms since the energy scale differ by a factor 1/4 fromthe maximal one. It should be noticed that the resting potential of neuronis around .64 meV (the value varies in considerable limits up to 80 meV).

c) The ionization wavelength for ordinary h would be ' 46.5X2 µm,which for X = 1 below the maximal size of neuron about 100 µm. Forhs = h/v0 the wavelength is given by factor 46.5X3/v0 and ' 9.4X3 cm,which happens to be the size scale of brain hemisphere for X = 1.

d) The Bohr radius of dark atom scales as h2 and is given by ad =(X/v0)2a0 ' .2X2 < .8 mm, (a0 = h/αemme). The size of basic multi-neuron modules in cortex is about 1 mm. These intriguing observationsgive hints about the possible role of dark atoms in the functioning of living

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matter and brain.

2.2.3 How dark photons transform to ordinary photons?

The transitions of dark atoms naturally correspond to coherent transitions ofthe entire dark electron BE condensate and thus generate Ncr dark photonswhich have complex conformal weights and are conformally confined andbehave thus like laser beams. Dark photons do not interact directly withthe visible matter.

The simplest guess is that the transformation of dark photon BE conden-sates to ordinary photons corresponds to a loss of coherence by conformalliberation in which conformal weights of photons become real. An openquestion is whether even ordinary laser beams could be identified as beamsof dark photons. Note that the transition from dark to ordinary photonsimplies the reduction of wave length and thus also of coherence length by afactor v0.

Dark ↔ visible transition should have also a space-time correlate. Theso called topological light rays or MEs (”massless extremals”) represent acrucial deviation of TGD from Maxwell’s ED and have all the propertiescharacterizing macroscopic classical coherence. Therefore MEs are excellentcandidates for the space-time correlate of BE condensate of dark photons.

MEs carry in general a superposition of harmonics of some basic fre-quency determined by the length of ME. A natural expectation is thatthe frequency of classical field corresponds to the generalized de Brogliefrequency of dark photon and is thus h/hs times lower than for ordinaryphotons. In completely analogous manner de Broglie wave length is scaledup by k = hs/h. Classically the decay of dark photons to visible photonswould mean that an oscillation with frequency f inside topological light raytransforms to an oscillation of frequency f/k such that the intensity of theoscillation is scaled up by a factor k. Furthermore, the ME in question couldnaturally decompose into 1 < Ncr ≤ 137 ordinary photons in case that darkatoms are in question. Of course also MEs could decay to lower level MEsand this has an interpretation in terms of hierarchy of dark matters to bediscussed next.

2.2.4 Hierarchy of dark matters and hierarchy of minds

The notion of dark matter is only relative concept in the sense that darkmatter is invisible from the point of view of the ordinary matter. Onecan imagine an entire hierarchy of dark matter structures corresponding to

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the hierarchy of space-time sheets for which p-adic length scales differ bya factor 1/v0 ∼ 211. The BE condensates of Ncr ordinary matter particleswould serve as dynamical units for ”dark dark matter” invisible to the darkmatter. The above discussed criticality criterion can be applied at all levelsof the hierarchy to determine the value of the dynamical interaction strengthfor which BE condensates of BE condensates are formed.

This hierarchy would give rise to a hierarchy of the values of hn/h com-ing as powers of v−n

0 as well as a hierarchy of wavelengths with same energycoming as powers or vn

0 . For zero point kinetic energies proportional to h2

this hierarchy would come in powers of v−2n0 , for magnetic interaction ener-

gies proportional to h the hierarchy would come in powers v−n0 whereas for

atomics energy levels the hierarchy would come in powers of v2n0 (assuming

that this hierarchy makes sense).The most interesting new physics would emerge from the interaction

between length scales differing by powers of v0 made possible by the decayof BE condensates of dark photons to ordinary photons having wavelengthshorter by a factor ∼ v0. This interaction could provide the royal roadto the quantitative understanding how living matter manages to build upextremely complex coherent interactions between different length and timescales.

In the time domain dark matter hierarchy could allow to understandhow moments of consciousness organize to a hierarchy with respect to thetime scales of moment of consciousness coming as 211k multiples of CP2 timescale. Even human life span could be seen as single moment of consciousnessat k = 14th level of the dark matter hierarchy whereas single day in humanlife would correspond to k = 12.

2.2.5 Wave-length hierarchy, coherent metabolism, and proton-electron mass ratio

The fact that a given wavelength length corresponds to energies related toeach other by a scaling with powers of v0 provides a mechanism allowing totransfer energy from long to short long scales by a de-coherence occurringeither in the standard or reversed direction of geometric time. De-coherencein the reversed direction of time would be associated with mysterious lookingprocesses like self-assembly allowing thus an interpretation as a normal decayprocess in reversed time direction.

It is perhaps not an accident that the value of v0 ' 4.6 × 10−4 is nottoo far from the ratio of me/mp ' 5.3× 10−4 giving the ratio of zero pointkinetic energies of proton and electron for a given space-time sheet. This co-

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incidence could in principle make possible a metabolic mechanism in whichdark protons and ordinary electrons co-operate in the sense that dark pro-tons generate dark photon BE condensates with wave length λ transformingto ordinary photons with wavelength v0λ absorbed by ordinary electrons.

Some examples are in order to illustrate these ideas.a) As already found, in the case of dark atoms the scaling of binding

energies as 1/h2 allows the coupling of ∼ 9 cm scale of brain hemispherewith the length scale ∼ 50 µm of large neuron. Ncr ≤ 137 ordinary IRphotons would be emitted in single burst and interacting with neuron.

b) For a non-relativistic particle in a box of size L the energy scale isgiven by E1 = h2π2/2mL2 so that the visible photons emitted would haveenergy scaled up by a factor (hs/h)2 ' 4 × 106. The collective dropping ofNcr dark protons to larger space-time sheet would liberate a laser beam ofdark photons with energy equal to the liberated zero point kinetic energy.For instance, for the p-adic length scale L(k = 159 = 3× 53) ' .63 µm thisprocess would generate laser beam of IR dark photons with energy ∼ .5 eValso generated by the dropping of ordinary protons from k = 137 atomicspace-time sheet. There would thus be an interaction between dark protonsin cell length scale and ordinary protons in atomic length scale. For instance,the dropping of dark protons in cell length scale could induce driving ofprotons back to the atomic space-time sheet essential for the metabolism[D3]. Similar argument applies to electrons with the scale of the zero pointkinetic energy about 1 keV.

c) If the energy spectrum associated with the conformational degreesof freedom of proteins, which corresponds roughly to a frequency scale of10 GHz remains also invariant in the phase transition to dark protein state,coherent emissions of dark photons with microwave wave lengths would gen-erate ordinary infrared photons. For instance, metabolic energy quanta of∼ .5 eV could result from macroscopic Bose-Einstein condensates of 58 GHzdark photons resulting from the oscillations in the conformational degreesof freedom of dark proteins. A second option is that the conformal energiesare scaled by hs/h (ω would remain invariant). In this case these coherentexcitations would generate ordinary photons with energy of about 1 keVable to drive electrons back to the atomic k = 137 space-time sheet.

d) Since magnetic flux tubes have a profound role in TGD inspired the-ory of consciousness, it is interesting to look also for the behavior of effec-tive magnetic transition energies in the phase transition to the dark matterphase. This transition increases the scale of the magnetic interaction energyso that anomalously large magnetic spin splitting hseB/m in the externalmagnetic field could serve as a signature of dark atoms. The dark transition

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energies relate by a factor hs/h to the ordinary magnetic transition energies.For instance, in the magnetic field of Earth with a nominal value .5×10−4

Tesla dark electron cyclotron frequency is 6 × 105 Hz and corresponds toordinary microwave photon with frequency ∼ 1.2 GHz and wavelength λ '25 cm. For proton the cyclotron frequency of 300 Hz would correspond toenergy of ordinary photon with frequency of 6 × 105 Hz and could induceelectronic cyclotron transitions and spin flips in turn generating for instancemagneto-static waves.

It is easy to imagine a few step dark matter hierarchy connecting EEGfrequencies of dark matter with frequencies of visible light for ordinary pho-tons. This kind of hierarchy would give considerable concreteness for thenotion of magnetic body having size scale of Earth.

2.2.6 Dark condensed matter and grey matter

The BE condensates containing Ncr dark atoms define what might be calleddark super-atoms. One cannot avoid asking whether these super-atomscould form molecular structures with a typical distance between super-atomsgiven by dark Bohr radius ad ' .2X2 mm, and whether also the dark coun-terpart of the condensed matter could exist. Even super-dark counterpartsof bio-molecules can be imagined. One can also wonder whether the 1 mmsized basic structural units of cortex might be visible matter quantum con-trolled by a dark condensed matter. This prelude motivates the followingsimple scaling arguments allowing to deduce the basic characteristics of thespectrum of dark super molecules.

a) The scale for the vibrational energy spectrum of dark super-moleculeswould be given by hs

√k/Ncrm with elastic constant behaving as k ∝ 1/a2

d sothat dark vibrational energy spectrum would relate by a factor (h/hs)/Ncr '2−11/Ncr to the ordinary spectrum of vibrational energies. The scale of therotational energy energy spectrum would be h2

s/a2dNcr being by a factor Ncr

smaller than for ordinary molecules. Since the ratio of scales for rotationaland vibrational energies is ∼ 10−3 for ordinary molecules, these scales wouldbe essentially same for dark molecules and about 10−4/Ncr eV correspondingto frequencies f ≤ 1 GHz for ordinary light.

b) Also the dark counterparts of condensed matter phases can be imag-ined. The lattice constant would be of order ad and the widths of elec-tronic energy bands would be below the maxima of electronic kinetic ener-gies h2

sπ2/2mea

2d ∼ 10−4 eV. A more precise estimate in the case of solids

is obtained from the scaling of the Fermi energy determined by the densityof electrons which is at most 2Ncr electrons per atomic volume at a given

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energy level. In free electron approximation the Fermi energy of electrondetermining the width of the band is given by

EF (dark) ∼ ( hhs

)2N2/3cr EF , EF = ( 3π2

21/2 )2/3 h2

a2me, (1)

and differs from the rough estimate by N2/3cr factor.

A fascinating question is whether the IR spectrum of say water couldreveal the presence of dark condensed matter. The IR spectrum of waterexhibits some structure in the wavelength region containing the wavelengthsλ(1 → ∞) = 46.5X2 µm and λ(1 → 2) = 62.0X2 µm (see Fig. 2.2.6).There are two peaks at wavelengths whose ratio is somewhat smaller thanλ(1 → 2)/λ(1 →∞) = 3/4. The value of v0 estimated from the lower peakis v0 ∼ 5.8 × 10−4 for X = 1 and suspiciously large. Since X depends onatom no sharp peaks are expected. Furthermore, if dark super-atoms forma condensed matter phase in the proposed sense, dark atom energy levelsare widened to energy bands with the maximal width of the band beingEmax ∼ N

2/3cr × 10−4 eV in free electron approximation so that one has

∆λ/λ ∼ .1×N2/3cr .

Figure 1: Infrared spectrum of water. The structure in the wavelengthinterval 30-50 µm might relate to the ionization energy spectrum ∆E(1 → n)of dark atoms.

It deserves to be noticed that ordinary radiation in the wavelength range.3 mm-30 cm is strongly absorbed by water molecules [52] whereas the fre-

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quencies below and above this range are either too slow or fast for a con-siderable absorption to occur. Therefore the ordinary photons generated inthe transitions of dark atoms and molecules would have a maximal effect onbio-molecules.

The size of the basic multi-neuron modules in the cortex is about 1 mmso that the idea that grey matter might be controlled by a dark variant ofcondensed matter with an atomic volume about 1 mm3 is irresistible. Vibra-tional, rotational, and electronic transitions would generate dark photon BEcondensates mediating coupling of the dark condensed matter with ordinarymatter at IR, microwave and radio frequencies. One can also play with thethought that linear dark super molecules could realize genetic code at thelevel of cognition: these super molecules would code for ”cognitive proteins”identifiable as linear strongly coupled structures formed by the structuralmodules of the cortex and serving as neural correlates for sequences of as-sociations.

Dark radiation with wavelengths coming as sub-harmonics of the darkatomic distance ad ∼ 1 mm is predicted. This radiation would correspondto visible and UV wave lengths for the ordinary photons: bio-photons haveenergies in this energy range [55]. This point will be discussed in more detailbelow.

2.2.7 A connection with bio-photons

The biologically active radiation at UV energies was first discovered by Rus-sian researcher Gurwitz using a very elegant experimental arrangement [53].Gurwitz christened this radiation mitogenetic radiation since it was espe-cially intense during the division of cell.

A direct proof for the biological activity of mitogenetic radiation con-sisted of a simple experiment in which either quartz or glass plate was putbetween two samples. The first sample contained already growing onionroots whereas the second sample contained roots which did not yet grow. Inthe case of quartz plate no stimulation of growth occurred unlike for glassplate. Since quartz is not transparent to UV light whereas the ordinary glassis, the conclusion was that the stimulation of growth is due to UV light.

The phenomenon was condemned by skeptics as a pseudo science andonly the modern detection technologies demonstrated its existence [55], andmitogenetic radiation became also known as bio-photons (the TGD basedmodel for bio-photons is discussed in [D3]). Bio-photons form a relativelyfeatureless continuum at visible wavelengths continuing also to UV energies,and are believed to be generated by DNA or at least to couple with DNA.

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The emission of bio-photons is most intense from biologically active organ-isms and the irradiation by UV light induces an emission of mitogeneticradiation by a some kind of amplification mechanism. It has been suggestedthat bio-photons represent some kind of leakage of a coherent light emittedby living matter.

According to Russian researcher V. M. Injushin [54], mitochondrios emitred light at wavelengths 620 nm and 680 nm corresponding to energies 2 eVand 1.82 eV. According to the same source, the nucleus of cell sends UV lightat wavelengths 190, 280 and 330 nm corresponding to the energies 6.5, 4.4and 3.8 eV. The interpretation as a kind of leakage of coherent light wouldconform with the identification in terms of BE condensates of dark photonswith hs/h ' 211 emitted at wavelengths varying in the range .3− 1.25 mmand decaying to photons with energies visible and UV range. For instance,1.82 eV radiation corresponds to a dark photon wave length of 1.4 mmfor v0(eff) = 2−11. A bio-control of ordinary bio-matter at sub-cellularlevel performed by dark matter from the millimeter length scale could be inquestion. This proposal conforms with the fact that 1 mm defines the scaleof the blobs of neurons serving as structural units in cortex.

The analysis of Kirlian photographs has shown that the pattern of visiblelight emitted by various body parts, for instance ear, code information aboutother body parts [56]. These bio-holograms for which a general model isdiscussed in [D4] could be realized as dark photon laser beams.

In phantom DNA effect [57] a chamber containing DNA is irradiated witha visible laser light and the DNA generates as a response coherent visibleradiation at same wavelength. Strangely enough, the chamber continues toemit weak laser light even after the removal of DNA. This effect could bedue to the decay of a dark photon BE condensate remaining in the chamber.Also the findings of Peter Gariaev [58] about the effects of visible laser lighton DNA, in particular the stimulated emission of radio waves in kHz-MHzfrequency range might also relate to dark photons somehow.

3 A more detailed theoretical view about darkmatter

In order to make progress it is necessary to try to find more precise definitionsof the notions related to dark matter.

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3.1 Characterization of dark matter as large h phase

The precise characterization of dark matter is far from obvious and it seemsthat darkness is a relative notion, that TGD actually predicts an infinitevariety of dark matters, and that the notions of relative and partial darknessmake sense.

The basic hypothesis have been that dark matter corresponds to a largevalue of h. This notion could be too narrow. A phase with given value of his dark relative to any phase with different value of h, large or small, sinceparticles in interaction vertices must have the same value of h. A secondimportant point is that in the proposed interpretation h characterizes sub-system-environment relationship described Jones inclusion characterized byBeraha number Bn = 4cos2(π/n). h is expressible in terms of logarithm ofBq, q > 3 rational. Large h phase corresponds to 3 < q < 4 [A4].

Second basic assumption is that the transition to dark matter phaseoccurs when the the interaction strength satisfies the criticality conditionQ1Q2α ' 1. A special case corresponds to self interaction with Q1 = Q2.This condition applies only to gauge interactions so that particles can becharacterized by gauge charges. A more general characterization would bethat transition occurs when perturbation theory ceases to converge. Thecriterion cannot be applied to phenomenological QFT description of strongforce in terms of, say, pion exchange.

The questions are whether all elementary particles have dark counter-parts in dark matter phase and whether the transition to dark phase occurssimultaneously for all p-adic primes characterizing the particle. The detailsof the answer depends on in which sense particles correspond to collectionsof p-adic primes [B5].

a) Suppose that each particle corresponds to a collection of space-timesheets at which it feeds gauge fluxes and also the space-time sheet corre-sponding to say, electro-weak gauge bosons, is involved. If each space-timesheet associated with the particle can make the phase transition separately,the minimal option is that the scaled up variants of space-time sheets as-sociated with the bosons mediating the interaction in question emerge inthe phase transition for each particle interacting via the exchange of thesebosons. Of course, it is also possible that all space-time sheets make thephase transition.

b) If the collection of p-adic primes corresponds to multi-p p-adicity. Inthis case the phase transition must occur for all primes simultaneously sothat criticality for single interaction implies transition for all interactions.Obviously this option makes much more stringent predictions.

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This picture also resolves the interpretational problems caused by theexistence of scaled up variants of QCDs and electro-weak multiplets.

Some warnings are in order.a) A phase for which particles have complex conformal weights need nothave a large h.b) Matter at larger space-time sheets need not have large h.c) Large h phase need not interact via exotic weak bosons. D ≥ 3 is theonly proper criterion for the presence of exotic weak bosons. Large h phaseneed not interact via exotic gluons. Anomalous hyper charge and isospincan save from long range color fields for D = 2.

Some examples are in order to test this view.a) Transition from perturbative phase in QCD to hadronic phase is the

most obvious application. The identification of valence quarks and gluonsas dark matter would predict for them QCD size (k = 107 space-time sheet)of about electron Compton length. This does not change the QCD crosssections in the lowest order perturbation theory but makes them excellentpredictions. It also provides completely new view about how color forcedetermines the nuclear strong force indeed manifesting itself as long rangedharmonic oscillator potential, the long range of which becomes manifest incase of neutron halos of size of 2.5× 10−14 m [33]. One can also understandtetraneutron in this framework. This criterion applies also in QCD plasmaand explains the formation of liquid like color glass condensate detected inRHIC [5]. A possible interpretation for QCD size would be as a length ofthe cylindrical magnetic walls defining the magnetic body associated with uand d type valence quarks, nucleons, and nuclei. There is no need to assumethat conformal weights are complex in this phase.

b) QCD size of quark must be distinguished from the electromagneticsize of quark associated with k = 113 space-time sheets of u and d quarksand assignable to the height of the magnetic body and defining the lengthscale of join along boundaries contacts feeding quark charges to k = 113space-time sheets.

c) In the case of atomic nuclei the criterion would naturally apply to theelectromagnetic interaction energy of two nucleon clusters inside nucleus orto to self energy (Q2αem = 1). The size of the electromagnetic k = 113space-time sheet would increase by a factor about 211n, n = 1, 2, ... in thetransition to large h phase. Similar criterion would apply in the plasmaphase. Note that many free energy anomalies involve the formation of coldplasma [B8].

The criterion would give in the case of single nucleus and plasma Z ≥ 12if the charges are within single space-time sheet. This is consistent with cold

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fusion involving Palladium nuclei [8]. Since u and d quarks have k = 113,they both and thus both neutrons and protons would make a transitionto large h phase. This is consistent with the selection rules of cold fusionsince the production of 3He involves a phase transition pnpd → pnp and thecontraction of pd to p is made un-probable by the Coulomb wall whereas thetransition nnpd →nnp producing tritium does not suffer from this restriction.

The transition to a phase in which conformal weights are complex andelectro-weak bosons become light is not necessary. Thus strong and weakphysics of nuclei would not be affected in the phase transition. Electromag-netic perturbative physics of nuclei would not be changed in the process inthe lowest order in h (classical approximation) but the height of the Coulombwall would be reduced by a factor 2−11/n by the increase in the electromag-netic size of the nucleus. Also Pd nuclei could make the transition and Pdnuclei could catalyze the transition in the case the deuterium nuclei.

3.2 Criterion for the presence of exotic electro-weak bosonsand gluons

Classical gauge fields directly are space-time correlates of quantum states.The gauge fields associated with massless extremals (”topological light rays”)decompose to free part and a part having non-vanishing divergence givingrise to a light-like Abelian gauge current. Free part would correspond toBose-Einstein condensates and current would define a coherent state of darkphotons.

The dimension D of the CP2 projection of the space-time sheet serves asa criterion for the presence of long ranged classical electro-weak and gluonfields. D also classifies the (possibly asymptotic) solutions of field equations[A5].

a) For D = 2 induced gauge fields are Abelian and induced Kahler formvanishes for vacuum extremals: in this case classical em and Z0 fields areproportional to each other. The non-vanishing Kahler field implies thatinduced gluon fields are non-vanishing in general. This raises the questionwhether long ranged color fields and by quantum classical correspondencealso long ranged QCD accompany non-vacuum extremals in all length scales.This makes one wonder whether color confinement is possible at all andwhether scaled down variants of QCD appear in all length scales.

The possibility to add constants to color Hamiltonians appearing in theexpression of the classical color gauge fields allows to have vanishing colorcharges in the case of an arbitrary space-time sheet. The requirement thatcolor quantum numbers of the generator vanish allows to add the constant

27

only to the Hamiltonians of color hyper charge and isospin so that for D = 2extremals color charges can be made vanishing. This might allow to under-stand how color confinement is consistent with long ranged induced Kahlerfield.

b) For D ≥ 3 all classical long ranged electro-weak fields and non-Abeliancolor fields are present. This condition is satisfied when electric and magneticfields are not orthogonal and the instanton density A∧J for induced Kahlerform is non-vanishing. The rather strong conclusion is that in length scales inwhich exotic electro-weak bosons are not present, one has D = 2 and gaugefields are Abelian and correspond trivially to fixed points of renormalizationgroup realized as a hydrodynamic flow at space-time sheets [A1].

Quantum classical correspondence suggests the existence of electro-weakgauge bosons with mass scaled determined by the size of the space-timesheets carrying classical long range electro-weak fields. This would meanthe existence of new kind of gauge bosons.

The obvious objection is that the existence of these gauge bosons wouldbe reflected in the decay widths of intermediate gauge bosons. The remedyof the problem is based on the notion of space-time democracy suggestedstrongly by the fact that it is very difficult to define interactions betweenspace-time sheets possessing different p-adic topologies. Rather, fundamen-tal interaction vertices are possible only between space-time sheets corre-sponding to the same p-adic number field or its algebraic extension. Ofcourse, phase transitions changing the value of p-adic prime are possiblebut these destroy quantum coherence.

This makes sense. p-Adic mass calculations have demonstrated thatparticles do not correspond to single space-time sheet but feed their gaugefluxes to several space-time sheets characterized by p-adic primes. Anyparticle which couples to say weak gauge boson, must feed its weak gaugeflux to a space-time sheet with a size characterized by electro-weak lengthscale. Electro-weak gauge bosons however need not have in their space-time sheet hierarchy space-time sheets corresponding to the p-adic primescharacterizing various elementary particles.

For light exotic electro-weak bosons also the corresponding leptons andquarks would possess a large weak space-time sheet but lack the ordinaryweak space-time sheet so that there would be no direct coupling to electro-weak gauge bosons. These space-time sheets are dark in weak sense butneed not have a large value of h. This picture implies the notion of partialdarkness since any space-time sheets with different ordinary of Gaussianprimes are dark with respect to each other.

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3.3 Do Gaussian Mersennes define a hierarchy of dark electro-weak physics?

Gaussian Mersennes are defined as Gaussian primes of form gn = (1+i)n−1,where n must be prime. They have norm squared gg = 2n − 1. The list ofthe first Gaussian Mersennes corresponds to the following values of n.

2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283,353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291,85237, 106693, 160423 and 203789.

The Gaussian primes k = 113, 151, 157, 163, 167 correspond to lengthscales which are of most obvious interest but in TGD framework one cannotexclude the twin prime 239, 241 corresponds to length scales L(k) ' 160km and 320 km. Also larger primes could be of relevant for bio-systemsand consciousness. Also the secondary and higher length scales associatedwith k < 113 could be of importance and their are several length scalesof this kind in the range of biologically interesting length scales. Physicsand biology inspired considerations suggests that particular Gaussian primescorrespond to a particular kind of exotic matter, possibly also to large hphase.

k = 113 corresponds to the electromagnetic length scale of u and dquarks and nuclear p-adic length scale. For dark matter these length scalesare scaled up by a factor ∼ 211n, where n is an integer. For k = 113 oneobtains atomic length scale .8 A for n = 1. k = 151, 153, 163, 167 correspondto biologically important p-adic length scales varying in the range 10 nm-2.5µm with the scaled up length scales varying in the range 2 µm- 5 mm.

On basis of biological considerations (large parity breaking in living mat-ter) there is a temptation to assign to these length scales a scaled down copyof electro-weak physics and perhaps also of color physics. The mechanismgiving rise to these states would be a phase transition transforming the or-dinary k = 89 Mersenne of weak space-time sheets to a Gaussian Mersenneand thus increasing its size dramatically.

If given space-time sheet couples only to space-time sheets characterizedby same prime or Gaussian prime, the bosons of these physics do not coupledirectly to ordinary particles, and one avoids consistency problems due tothe presence of new light particles (consider only the decay widths of in-termediate gauge bosons [B4]) even in the case that the loss of asymptoticfreedom is not assumed.

One might argue that there must be some quantum number distinguish-ing between various Gaussian primes such as k = 113, 151, 157, 163, 167. The

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basic prediction of TGD is that elementary particles can possess complexconformal weights closely related to the non-trivial zeros of Riemann Zeta.For physical states the net conformal weight is however always real, whichgives rise to conformal confinement.

On basis of mathematical considerations alone, it would not be too sur-prising if complex conformal weights would correspond to complex primes sothat fermions with complex conformal weights would correspond to Gaus-sian primes with norm corresponding to the same value of k as for ordinaryfermion (this is suggested by the fact that p-adic thermodynamics gives thedominating contribution to fermion mass). This would guarantee darknesswith respect to ordinary fermions without assuming large h.

The simplest assumption would be that a given dark sector of this kindcorresponds to a single zero of Zeta and its conjugate. For instance, protonsand electrons could have complex conjugate conformal weights so that con-formal confinement would mean charge neutrality. This option would notimply apparent breakings of Fermi statistics. If gauge bosons with complexconformal weights correspond to Gaussian Mersennes, a phase transitionchanging k = 89 is necessary.

A good working hypothesis to start with are following.a) All particle species can be in phases with large h and in phases charac-

terized by ordinary primes or primes in algebraic extensions of p-adic num-bers, in particular Gaussian primes. Space-time sheets with different valueof h and p-adic primes (in general in algebraic extension of p-adic numbers)do not couple together quantum coherently. Each particle species, perhapseven gauge bosons, would be characterized by a collection of primes deter-mining with which particles it can interact. If so, not only theory wouldrepresent particle interactions, but also particles would represent the theoryabout their interactions in their own structure.

b) The transition to a phase with a given complex conformal weightreplaces ordinary prime with a Gaussian prime: p ' 2k → g, gg ' 2k.An interesting further working hypothesis is that ordinary gauge bosons arecharacterized by Mersenne primes and gauge bosons with complex conformalweights by Gaussian Mersennes.

c) These two phase transitions can occur independently.A question arises about the interpretation of structures of the predicted

size. The strong interaction size of u and d quarks, hadrons, and nuclei issmaller than L(k = 113) ' 2×10−4 m for even heaviest nuclei if one acceptsthe formula R ∼ A1/3 × 1.5 × 10−15 m. A natural interpretation for thislength scale would be as the size of the field body/magnetic body of systemdefined by its topologically quantized gauge fields/magnetic parts of gauge

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fields. The (possibly dark) p-adic length scale characterizes also the lengthsof join along boundaries bonds feeding gauge fluxes from elementary particleto the space-time sheet in question. The delocalization due these join alongboundaries bonds in p-adic length scale in question would determine thescale of the contribution to the mass squared of the system as predicted byp-adic thermodynamics.

3.4 Interpretations of the dark matter hierarchies

The basic prediction is a hierarchy of dark physics in which many-particlestates of the previous level become particles at the next level. The sizes ofthe corresponding space-time sheets would be scaled up by a factor of 211nin each transition to a new dark phase. Also the Gaussian Mersennes allowdark matter hierarchy. At the first level of hierarchy and for scaling factorn/v0 corresponding to n = 1, k = 151 would correspond to 2 µm, k = 157to 16 µm, k = 163 to 1.25 mm, and k = 167 to 5 mm. There are severalinterpretations of dark matter hierarchy based on values of h.

a) The hierarchy of dark matters would correspond at the level of con-sciousness to kind of abstraction hierarchy (thoughts about thoughts about....).The durations of quantum jumps coming as powers of n/v0 would correspondto this hierarchy.

b) Dark matter hierarchy would provide an interpretation for the hier-archy of infinite primes [A3] also obtained by repeatedly quantizing super-symmetric arithmetic quantum field theory using the many particle statesof given level as single particle states of the next level.

The second hierarchy is based on the zeros of Riemann Zeta.a) At the level of biology the series k = 113, 151, 157, 163, 167 could

correspond to DNA double strand and a hierarchy of coilings of DNA (coilsformed from cables with are coils formed from...) [D4]. This hierarchy doesnot correspond to a hierarchy of dark matters but to a hierarchy formedby the increasing size of the imaginary part of complex zeros of RiemannZeta with the lowest level corresponding to small h but complex conformalweights corresponding to a fixed zero or its conjugate.

b) Also these conformally confined phases are expected to define a hier-archy of dark variants since for sufficiently large number of dark particles theQ2α ' 1 criterion would force the transition to the next level of darkness.In the simplest scenario involving single conformal weight the compositesat the next level of hierarchy would necessarily have complex conformalweights since otherwise their charges would vanish. Obviously there is aclose analogy with nuclear physics. The controlling dark matter hierarchy

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would have interpretation as levels in intentional hierarchy which could con-tinue beyond k = 167 to k = 239, 241, 283 in accordance with the view thatmagnetic bodies responsible for our intentional action have astrophysicalsize.

c) The algebraic extensions of p-adic numbers define a hierarchy of darkmatter like phases and allow the analog of complex conjugation. The notionof Mersenne might generalized. For instance, for extension containing

√3

as unit Mersennes would be defined as primes of form m = (1 +√

3)n − 1having norm mm = 4n−1. Any number of form m = (1+

√Mk)n−1, where

Mk is Mersenne, would define a generalized Mersenne with norm 2kn − 1.In fact, it seems that any p-adic number field or its extension could define

a phase of matter and whether these phases are dark with respect to eachother in the sense that coherent interactions are possible only between phasecorresponds to the same extension of given p-adic number field. This leadsnaturally to the notion of partial darkness and to the notion of topologicaldemocracy in which each particle becomes a collection of p-adic primes suchthat particles having common primes in this collection can have quantumcoherent interactions with each other.

Super-canonical conformal weights are complex and the conjugation forthe algebraic extension could correspond to the conjugation of the conformalweight (note however that only primes p mod 4=3 allow an extension basedon√−1: for Mersennes this condition is satisfied).

Beraha phases exp(inπ/3), n ≥ 3, associated with the hierarchy of Jonesinclusions are always complex and would naturally relate (to at least thecomplex) algebraic extensions of p-adic numbers. n = ∞ corresponds to afull Kac Moody symmetry instead of quantum group symmetry and to atrivial quantum group phase, and to the minimal value of h. The identifi-cation of the ordinary particles as n = ∞ phase is suggestive and supportedalso by the correspondence with string model.

The complexity of Beraha phases forces to consider the possibility thatall phases with n 6= ∞, and thus also large h phases, correspond to Gaussianprimes or primes associated with the algebraic extensions of p-adic numbers.

3.5 Conformal weights of dark matter

At this stage one can only make guesses about the spectrum of conformalweights of dark matter phase.

a) Assume that the spectrum of the conformal weights is same at thehigher levels of dark matter hierarchy and corresponds to free many-particlestates and bound states of elementary particles. Free many particles of these

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many-particle states and bound states of them, etc...b) Free many-particle states of conformally confined elementary particle

families could correspond to various super-canonical subalgebras of the alge-bra whose conformal weights are expressible in terms of linear combinationsof a finite number of non-trivial zeros of Riemann Zeta. There is a naturalordering of the imaginary parts of the conformal weights and correspondingsuper-canonical sub-algebras.

The simplest working hypothesis is that a minimum number of complexzeros of Zeta and their conjugates consistent with the conservation of emcharge, color charges, and conformal weight itself characterize a given exoticparticle. This would allow a natural orthogonalization of various conformallyconfined phases and order them hierarchically. This also means that thateffective fermion statistics is not changed.

Obviously bosonic conformal weights would be differences of fermionicconformal weights appearing in the vertices so that conformal weights wouldprovide representations for diagonal quantum numbers. Neutral gauge bosonswould necessarily have real conformal weights. Contrary to the first expec-tations, dark neutral bosons, and in particular, dark photons with complexconformal weights would not be possible. This would suggest h 6= h(n = ∞)is necessary for Gaussian primes. Suitable integer multiples of single zero ofZeta would allow this in the case of electromagnetic charge.

c) Also bound many-particle states must be considered. In [A2] I intro-duced the notion of bound state conformal weight generalizing the notion ofbinding energy. The zeros of polyzetas generalizing Riemann Zeta to func-tions of N complex arguments define a candidate for the building blocks ofcomplex conformal weights of N -particle bound states at the first level ofthe hierarchy and perhaps also at higher levels of hierarchy.

The basic question is whether and how the mass of the state dependson the imaginary part of the super-canonical conformal weight. If the phasetransition in the case of fermions only changes ordinary p-adic prime to itsGaussian counterpart with the same norm, the dependence can be neglected.If in the case of massive gauge bosons p-adic prime not only becomes complexbut also preserves its Mersenne property, the p-adic length scale must changeand k = 113, 15, 157, 163, 167 are the most interesting candidates for theconformally non-trivial scales.

3.6 How dark and visible matter interact?

An important application of matter dark matter interaction is the transfor-mation of ordinary matter to dark matter and de-coherence of dark matter

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to ordinary matter.

3.6.1 Constraints on matter dark matter interaction

De-coherence transitions and formation of dark matter phase are importanceinstances of matter dark matter interaction. For instance, one can ask howbeam of gauge bosons de-coheres to ordinary matter. One can make thefollowing assumptions.

a) The transition of conformally confined dark matter involve ”laserbeams” of bosons. De-coherence of these beams is involved with the phasetransition from ordinary to dark matter. In the case of possible existinglarge h matter with real prime, laser beams are not necessary.

b) Conformal weight is conserved in vertices. For coherent interactionsof particles in the same vertex are either ordinary or dark: this means thath is same for all particles in the same vertex. Obviously, matter → darkmatter phase transition cannot cannot be described by ordinary Feynmandiagrams.

c) One can consider a description of the phase transition from perturba-tive to non-perturbative regime using generalized Feynman diagrams. De-coherence can be localized to a vertex and can be regarded as a phase transi-tion changing the real p-adic prime to a Gaussian prime with the same norm.This means that space-time sheet contracts by a factor 211n in de-coherence.

d) The phase transition corresponds to the operation p ↔ g(p) meaningto an expansion or contraction of the space-time sheet by a factor of 211nsuch that p-adic prime p ' 2k is replaced with Gaussian prime g, |g| ' 2k.Propagator lines would contain dark particles. This characterization allowsreal conformal weights also in dark phase as a special case.

3.6.2 Space-time correlates for matter dark matter interaction

One can ask what is the space-time correlate for matter dark matter inter-actions via de-coherence and what is the space-time correlate for the darkmatter hierarchy?

The de-coherence transition g(p) → p corresponds naturally to a con-traction of space-time sheet. Coherent interactions between particles at thesame p-adic level would correspond to interactions mediated via exchangesof space-time sheets characterized by the same value of p. These exchangesdefine 4-D join along boundaries bond contacts (JABs). For bound statesthese would correspond 3-D JABs. They should not be present for visibleto dark topological condensate. This would suggests that topological con-

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densation involving only wormhole contacts is the space-time correlate forthe static interaction via de-cohering boson beams.

This picture can be applied to the interaction of dark protons and or-dinary electrons. Dark protons create harmonic oscillator potential, whichis not affected by the ordinary electrons since there are only wormhole con-tacts present. The interaction mediated by de-cohering dark photon beamsis modellable by using a harmonic oscillator potential since the interactionwith dark matter is uni-directional. Quite generally, dark matter system isin some sense in the role of external observer characterizable by externalfield.

4 k = 113 dark matter and light weak bosons

A detailed model for k = 113 dark nucleon phase allows to concretize thegeneral ideas.

4.1 Stability of the large h nucleon condensate

The minimization of ground state energy could serve as a criterion forwhether the nucleon (or quark) phase with large h for k = 113 space-timesheet is formed. In the electromagnetic case the energy difference per pro-ton would define the temperature below which the generation of the phaseis possible. If the evaporation of water corresponds to the decay of darkproton phase, the boiling point 100 ◦C of water would give the first estimatefor the energy involved as 3.7× 10−2 eV. If the denaturation of DNA is duethe same mechanism, the energy in case of DNA would be around 40 ◦C.The lower temperature could due to the fact that the situation is effectively1-dimensional and the fact that thermal energy is in 3-D case equal to 3T/2.

4.2 What is the value of integer n?

Besides the canonical value v0 ' 2−11 roughly equal to the ratio of Plancklength and CP2 length also harmonics and sub-harmonics of v0 appear in theformula for h so that the scaling factor implied by the criticality conditioncome as n/v0 for sub-harmonics. The fact that the value of h before scalingup varies in the range [h(∞), 2h(∞)], gives additional flexibility due to afactor x ∈ [1, 2].

mW ∼ 100 GeV gives for the weak screening length scale Lw(89) =2π/mW ' 2.56L(89). The weak length scale for a system which has suffered

35

k = 89 → 113 weak transition followed by a phase transition to dark phaseis given by

Lw =nx

v0× 2

113−892 × Lw(89) ' 2.56× xn× L(135) ' xn Angstrom.

The scale varies in the range 1-2 Angstrom for x = 1, n = 1. The lengthof hydrogen bonds is about .9 A. n = 3 predicts that k = 113 p-adic lengthscale defining electromagnetic and weak quark size is scaled up to 3-6 A.n = 3 is favored by van der Waals radii for atoms and interatomic distancesof atoms inside bio-molecules and also by the scaling law of homeopathy[D1]. The distance between between DNA base pairs 3.4 A and the distancebetween oxygen atoms in water is about 3 A. The X-ray diffraction of watersuggests the presence of the length scale 3.7 A [1].

4.3 Large h phase with real conformal weights and ordinaryweak bosons

For the minimal option strong and weak interaction physics are not changedappreciably and thus it is the most conservative option concerning the ex-planation of known anomalies. It the case of water the minimum assumptionis that only some fraction of hydrogen nuclei go to the large h phase. Onlyone proton per two water molecules should be in this phase if H1.5O formula[2] is taken seriously. For cold fusion using Pd targets the minimum optionis that Pd is in this phase and induces the transition of Deuterium to thisphase.

Exotic chemistries [11] suggest that the electromagnetic phase transitionto large h phase can occur also for more general nuclei. That the exoticchemistries are induced via laser induced explosions fits with the fact thatthe nucleons expand by a factor of order 211. In sono-fusion [10] the rapidcontraction could induce critical value of αZ1Z2 inside single space-timesheet, and induce a phase transition increasing h so that the explosion of thebubble would not increase p-adic prime characterizing the bubble. Kind ofover-cooling phenomenon would be in question. The scaled up dark nucleushas same weak and strong interaction physics as the ordinary nucleus sinceonly electromagnetic space-time sheets are scale dup.

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4.4 Could also weak bosons make a transition to conformallynon-trivial large h phase?

The presence of conformally non-trivial weak bosons with large h wouldexplain at least partially the large parity breaking effects in living matter.The change k = 113 → k = 89 suggested by complex Mersenne propertywould introduce a scaling factor of 215/4 to the free neutron decay rateirrespective of whether bosons correspond to large h or not. A furtherphase transition changing real prime p ' 2113 to the corresponding GaussianMersenne would induce also increase the bosonic h. The n-p mass difference.78 MeV is rather near to the electron mass .51 MeV. Hence the stabilizationof neutrons would be achieved by the reduction of n-p mass difference by thepositive electromagnetic contribution to the proton’s rest mass by Coulombenergy so that the direct decay to conformally confined proton would beprevented. Note that the direct decay to ordinary proton is not possible indark phase.

Light weak bosons would also induce parity breaking effects larger thanthe normal parity breaking effects by the same factor with the scale of paritybreaking being about Lw = 2.4 A for n = 3. One can estimate the life timesof dark weak bosons by scaling since the decay rates do not depend on hin the lowest order. The lifetimes are by a factor 212 longer than those forordinary weak bosons.

4.5 Formation and de-coherence of k = 113 dark nucleonphase

Suppose that only k = 113 space-time sheets are replaced by their variantscorresponding to Gaussian Mersenne and that the transition possibly occursalso for electron. There are two options.

4.5.1 Option I: Dark weak bosons are not present in k = 113 phase

Assume that dark protons have same value of imaginary part of conformalweight opposite to that of dark electrons. In this case the phase transitioncan occur via dark photon exchange plus expansion of space-time sheet by211n factor. In sufficiently rare dark matter phase, dark electrons and darknuclei could form dark atoms of Bohr radius in the range .2− .8 mm. Theanalogs of liquids and solids with lattice constant of order dark Bohr radiuscan be imagined.

If the density of dark nucleons in condensed matter phase is high enough,harmonic oscillator approximation becomes excellent assuming that the num-

37

ber of dark electrons in low energy states is so small that it does not neutral-ize the system. The invariance of harmonic oscillator energies in the scalingof h and ω would mean that energy spectrum would remain invariant.

Ordinary electrons could behave like point charges in the external har-monic oscillator potential created by the dark proton jellium. The incoherentinteraction with ordinary electrons does not involve the Coulombic feedbackflattening the harmonic oscillator potential. Hence the resulting many nu-cleon system is completely analogous to a charged atomic nucleus, allowsmagic numbers, and is expected to be stable below some critical value of thetotal em charge. Exotic chemistries could correspond to either ordinary ordark electrons ion this kind of jellium background [11].

The over-all charge neutrality of dark nucleon phase could mean thatordinary matter carries a negative charge. DNA and cells are indeed neg-atively charged, and the mechanism generating the negative charge is thebasic mechanism for the generation of plasmoids as primitive life forms inTGD Universe and involves dark matter and magnetic walls surroundingthe source of the magnetic field. The anomalies associated with rotatingmagnetic systems of size scale of ∼ 1 meter [23, B8] involve the formationof magnetic walls up to distances of about 10 m: their height is known tobe considerably larger and one cannot exclude that that it is about 211

m. This would suggest that the heights of the magnetic walls surroundingdark nuclei are of order dark length scale 211 × L(113) = L(135) ' .4 A.Note that the height for dark dark super nuclei would be 211 × L(113) =L(157) ' .8 µm.

4.5.2 Option II: Dark weak bosons are present in k = 113 darkphase

This kind of phase would explain the large parity breaking effects in bio-matter. The requirement that vectorial Z0 charge is conserved in the phasetransition allows phase transition to occur via proton electron scattering byexchange of dark electro-weak boson, which are massless below the atomiclength scale.

The free dark neutrons resulting in dark W exchange decay rapidly todark protons if this is kinematically possible (the rate is enhanced by afactor 2(89−113)2 ' 1015/4 in proton and neutron masses do not change.Dark W− bosons would decay to electrons and dark anti-neutrinos withrate determined by the mass scale. It is however quite plausible that elec-tromagnetic repulsion between dark protons reduces the n-p mass differencebelow electron mass so that at least some subset of nuclei have stable enough

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counterparts but with anomalous weak interaction properties. The occur-rence of nuclear transmutations in living matter [9] suggests the presence ofthis kind of phase.

4.6 Dark k = 113 weak bosons and doubly dark k = 107 va-lence quarks as a minimal option explaining anomalies

One can ask what could be the minimal identification of k = 113 darkmatter explaining maximal number of anomalies, in particular the anomaliesof water. The model for atomic nuclei as nuclear strings [B6] combined withthe model for cold fusion gives important insights in this respect.

The model assumes that nuclei are nuclear strings with nucleons con-nected by long color magnetic flux tubes each having exotic quark andanti-quark at their ends. On basis of the basic criterion applied to colorinteractions of quarks, valence quarks are assumed to have large h with re-spect to color interactions at QCD level so that k = 107 color magnetic fluxtubes have length corresponding to L(keff = 129) ' .05 Angstrom. Exoticquarks have ordinary h with respect to em interactions and the space-timesheet determining the mass scale of the exotic quark corresponds to k = 127.Weak space-time sheet of the exotic quark corresponds to large h and scaledup k = 113 space-time sheet with Lw = nL(113)/v0, which is in the range 1-2 Angstrom for n = 1. This means that weak coupling strength satisfies thecriticality condition so that weak interactions become not only long rangedbut also strong.

The weak magnetic flux tubes have length of order atomic size and alsocolor magnetic flux tubes are relatively long. If criticality condition is sat-isfied, valence quarks could make a phase transition to doubly dark phasein which case L(keff = 107 + 44 = 151) = 10 nm, the thickness of cellmembrane, would characterize the size of the color magnetic bodies. In thisphase nucleons would not be visible with respect to nuclear strong inter-actions: the H1.5O formula for water suggests that this is the case for onefourth of protons in water.

This raises the question whether nuclei could combine to super-nucleisuch that color and weak magnetic flux tubes would have long straight por-tions connecting neighboring nuclei so that nuclei would correspond to highlyknotted and linked portions of single string located at interatomic distancesalong the string. This picture would be a fractally scaled down version aboutthe model for how galaxies organize into linear structures in TGD Universe[A6, A7] and would realize to some extent the ideas proposed by Thompsonbefore the advent of quantum mechanics.

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This view would dramatically modify the existing ideas about the struc-ture of condensed matter and chemistry. For instance, chemical bonds couldinvolve color and weak bonds with em and weakly charged bonds bringingin a repulsive weak interaction between the exotic quark and anti-quark ofneighboring atoms compensated by the attractive color interaction in equi-librium. Ordinary chemistry could perhaps correspond to n = 1 with Lw inthe range 1-2 Angstrom and biochemistry to n = 3 with Lw in the range3-6 Angstrom. The atoms of nuclei forming super-nuclei could have frac-tional charges. Large parity breaking effects in molecular length scale wouldemerge naturally. The reported nuclear transmutations in living matter[29, 30] could correspond to situations in which weakly charged color bonddecays weakly to a non-charged one so that stability against collapse tosingle nucleus is lost.

This picture relates also interestingly to the question whether nuclei canmake a transition to electromagnetic large h phase. When the em chargeof nucleus, nucleon cluster inside nucleus, or of a nuclear shell is so highthat the criticality condition Z1Z2αem ' 1 or its variant for self interactionenergy is satisfied, one expects something to happen.

a) The first possibility discussed in [B6] is that k = 113 em space-timesheet is scaled up so that dark em phase with Lem = nL(113)/v0 results.The weak length scale Lw for exotic k = 127 quarks would be equal to Lem,and one might wonder whether this is a consistent option.

b) The surplus positive charge of nucleons could be also transferred byan exchange of dark W (113) boson between valence quark and exotic quarkto exotic quark (W (113) would transform to W (89) by mixing during prop-agation) so that the net em and weak charges of quarks at the ends of colorbond would become non-vanishing. Tetra-neutron identified as an alpha par-ticle with two negatively charged color bonds would example of an exoticnucleus [B6]. Chemically and with respect to nuclear strong forces exoticnuclei would not differ appreciably from ordinary nucleus and no radicalmodifications of chemistry might be needed as far interactions of electronswith nuclei are considered. The resulting repulsive weak force in color bondscould however induce the fission of the nucleus or its expansion to super-nucleus. This general picture explains various anomalies like nuclear halos,tetra-neutron, tritium beta decay anomaly, cold fusion and related anomalies[B6], and as will be found, is also consistent with the anomalies of water.

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4.7 Exotic chemistries and electromagnetic nuclear darkness

The extremely hostile and highly un-intellectual attitude of skeptics stimu-lates fear in anyone possessing amygdala, and I am not an exception. There-fore it was a very pleasant surprise to receive an email telling about an articlepublished in April 16, 2005 issue of New Scientist [11]. The article gives apopular summary about the work of the research group of Walter Knightwith Na atom clusters [12] and of the research group of Welford Castlemanwith Al atom clusters [13].

The article tells that during last two decades a growing evidence for anew kind of chemistry have been emerging. Groups of atoms seem to beable to mimic the chemical behavior of single atom. For instance, clustersof 8, 20, 40, 58 or 92 sodium atoms mimic the behavior of noble gas atoms[12]. By using oxygen to strip away electrons one by one from clusters ofAl atoms it is possible to make the cluster to mimic entire series of atoms[13]. For aluminium cluster-ions made of 13, 23 and 37 atoms plus an extraelectron are chemically inert.

The proposed explanation is that the valence electrons form a kind ofmini-conductor with electrons delocalized in the volume of the cluster. Theelectronic analog of the nuclear shell model predicts that full electron shellsdefine stable configurations analogous to magic nuclei. The model explainsthe numbers of atoms in chemically inert Al and Ca clusters and generalizesthe notion of valence to the level of cluster so that the cluster behave likesingle super-atom.

TGD based model would involve the formation of a large h phase inwhich the nuclear em space-time sheets would have scaled up p-adic size.The electromagnetic k = 113 space-time sheets (em field bodies) of quarkswould have scaled up size nL(113)/v0 ' nx211 × 2 × 10−14 m, n = 1, 2, 3,x ∈ [1, 2], would have atomic size 1-2 Angstrom for n = 1. A suggestiveinterpretation is that the electric charge of nuclei is delocalized quantummechanically to atomic length scale. Hence electrons would in a good ap-proximation experience quantum mechanically the nuclear charges as a con-stant background, jellium, whose effect is indeed modellable using harmonicoscillator potential.

The general vision about nuclear dark matter suggests that the systemconsists of super-nuclei analogous to ordinary nuclei such that electrons areordinary and do not screen the Coulomb potentials of atomic nuclei. Inthis case the unscreened nucleus-nucleus electromagnetic interaction energywould define the relevant parameter as r ≡ Z2α ' 1. For Na with Z = 11one would have r = 121/137 whereas Al with Z = 13 gives r = 169/137 so

41

that the condition for the transition to em dark phase would be approxi-mately satisfied in both case.

This would support the view that atomic nuclei combine to form super-nuclei with atomic nuclei connected by color bonds and valence quarks indoubly dark state, exotic quarks in k = 113 large dark state with hw =nwh/v0, and electromagnetic space-time sheets of quarks in k = 113 in darkstate with hem = nemh/v0, nem 6= nw from internal consistency.

It would seem that ordinary conductors cannot be identified as elec-tromagnetically dark nuclear phases. The reason is that not only conduc-tion electrons but also bound electrons would experience the smoothed outCoulomb potential and the smoothing out of Coulomb potential would dra-matically affect the bound state energies.

For this option electrons themselves would be in the ordinary state andexperience this harmonic oscillator potential as a fixed background withoutaffecting it. Since the energy spectrum of harmonic oscillator potential isinvariant under the scaling of h accompanied by the opposite scaling of theoscillator frequency ω, one must consider also the electrons are in large hphase (one can of course ask whether they could be observed in this phase!).In this case electrons would however screen the nuclear charge and modifythe harmonic oscillator potential. The Compton wavelength of electronswould be scaled up by a factor n/v0 and correspond to ∼ n × 5 nm. Theatomic cluster of this size would contain roughly n×104(a0/a)3 atoms wherea is atomic volume and a0 = 1 Angstrom is the natural unit.

5 TGD based model for water based on magicdark nuclei and magic dark dark super-nuclei

5.1 The 41 anomalies of water

The following list of 41 anomalies of water taken from [1] should convincethe reader about the very special nature of water. The detailed descriptionsof the anomalies can be found in [1].

1. Water has unusually high melting point.2. Water has unusually high boiling point.3. Water has unusually high critical point.4. Water has unusually high surface tension and can bounce.5. Water has unusually high viscosity.6. Water has unusually high heat of vaporization.7. Water shrinks on melting.

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8. Water has a high density that increases on heating (up to 3.984◦C).9. The number of nearest neighbors increases on melting.10. The number of nearest neighbors increases with temperature.11. Pressure reduces its melting point (13.35 MPa gives a melting point

of -1◦C)12. Pressure reduces the temperature of maximum density.13. D2O and T2O differ from H2O in their physical properties much more

than might be expected from their increased mass; e.g. they have increasingtemperatures of maximum density (11.185◦C and 13.4◦C respectively).

14. Water shows an unusually large viscosity increase but diffusion de-crease as the temperature is lowered.

15. Water’s viscosity decreases with pressure (at temperatures below33◦C).

16. Water has unusually low compressibility.17. The compressibility drops as temperature increases down to a mini-

mum at about 46.5◦C. Below this temperature, water is easier to compressas the temperature is lowered.

18. Water has a low coefficient of expansion (thermal expansivity).19. Water’s thermal expansivity reduces increasingly (becoming nega-

tive) at low temperatures.20. The speed of sound increases with temperature (up to a maximum

at 73◦C).21. Water has over twice the specific heat capacity of ice or steam.22. The specific heat capacity (CP and CV ) is unusually high.23. Specific heat capacity; CP has a minimum.24. NMR spin-lattice relaxation time is very small at low temperatures.25. Solutes have varying effects on properties such as density and vis-

cosity.26. None of its solutions even approach thermodynamic ideality; even

D2O in H2O is not ideal.27. X-ray diffraction shows an unusually detailed structure.28. Supercooled water has two phases and a second critical point at

about -91◦C.29. Liquid water may be supercooled, in tiny droplets, down to about

-70◦C. It may also be produced from glassy amorphous ice between -123◦Cand - 149◦C and may coexist with cubic ice up to -63◦C.

30. Solid water exists in a wider variety of stable (and metastable) crystaland amorphous structures than other materials.

31. Hot water may freeze faster than cold water; the Mpemba effect.

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32. The refractive index of water has a maximum value at just below0◦C.

33. The solubilities of non-polar gases in water decrease with tempera-ture to a minimum and then rise.

34. At low temperatures, the self-diffusion of water increases as thedensity and pressure increase.

35. The thermal conductivity of water is high and rises to a maximumat about 130◦C.

36. Proton and hydroxide ion mobilities are anomalously fast in anelectric field.

37. The heat of fusion of water with temperature exhibits a maximumat -17◦C.

38. The dielectric constant is high and behaves anomalously with tem-perature.

39. Under high pressure water molecules move further away from eachother with increasing pressure.

40. The electrical conductivity of water rises to a maximum at about230◦C and then falls.

41. Warm water vibrates longer than cold water.

5.2 The model

Networks of directed hydrogen bonds H − O − H · · ·OH2 with positivelycharged H acting as a binding unit between negatively charged O (donor)and OH2 (acceptor) bonds explaining clustering of water molecules can beused to explain qualitatively many of the anomalies at least qualitatively[1].

The anomaly giving evidence for anomalous nuclear physics is that thephysical properties D2O and T2O differ much more from H2O than onemight expect on basis of increased masses of water molecules. This suggeststhat dark protons of various sizes are responsible for the anomalies. Thatheavy water in large concentrations acts as a poison is consistent with theview that the macroscopic quantum phase of dark protons is responsible forthe special biological role of water.

What proton darkness actually means? One fourth of protons of waterare not seen in neutron scattering so that they should be dark with respectto nuclear strong interactions. Valence quarks are dark at QCD level butnot at the nuclear level. The previous considerations inspired by the modelof nuclei as nuclear strings suggests a minimal answer. Dark protons shouldform super nuclei in which valence quarks connected by color and weak

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bonds are doubly dark with respect to color since otherwise they wouldbe visible in neutron scattering. This would also allow to understand whynucleons with atomic distances can belong to single super-nucleus.

Electromagnetic darkness of the protons can be explained in two man-ners.

a) k = 113 space-time sheets or valence quarks are scaled up in sizeto Lem = Lw = nL(113)/v0: Lem = Lw. Em space-time sheet of valencequarks would be identical with the weak space-time sheet of exotic k = 127quarks. This might be excluded by internal consistency.

b) Protons are transformed to neutrons connected by long positivelycharged color bonds of length of order L(keff = 151) = 10 nm correspondingto hs = n2L(107)/v2

0, n = 1, and the neutrons in the string have mutualdistances of order Lw ' nL(113)/v0, n = 3. Cell membrane thickness wouldbe coded into the structure of water. The weak repulsion between quarkand anti-quark at the ends of the bond would compensate for the attractivecolor interaction so that the system does not collapse to single nucleus norsuffer an explosion. This option looks more plausible since it explains a widerange of nuclear physics related anomalies.

5.3 Dark and ordinary hydrogen bonds

The first question is whether hydrogen bond corresponds actually to a darkproton or whether there are two different hydrogen bonds, dark and ordi-nary. There is experimental evidence for two hydrogen bonds. Li and Rossrepresent experimental evidence for two kinds of hydrogen bonds in ice in anarticle published in Nature 1993 [32], and there is a popular article ”WackyWater” in New Scientist about this finding [31]. The ratio of the force con-stants associated with the bonds is 1:2 which suggests that binding energiesscale as 2:1. This finding plus the fact that ice is also transparent excludesthe possibility that all hydrogen bonds are ordinary for ice.

If one takes seriously the evidence for H1.5O formula [2] and its expla-nation, about one fourth of hydrogen atoms should be associated with darkhydrogen bonds. This would give rise to crystal like structures, which be-sides hydrogen bond networks, could allow to understand why water is sodifferent from other molecular liquids containing also hydrogen atoms. Thelong range order of water molecule clusters would reflect basically the longrange order of the dark nuclei rather than that of hydrogen bonding net-work alone. As a matter fact, it is perhaps the presence of dark dark matterrealized as icosahedral clusters consisting of tedrahedral clusters, which dis-tinguishes water from other liquids.

45

This model is consistent with the model for hexagonal ice which assumesfour hydrogen bonds per water molecule: for two of them the molecule actsas a donor and for two of them as an acceptor. Each water molecule in thevertices of a tedrahedron containing 14 hydrogen atoms has a hydrogen bondto a water molecule in the interior, each of which have 3 hydrogen bonds tomolecules at the middle points of the edges of the tedrahedron. This makes16 hydrogen bonds altogether and if half of them is dark there are 8 darkbonds so that a fraction r = 2/7 ∼ 1/4 of protons would be dark. The magicnumbers in harmonic oscillator model are given N = 2, 8, 20, 28, 50, 82, 126so that 8 dark protons corresponds to a magic dark nucleus. H1.5O formula,if taken at face value, would predict 7 dark protons: the experimental dataare too inconclusive to even decide for the darkness [3, 4], and certainly donot allow to distinguish 7 from 8.

The possibility to divide the bonds to ordinary and dark bonds in anarbitrary manner brings in a large ground state degeneracy given by D =16!/(8!)2 unless additional symmetries are assumed and give for the systemspin glass like character and explain large number of different amorphousphases for ice [1]. This degeneracy would also make possible informationstorage and provide water with memory.

5.4 Icosahedral clusters and dark dark matter

The icosahedral model [1] for water clusters assumes that 20 tedrahedralclusters each of them containing 14 molecules combine to form icosahe-dral clusters containing 280 water molecules. Concerning the explanationof anomalies, the key observation is that icosahedral clusters have a smallervolume per water molecule than tedrahedral clusters but cannot form a lat-tice structure. What is remarkable from TGD point of view, is that thenumber 20 for the dark magic dark nuclei forming the icosahedron is also amagic number.

A possible interpretation is as a magic super-nucleus at the next levelof the dark matter hierarchy consisting of 20 first level dark nuclei withnon-vanishing imaginary part of conformal weight. The corresponding weakbosons correspond to the length scale k = 151, which means 10 nm scale forordinary matter and 2 µm length scale for corresponding dark weak bosons.Z0 magnetic bodies could induce large parity breaking effects in cell lengthscale.

The size of the magnetic body of dark dark nuclei would be 222L(113) =L(157) ' 80 nm, which happens to correspond to Gaussian Mersenne MG

157.If the height of the cylindrical magnetic walls possibly associated with dark

46

dark super nuclei corresponds to this length scale, these walls already presentin water would serve as templates for the formation of linear molecules likeDNA.

If this picture is correct, dark hydrogen bonds could be present also inother molecular liquids containing hydrogen and play a key role in under-standing the basic properties of the molecular condensed matter contain-ing hydrogen. It would be the presence of dark dark icosahedral clusterswhich, using the terminology of consciousness theory, would bring in ad-ditional ”reflective” level, and thus raise water to a unique position in thesense that biological evolution as emergence of additional reflective levelsk = 157, 163, 167, ... would rely on the special properties of water. Thiscould mean that water is to some extent a conscious intelligent entity ableto remember, learn, and form cognitive representations. This would provideadditional support for the TGD based explanation of homeopathy in termsof the ability of water to mimic various molecules [D5].

5.5 Comments on 41 anomalies

Some clarifying general comments about the anomalies are in order. Quitegenerally, it seems that it is the presence of new degrees of freedom, thepresence of icosahedral clusters, and macroscopic quantum coherence of darkmatter, which are responsible for the peculiar properties of water.

1. Anomalies relating to the presence of icosahedral clusters

Icosahedral water clusters have have a better packing ratio than tedra-hedral lattice and thus correspond to a larger density. They also minimizeenergy but cannot cannot form a lattice [1].

a) This explains the unusually high melting point, boiling point, criticalpoint, surface tension, viscosity, heat of vaporization, shrinking on melting,high density increasing on heating, increase of the number of nearest neigh-bors in melting and with temperature. It is also possible to understand whyX-ray diffraction shows an unusually detailed structure.

The presence of icosahedral clusters allows to understand why liquidwater can be super-cooled, and why the distances of water molecules increaseunder high pressure. The spin glass degeneracy implied by dark and ordinaryhydrogen bonds could explain why ice has many glassy amorphous phases.The two phases of super-cooled water could correspond to the binary degreeof freedom brought in by two different hydrogen bonds. For the first phaseboth hydrogen atoms of a given water molecule would be either dark orordinary. For the second phase the first hydrogen atom would be dark and

47

second one ordinary.Since icosahedral clusters have lower energy than a piece of ice of same

size, they tend to super-cool and this slows down the transition to the solidphase. The reason why hot water cools faster would be that the number oficosahedral clusters is smaller: if cooling is carried with a sufficient efficiencyicosahedral clusters do not form.

b) Pressure can be visualized as a particle bombardment of water clusterstending to reduce their volume. The collisions with particles can inducelocal transitions of hexagonal lattice to icosahedral structures with a smallerspecific volume and energy and induce local melting. This would explainthe low compressibility of water and why pressure reduces melting point andthe temperature of maximum density and viscosity.

c) The increase of temperature is expected to reduce the number oficosahedral clusters so that the effect of pressure on these clusters is not solarge. This explains the increase of compressibility with temperature below46.5◦C. The fact that the collapse of icosahedral clusters opposes the usualthermal expansion is consistent with the low thermal expansivity as well asthe change of sign of expansivity near melting point. Since the square ofsound velocity is inversely proportional to compressibility and density, alsothe increase of speed of sound with temperature can be understood.

2. The presence of dark degrees of freedom and spin glass degeneracy

The presence of dark degrees of freedom and the degeneracy of darknucleus ground states could explain the high specific heat capacity of water.The reduction of dark matter degrees of freedom for ice and steam wouldexplain why water has over twice the specific heat capacity of ice or steam.The possibility to relax by dissipating energy to the dark matter degrees offreedom would explain the short spin-lattice relaxation time. The fact thatcold water has more degrees of freedom explains why warm water vibrateslonger than cold water.

Also the high thermal and electric conductivity of water could be under-stood. The so called Grotthuss mechanism [1, 34] explaining OH− and H+

mobilities (related closely to conductivities) is based on hopping of electronof OH− and H+ in the network formed by hydrogen bonds and generalizesto the recent case. The reduction of conductivity with temperature wouldbe due to the storage of the transferred energy/capture of charge carriers tothe water molecule clusters.

3. Macroscopic quantum coherence

The high value of dielectric constant could derive from the fact that

48

dark nuclei and super-nuclei are quantum coherent in a rather long lengthscale. For curl free electric fields potential difference must be same alongspace-time sheets of matter and dark matter. The synchronous quantumcoherent collective motion of dark protons (and possible dark electrons)in an oscillating external electric field generates dark photon laser beams(it is not clear yet whether these dark laser beams are actually ordinarylaser beams) de-cohering to ordinary photons and yield a large dynamicalpolarization. As the temperature is lowered the effect becomes stronger.

6 Dark matter, long ranged weak force, condensedmatter, and chemistry

The challenge of understanding the effects of dark weak force in condensedmatter and chemistry is not easy since so many options are available. Theguidelines to be used are maximal conservatism, the model for the nuclearphysics, the general criterion for the transition to dark phase, and intriguinghints that dark weak force could play important role not only in biochemistrybut also in ordinary condensed matter physics contrasted with the fact thatisotopic independence is not visible in the physics of condensed matter andin chemistry.

6.1 What is the most conservative option explaining chiralselection?

The long range weak interactions should produced parity breaking respon-sible for the chiral selection. The first thing that comes in mind is thatordinary nucleons suffer a phase transition in which the p-adic prime char-acterizing weak space-time sheets increases, perhaps to one of the GaussianMersennes k = 113, 151, ...

There are objections against this idea.a) The criterion αwQ1Q2 ' 1 for the transition to dark phase does not

apply at weak space-time sheets so that ordinary quarks should not performthis transition.

b) If ordinary nucleons make the transition to the dark weak phase withk ≤ 113, very large Z0 Coulombic interaction results and isotopic depen-dence of chiral symmetry breaking is predicted.

c) Repulsive weak interaction would provide a nice explanation for thehard core of the interaction potential in van der Waals equation for liquidphase. Isotopic dependence is again the problem.

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The view about nuclei based on dark valence quarks and exotic quarkswith ordinary value of h but with k = 127 electromagnetic space-time sheet[B6] suggests a maximally conservative model for chiral selection consistentwith these objections.

a) Assume that nucleons are not affected at all in the transition and thatnothing happens in the transition even at the level of exotic quarks so thatthey must have weak space-time sheets with size at least of order atom size.

b) The weak space-time sheet of exotic quarks cannot correspond tok = 89 since this would be seen in the decay width of the ordinary electro-weak gauge bosons. Neither can the weak space-time sheet of exotic quarkscorrespond to any space-time sheet of ordinary quarks or their dark coun-terparts. This leaves as a scaled up k = 113 ordinary em space-time sheetwith atomic size or k = 151 space-time sheet with ordinary value of h. Itseems natural to assume that all leptons and quarks appear with large weakspace-time sheets. These leptons need not correspond to the p-adic lengthscales of ordinary leptons although this is possible.

b) It would be nice to have weakly charged nuclei. Weak charges shouldnot be however too large. This is achieved if some of the color bonds contain-ing exotic quark and anti-quark at their ends carry net em charge and thusalso weak charge. This hypothesis allows to understand tetra-neutron as analpha particle containing two negatively charged color bonds and predictsentire spectroscopy of exotic nuclei containing charged color bonds [B6].

Pseudo-deuteron would be second example of exotic nucleus and wouldconsist of two neutrons connected by a positively charged color bond. Coldfusion could be understood in terms of absence of Coulomb wall in thecollision of ordinary proton with pseudo-deuteron since em space-time sheetsof valence quarks in proton and exotic space-time sheets in pseudo deuteronwould be different.

c) Instead of ordinary neutrinos transformed to dark neutrinos in weaksense, neutrino species with weak space-time sheets would be present andparticipate in the weak screening together with exotic W+ bosons and pos-sible exotic counterparts of electrons. The Gaussian Mersennes associatedwith k = 151, 157, 163, 167 define good candidates for the space-time sheetsof exotic leptons.

d) Also higher levels of darkness would be allowed by the standard cri-terion applied to say molecules. Also a hierarchy of colored dark matterscould emerge as nuclei get net color charge and combine to form moleculeswhich are color singlets.

Consider now the implications of this picture.a) The repulsive weak interaction between exotic quarks with net em

50

and weak charge would explain the hard core of the interaction potential invan der Waals equation without isotope dependence.

b) Bio-control could occur by the variation of weak screening using W+

bosons and exotic neutrinos. The resulting parity breaking effects would belarge below the length scale Lw. Chiral selection would not have isotopedependence.

6.2 Questions related to ordinary condensed matter and chem-istry

Consider first some questions related to ordinary condensed matter andchemistry.

1. Could electromagnetic darkness relate to the properties condensedmatter?

The purely electromagnetic dark phase for k = 113 space-time sheetswithout dark weak bosons implies that atomic nuclei possess field bodies ofatomic size, and one can wonder how this might relate to the basic proper-ties of condensed matter. For instance, the linking of magnetic flux tubesof field bodies of different nuclei might have some role in quantum infor-mation processing, if the general vision of [C1] about topological quantumcomputation in terms of linking of magnetic flux tubes is taken seriously.

2. Does repulsive weak force relate to the stability of condensed matter?

The Coulomb repulsion of electrons could be enough to explain van derWaals equation of state. One can still wonder whether the dark weak forceeffective below the length scale Lw(dark) could have something to do withthe repulsive core in van der Waals equation of state and with the sizes ofatoms in condensed matter.

The low compressibility of condensed matter indeed suggests that repul-sive Z0 force between constituent molecules is present or at least appearswhen one tries to compress condensed matter. The long ranged weak inter-actions between exotic quarks associated with color bonds of condensed mat-ter nuclei would explain this without predicting non-trivial isotopic effectsin van der Waals equation. The most conservative option is that compres-sion induces a phase transition to a phase in which nuclei contain chargedcolor bonds and generates strong Z0 repulsion in the length scale of atomicradius. The fact that the density of water is reduced above freezing pointwhen pressure is increased or temperature reduced might have explanationin terms of this mechanism.

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The orthodox physicist would presumably argue that the mere electro-magnetic interactions allow to understand the value of the atomic radius.The following argument challenges this belief in case of heavy atoms.

The size of atom in the absence of the classical dark weak forces canbe estimated from the expression of the radius of the orbital n given byrn = n2a, where a = a0/Z is the radius of the lowest electronic orbital, andfrom the fact that a given orbital contains 2n2 electrons. In a reasonableapproximation one has Z = 2n3

max/3 and nmax = (3Z/2)1/3. In this approx-imation the radius of the largest orbital identifiable as the atomic radius rZ

is

rZ = (32)2/3 a0

Z1/3. (2)

Indeed, at distances above this radius the atom looks more or less neutralsince electrons screen the nuclear charge completely. This gives an estimatefor the density of the condensed phase consisting of atoms with nuclearcharge Z.

ρ =49AZ × mp

a30

. (3)

In case of iron (A = 55, Z = 26) one would have ρ ' 635 kg/dm3: the valueis roughly 100 times higher actual value ρ = 7.8 kg/m3 at room temperature!

Thus the radii of heavy nuclei seem to be too large in the standardphysics framework. The transition to a phase in which charged color bondsare present is expected to be especially probable in the case of heavy nucleiand a generation of repulsive Z0 force might explain the radii.

3. Could the repulsive weak core relate to the stability of chemical com-pounds?

Could the long ranged repulsive weak force relate the typical lengths ofchemical bonds? Could it even make possible valence quark approximation?Since the generation of weakly charged color bonds and even color bondsconnecting different atomic nuclei does not involve isotopic dependence, onemust consider the possibility that these forces might be involved even withthe physics of chemical bonds.

For instance, the generation of a chemical bond might involve formationof state containing a component in which the two nuclei have generated colorbonds with opposite charges creating additional attractive force. One can

52

also consider the possibility that nuclei generate anomalous electromagneticcharge of same sign so that a repulsive weak force between atoms results.This would give rise to a hard sphere behavior essential for the notion ofvalence.

At least at classical level one can question the hard sphere behavior ofatoms assumed implicitly in the models of molecules based on molecularorbitals and allowing to treat full electronic shells as rigid structures so thatonly valence electrons are dynamical and give rise to shared orbitals. Onecan argue that purely electromagnetic atoms/molecules do not behave likehard spheres and that all electrons should be treated like valence electronsmoving in the combined Coulomb field of the two nuclei whose distance isnot fixed by the molecular size.

Since electrons are very light, one could classically regard the electroniccloud as a perfectly conducting rapidly deformable shell. When atoms ap-proach each other the electronic charge density arranges in such a mannerthat it minimizes the Coulombic interaction energy between nuclei by pre-venting the penetration of the nuclear electric field of the other atom throughthe electronic shell. There the encounter of atoms would be more like acollision of point nuclei surrounded by highly deformable smooth electronmattresses than that of hard spheres.

What could go wrong with this argument? Fermi statistic might changethe situation and make closed electronic shells to behave like rigid chargedspheres.

6.3 Dark-to visible phase transition as a general mechanismof bio-control

Dark-to-visible phase transition reduces the de-Broglie wave lengths by afactor v0/n. This would essentially code patterns in dark length scale topatterns of visible matter in much shorter length scale and make possiblelong length scales to control short length scales in a coherent manner. Thisphase transition could occur separately on em, weak, and color space-timesheets. For instance, the dark phase of hydrogen ions in case of proton neednot involve dark weak phase.

The hierarchy of dark matters defines naturally a control hierarchy or-dered with respect to time and length scales. Dark electrons would be func-tional at the lowest level of the control hierarchy whereas dark neutrinoswould naturally appear at the higher levels.

The strange properties of water can be understood to a great extent if afraction of protons has made a transition large h phase in electromagnetic

53

sector. This does not require anything anomalous in the weak and coloredsectors.

The criterion for the transition is that a system consisting of sub-systemswith charges Z1 and Z2 makes a transition to dark matter phase reads asαemZ1Z2 ' 1.

Option I: If this criterion applies to self interactions as such, it wouldgive in the case of atomic nuclei Zcr = 12 (Mg).

Option II: If full nuclear shells are non-interacting, as one expects onbasis of Fermi statistics, the criterion could be interpreted as stating thatonly nuclei having Z = 2 + 6 + 12 = 20 (the self interaction of the full thirdshell would induce the transition) can make this transition [B6]. That Caions (Z = 20) satisfy this condition would conform with the fact that playa unique role in bio-chemistry and neurophysiology.

Option III: If the criterion does not apply to self interactions and onlyfull shells interact, the condition would be that the nucleus contains nucleonclusters with charge Z1 = Z2 = 20 giving Zcr = 40 if the critical interactionis between separate Z = 12 shells. TGD inspired view about nuclear physics[B6] based on dark valence quarks and k = 127 exotic quarks with ordinaryvalue of h at the ends of long color bonds responsible for nuclear strong forcesuggests that nuclei could be regarded as collections of linked and knottednuclear strings and that the linking of magic nuclei produces new especiallystable nuclei.

Cold fusion with Pd catalyst [28] having Z = 46 could involve localtransitions of Pd catalyst to k = 113 dark matter phase and perhaps alsothe transition k = 89 → 113.

For option III trace elements with Z ≥ 40 should play a key role in livingmatter inducing phase transitions of lighter nuclei to dark phase as the modelfor cold fusion suggests. There is some support for this interpretation.

a) DNA is insulator but the implantation of Rh atoms in DNA strandsis known to make it super-conductor [59], perhaps even super-conductor.Dark electrons obviously define a good candidate for the current carriers.

b) The so called mono-atomic elements [6] claimed by Hudson to possessvery special physical properties have explanation in terms of dark matterphase transition [D1] and have Z ≥ 44. Interestingly, Hudson claims thatmono-atomic elements have not only very special biological effects but alsoaffect consciousness, and that 5 per cent of brain tissue of pig by dry matterweight is Rhodium (Z = 45) and Iridium (Z = 77).

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6.4 Long ranged weak forces in chemistry and condensedmatter physics

According to the model of water, one fourth of hydrogen ions would be indark phase such that k = 113 space-time sheet has transformed to largeh phase and would have size of order atomic radius. This would suggeststhat that the atomic size could be understood in terms of large h associatedwith k = 113 electromagnetic space-time sheet. Weak interactions in thisphase could be normal. Quantum classical correspondence forces howeverto consider the possibility for which also long range weak force is present-

6.4.1 Exotic nuclear quarks as sources of long ranged weak force

One can a consider a copy of weak physics for which weak space-time sheetsof particles have k > 89, say k = 113. This would imply strong paritybreaking effects in k = 113 length scale. If this transition is followed bya transition of k = 113 space-time sheet to dark matter phase with largevalue of h, the length scale Lw(dark) = nL(113)/v0 in which strong paritybreaking effects occur corresponds to atomic length scale. This kind ofphase could explain chiral selection in living matter and dark weak bosoncondensates and dark quarks and leptons might play a fundamental role inbio-control.

The criterion for the transition to the large h phase does not suggestthat this transition could happen to ordinary quarks and leptons. Alsothe fact the absence of non-trivial isotopic dependence in chemistry andcondensed matter supports the conservative view ”once vacuum screened-always vacuum screened”.

The TGD based model of atomic nuclei however involves besides darkvalence quarks color bonds having k = 127 quarks at their ends and theirweak space-time sheets cannot correspond to k = 89 since this would bereflected in the decay widths of weak bosons. One possibility is that theweak space-time sheet corresponds to k = 113, possibly with large h.

TGD based identification of tetra-neutron is as an alpha particle con-taining two negatively charged color bonds [B6]. Since there is no reason toexpect that tetra-neutron would be a rare exception, one expects that or-dinary nuclei of condensed matter can make transition to a phase in whichsome color bonds are em charged and thus carry also weak charges creat-ing long ranged weak forces and parity breaking without the un-acceptableisotopic independence. The unavoidable long ranged weak and color fieldsassociated with non-vacuum extremals suggest even more radical possibil-

55

ity. The nuclear strings associated with neighboring condensed matter nucleicould fuse to single nuclear string so that nuclei would be color and weaklycharged and could carry fractional em charges.

Below Lw(dark) atoms whose nuclear color bonds carry net weak chargeswould look like Z0 ions and condensed matter in this phase would be kindof Z0 plasma. The weak forces could be screened by vacuum charges abovethe length scale Lw(dark) just as they are screened usually. Dark weakbosons would have mass obtained by scaling down the intermediate gaugeboson masses by a factor 2−12 for k = 113. An essential point is that theZ0 charge density of nuclei would be constant below Lw rather than thatcorresponding to Z0 charges with nuclear size. This makes Z0 screeningby particles much more easier and the question is not whether to achieveprecise enough screening in say nuclear length scale but in what scale it ispossible to vary the degree of screening.

6.4.2 Could long ranged weak forces be key players in bio-catalysis?

Bio-catalysis involves chiral selection in an essential manner which suggeststhat weak force is involved. This inspires the question about the underlyingmechanisms controlling the assembly and de-assembly of bio-molecules.

1. Bio-catalysis and phase transition to a phase containing charged colorbonds?

The considerations related to van der Waals equation and the fact thatcolor bonds could be unstable against beta decay via the emission of light Wboson nucleon suggest that nuclei could tend to develop color bonds with thesame sign of Z0 charges. Anomalous em charges could vanish if the transitioninvolves an emission of a dark W boson charging color bond transforming toordinary weak boson by de-coherence and absorbed by nucleon. This kindof transition could proceed spontaneously as a two-nucleon process if thenuclei are close enough as in the situation when liquid is compressed.

If so, the resulting weak forces tend to de-stabilize these molecules. Therange Lw ' 2.56L(89) gives for this force a scale about 2.56 × L(keff =133) ' 1.3n Angstrom if scaled directly from the Compton length of in-termediate gauge boson assuming the scaling h → nh/v0. n = 3 gives thelength scale of the typical chemical bond in DNA.

The molecules need not become un-stable in the phase transition tothe phase containing charged color bonds. The phase transition could onlyreduce the binding energies of the chemical bonds and facilitate chemicalreactions serving thus as a catalyst.

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Dark molecules of form AHn, where A is arbitrary atom and Hn refersto n hydrogen atoms be in the role of biological hardware since they are notaffected appreciably by this kind of phase transition. The basic moleculesof life are indeed molecules of type CHn, OHn, NHn, which could of coursebe also partially dark.

2. The variation of the strength of the Z0 force as a control mechanism

The variation of the strength of the repulsive Z0 force could be achievedby varying the density of screening particles. To be effective this tool shouldallow sharp enough length scale resolution and the resolution is determinedby the p-adic length scale of the screening particle. The situation is dra-matically improved by the fact that the Z0 charge density to be screened isconstant below Lw. Hence a constant Z0 charge density of screening chargesis enough to achieve a complete screening. The control of the degree of Z0

ionization rather than control of Z0 charge density would be in question.

3. What distinguishes between ordinary condensed matter and livingmatter?

If weakly charged color bonds appear already in ordinary condensedmatter and give rise to the repulsive core in van der Waals equation ofstate, one can wonder what is the real distinction between living matter andordinary condensed matter. The difference might relate to the value of nfor the transition h → nh/v0 for electromagnetic space-time sheets. n = 1could correspond to ordinary condensed matter with Lw in the range of 1-2Angstrom and n = 3 to living matter with Lw in the range 3-6 Angstrom.Water could differ from other condensed matter systems in that it wouldhave n = 3 for one fourth of hydrogen ions.

A second question relates to the identification of the weak space-timesheet of exotic quarks. Can one assume that the weak space-times sheet ofexotic quarks and em space-time sheet of valence quarks in dark em phaseboth correspond to k = 113 with large h? This hypothesis can be defended:below Lw dark electro-weak symmetry is not broken so that em and weakinteractions should take place at the same space-time sheet.

6.5 Z0 force and van der Waals equation of state for con-densed matter

Most physicists probably think that van der Waals equation of state repre-sents those aspects of condensed matter physics which have been thoroughlyunderstood for long time ago. Approximate isotopic independence of the ba-

57

sic parameters of the state equation provides support for this belief. Isotopicindependence does not however exclude the role of long ranged weak forces ifthey are associated with exotic k = 127 quarks appearing in the TGD basedmodel of nucleus [B6]. The decay widths of weak bosons require that exoticweak bosons correspond to some other p-adic length scale than k = 89, saykeff = 113 + 24 = 137 for large h or k = 151 for ordinary h. The pres-ence of em charged color bonds in ordinary nuclei would provide them withanomalous em and weak charges and bring in long ranged weak force.

One can imagine various scenarios for how dark weak forces might entercondensed matter physics.

a) It might be energetically favorable for the ordinary condensed matternuclei to be in a phase containing charged color bonds. By the chargeindependence of strong interactions this would not considerably affect thenuclear physical properties of condensed matter nuclei. The hard core of theinteraction potential in van der Waals equation could be seen as a signatureof dark weak force.

b) The nuclei could be ordinary in the ordinary liquid phase (waterforming a possible exception) so that long ranged weak forces need not bepresent. The low compressibility of the liquid phase could however be dueto a phase transition of nuclei inducing charged color bonds by exotic weakdecays of exotic quarks. This would induce a repulsive weak force felt inthe length scale Lw of order 3 − 6 Angstrom for k = 113 and h → nh/v0,n = 3. The dark weak force becoming visible only when liquid is compressedwould explain the hard core term in van der Waals equation. The energyprovided by the compression would feed in the energy making possible thephase transition not occurring spontaneously. Sono-luminescence [47] couldrepresent a situation in which the phase transition occurs.

The phase transition generating charged color bonds could be inducedby the direct contact of the nuclear em field bodies of exotic quarks andanti-quarks with size associated with any nucleus having A > 1 and havingfield em field body with size L ∼ nL(113)/v0 of order atomic radius (thispoint is discussed in detail in the model of nuclei based on color bonds [B6]).

Both options predict isotopic independence of compressibility and essen-tially standard nuclear physics. The explanation for the anomalous behaviorof water above its freezing point, in particular the reduction of density as thetemperature is lowered or pressure increases, could be basically due to theappearance of additional color bonds in oxygen nuclei during compression.

These considerations raise the question how weak forces reveal their im-plicit presence in the basic argumentation leading to van der Waals equationof state. In the sequel the deduction of van der Waals discussed in more de-

58

tail to make more explicit the origin of the hard core term.

1. Van der Waals equation of state

Van der Waals equation of state provides the simplest thermodynamicalmodel for gas-liquid phase transition. The equation can be derived fromthermodynamics using the following assumptions.

a) The partition function ZN for a condensed matter system consistingof N identical particles codes the thermodynamical information and can bededuced once the Hamiltonian of the system is known.

b) It is assumed that the Hamiltonian separates into a sum of singleparticle Hamiltonians H =

∑Hi = T + U =

∑Ti +

∑Ui. Single particle

Hamiltonian consists of a sum of the kinetic energy Ti, the energy associatedwith internal degrees of freedom (such as rotational degrees of freedom ofthe molecule), and the potential energy Ui =

∑j 6=i uij .

c) The potential energy uij is assumed to depend on the relative coordi-nate ri − rj only and to be large and positive at short distances and vanishrapidly at large distances. Also spherical symmetry can be assumed in agood approximation. Above 2r0, r0 molecular radius, u is assumed to besmall and negative and in this manner generate an attractive force, whichcan be assumed to be of electromagnetic origin.

Consider now the approximate deduction of the equation of state.a) The partition function factors into a product of the partition function

ZidN of ideal gas and a term defined by the potential energy terms in the

Hamiltonian of the whole system.

Z = ZidN (T )×QN (T, V ) ,

QN (T, V ) =1

V N

∫ ∏i

dViexp(−U/T ) . (4)

b) The standard manner to derive an approximate form of the partitionfunction, free energy and pressure in turn providing the equation of stateis based on the so called virial expansion using the elementary multiplica-tive properties of the exponential function exp(−U/T ) =

∏i,j exp(−uij/T )

appearing in QN . In the lowest non-trivial order one has

QN (T, V ) ' N2

VI2 ,

I2 =∫

dV λ(r) ,

λ(r) = exp(−u12(r)/T )− 1 . (5)

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The integrand in this expression is in a good approximation equal to −1inside the sphere of radius 2r0 defined by the minimal distance between themolecules of radius r0 and positive outside this sphere and approaches zerorapidly.

c) Quite generally, one can write QN as

QN (T, V ) ' 1 + N × n

2× I2 ' (1 +

nI2

2)N ,

n =N

V. (6)

The improved approximation is dictated by the fact that free energy mustbe an extensive quantity. For the free energy F = −T ln(Z) one obtains anapproximate expression

F = NF id −NTnI2 . (7)

For the pressure P = −(∂F/∂V )T,N one obtains

P = nT (1− nI2/2 + · · ·) . (8)

d) The value of I2 can be calculated approximately by dividing the inte-gration region to two parts. The first part corresponds to a sphere of radius2r0 (r0 is the radius of molecule) inside which λ12 = −1 could be interpretedin terms of the approximate vanishing of the exponential of the interactionpotential behaving like 1/r. The second part corresponds to the exteriorof the sphere of radius 2r0, where λ is assumed to have positive but smallvalues so that the exponential can be approximated by the first two termsof the Taylor series with respect to u12 This gives

I2 ' −4π3 (2r0)3 + 4π

T

∫drr2u12(r) ≡ 2b− 2a/T . (9)

Note that a > 0 implied by u12 ≤ 0 holds true.e) The resulting equation of state is

P + n2a = nT (1 + nb) . (10)

This equation is second order in n and does not give the characteristic cuspcatastrophe associated with the van der Waals equation.

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e) The approximation

1 + nb ' 11− nb

(11)

holding true for nb << 1 and then extrapolating to a region where thiscondition does not hold true. This gives the van der Waals equation of state

(P + n2a)(1− nb) = nT (12)

allowing a simple description of gas-to liquid phase transition requiring thatat least third power of n appears in the equation of state. The equationallows an attractive physical interpretation. Pin ≡ n2a can be identified asinternal pressure mainly due to the attractive van der Waals force and 1-nbtells the fraction of free volume so that PtotVfree = NT holds true.

This trick is believed to take into account the neglected higher orderterms in the virial expansion. The proper justification comes from the catas-trophe theory [40]. The virial expansion gives all orders in n to the righthand side of Eq. 10 and by the general theorems of catastrophe theorycusp catastrophe is the singularity associated with a state equation withtwo control variables a and b. What the cusp catastrophe means is thatthree values of n satisfy the equation of state for given values of P and T .Two of these values correspond to stable phases, liquid and gas, the lowerand upper sheets of the cusp, whereas the intermediate sheet of the cuspcorresponds to an unstable phase.

In TGD framework a could be interpreted as characterizing purely elec-tromagnetic interactions above the critical radius r0 and and b both em andlong ranged interactions below r0. The emergence of repulsive Z0 interac-tions below the critical radius r0 would serve as a physical definition for r0.The fraction of free volume 1−nb would differ from unity because repulsivedark weak forces enter in play when the number density n tends to becomelarger than 1/b.

In a very optimistic mood one might provocatively claim that the classi-cal Z0 Coulombic force allows to understand why the hard core approxima-tion behind van der Waals equation works and that the setting on of darkweak force provides a precise first principle definition for the notion of themolecular radius. The criticality implied by the Z0 Coulombic force wouldreflect itself as the criticality of the liquid-gas phase transition. Obviouslythe parameter b contains very little information about the details of the

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Z0 Coulombic interaction energy besides the fact that the phase transitioncharging some color bonds weakly occurs when molecules are at distancer < r0. The calculation of the value of the parameter a should reduce tostandard electromagnetic interactions between molecules.

6.6 Z0 force and chemical evolution

Although long ranged weak forces manages to hide themselves very effec-tively, they leave some tell tale traces about its presence. The most spectac-ular effect is chiral selection which is extremely difficult to understand in thestandard model. Also the mysterious ability of noble gases to act as anes-thetes [60] could be understood as being due to dark weak forces. If a phasetransition charging some color bonds of the noble gas nuclei increasing orreducing Z occurs, noble gas atoms behave chemically as ions. A discussion(somewhat obsolete now) of the mechanism can be found in [D7].

Classical Z0 force might also make itself visible by delicate chemicaleffects due to the fact that the classical Z0 charge of the hydrogen atomvanishes. Since the exotic Z0 charges of proton and electron necessarilyvanish by the absence of color bonds the prediction is that proton and elec-tron are in a completely exceptional role in chemistry, and in biochemistryin particular. Certainly this is the case: consider only the role of protonand electron in biochemistry (say in metabolic cycles and in polymeriza-tion). Furthermore, Z0 force seems to be the key player in the biochemicalevolution in TGD Universe: molecular stability could be controlled by thethe possibility to generate charged color bonds and by the screening of longranged weak forces.

Enzymatic action, known to involve chiral selection, can be based on thecontrol of the strength of the classical Z0 force by varying the densities ofthe Bose-Einstein condensates responsible for the Z0 screening. Metabolisminvolves basically the chopping of the nutrient molecules to pieces and theirre-assembly. The chopping into pieces could be partially achieved by weak-ening the screening of the classical Z0 force locally. The sizes of the enzymesand ribozymes are rather large and vary in the range 10-20 nm. This is noteasily understood in the standard chemistry context but is what one expectsif k = 151 weak bosons are involved.

An interesting hypothesis is that chemical evolution has proceeded viaa sequence of phase transitions producing dark weak bosons correspondingto Gaussian Mersennes Gk = (1 + i)k − 1, k = 113, 151, ... as k = 89 → 113followed by k = 113 → 151 → 157 → 163 → 167 → ....

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6.7 Parity breaking effects at molecular level

The observed parity breaking effects at molecular level are large: a naturalunit for molecular dipole moments is one Debye: e10−10 m ∼ eL(137). Thisscale compares favorably with the k = 113 weak length scale Lw = nxAngstrom, x ∈ [1, 2], n = 1, 2, 3. The larger the value of n, the larger thescale of parity breaking. The breaking of the mirror symmetry appears atgeometric level and this kind of symmetry breaking does not require largeparity breaking at the level of physics laws. The parity breaking howevertakes place in a much deeper manner: only second chirality of two mirrorimage molecules appears in Nature and an unsolved problem is to understandthis selection of the molecular chirality.

The axial part of weak forces, in particular Z0 force, suggests a firstprinciple explanation for the molecular parity breaking. A phase transitiongenerating dark weak force below length scale Lw would induce axial forceimplying different energies for mirror images of molecule.

6.7.1 Mechanism of parity breaking

One can imagine two mechanisms of chiral selection. For the first mecha-nism the classical Z0 interactions between the atoms of the molecule leadto a chiral selection. If equilibrium positions correspond to the minima ofZ0 Coulomb energy, the parity breaking effect, being proportional to thegradient of Z0 scalar potential, however vanishes. Of course, the net forceinvolves both electromagnetic and Z0 contributions so that the equilibriumpositions do not actually correspond to the minima of Z0 Coulomb poten-tial. Proton is an exception because of its small vectorial Z0 charge andby the fact that it is the only nucleus not containing color bonds (assumingthat self bonding does not occur).

Second mechanism is based on the presence of an external Z0 electricfield and to the fact that the energies of a chiral molecule and its mirrorimage in an external Z0 electric field are different. In this case the paritybreaking contributions of the individual atoms of the molecule to the energyare in general non-vanishing and lead to chiral selection. The presence ofclassical Z0 electric fields in bio-matter would not be surprising since bio-matter is also ordinary electret. Spontaneous Z0 electric polarization mightbe an essential element of chiral selection and lead to energy minimization.This kind of phase transition might be induced by a rather small externalperturbation such as bombarding of a system containing both chiralitieswith neutrinos or electrons.

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6.7.2 Detailed form of the parity breaking interaction

Consider first in more detail the form of parity breaking interaction.a) In molecular physics the minimization of the energy for electronic con-

figurations selects the ground state configuration for atoms in the molecule(this is essentially due to the small mass ratio me/mp).

b) The parity breaking force is proportional to the axial part of weakisospin, which is of same magnitude for all particles involved. Axial force isproportional to the gradient of Z0 scalar potential created by exotic quarksin color bonds. Axial force is also inversely proportional to the mass of theparticle involved.

The mass scale of exotic quarks is determined by k = 127. The hypoth-esis that lepto-hadrons are bound states of colored excitations of leptonspredicts also k = 127 for their mass scale and colored electrons would haveessentially the same mass as electrons. One can make only guesses aboutthe p-adic mass scale of exotic (possibly dark) neutrinos and electrons. Themaximally non-imaginative hypothesis is that the scales are same as for or-dinary leptons. In this case the mass would by a factor of about 10−6 smallerfor dark k = 169 neutrinos with mass about .1 eV than for exotic quarkswith mass ∼ .1 MeV if p-adically scaled down from that of ordinary quarks[B6]. Therefore the presence of dark neutrinos could induce the dominatingparity breaking effects. For this option the Z0 binding energy would bemuch larger than neutrino mass for reasonable values of nuclear Z0 charge,which would favor the Z0 screening by neutrinos.

c) The parity breaking Z0 interaction energies of exotic k = 127 quarkand anti-quark at the ends of color bond are of same sign in three casescorresponding to pion type color singlet bonds q↑q↓ and em and color chargedbonds u↑d

↑ and d↑u↑. Thus the parity breaking interaction does not requirethe presence of color charged bonds and is in principle present for all nucleibut can of course cancel in good approximation if the net spins of k = 127quarks and anti-quarks do not cancel separately.

d) For Fermi sea of dark neutrinos the parity breaking effects on energyare proportional to spin and sum up to zero if the number of neutrinos iseven. Note however that complete screening is not required.

Consider now a more quantitative estimate.a) The axial part of the Z0 force acting on neutrinos is given by

VNPC ' ±αZQAZ(ν)QV

Z (ν)1

m(ν)S · ∇VZ(r) . (13)

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b) The order of magnitude for the energy difference of a configurationand its mirror image is obtained as the difference of axial interaction energiesfor configurations related by reflection. Consider a particle with Z0 chargeQZ,1 and mass m experiencing the axial Z0 field created by a nucleus withanomalous Z0 charge QZ,2. In this case the contribution to energy differencehas order of magnitude

|∆E| ∼ αZ(QZ,1QZ,2

4mL2,

(14)

where L ≤ Lw is the typical distance between nucleus and the particleinvolved.

c) Consider now various options for the parity breaking assuming firstk = 113 dark weak matter so that L is of order of size of atom.

i) For k = 169 neutrino one would have ∆E ∼ 1 MeV, which doesnot sound reasonable. If partial neutrino screening is present for k = 113at all, it must involve spin pairing. As already found, neutrino screeningcannot be ideal for k = 113 since the Fermi energy would be rather high.Partial screening favored by the negative energies of dark neutrinos cannotbe however excluded since single neutrino could be shared between severalconstituents of, say, linear molecule. For k = 151 for neutrino and electronone would have ∆E ∼ 2 keV.

ii) For an exotic electron with ordinary mass but k = 113 weak space-time sheet the order of magnitude is ∆E ∼ 2 eV, which corresponds tovisible frequencies. For exotic quarks with mass m ∼ .1 MeV one wouldhave ∆E ∼ 10 eV. For both cases it would not be chiral selection whichwould thermally unstable but the dark weak phase itself, and the selectionwould be absolute in the temperature range were dark weak phase is possible.

iv) For dark W+(113) bosons having mass ∼ 25 MeV one would have∆E ∼ 10−2 eV, which corresponds to the scale of room temperature. Un-fortunately, the large mass and short lifetime of W+(113) do not favor thisidea.

d) Consider now k = 151 weak bosons. The difficulties of W+(113) op-tion are circumvented in the case of W (151) with mass of ∼ 50 eV sinceleptonic decays become impossible. The generation of W+(151) BE con-densate is also energetically favorable due to the large Z0 binding energy.L(151) corresponds to the thickness of the cell membrane and to a minimallength of DNA double strand giving rise to an integer multiple of 2π twistwith integer number (10) of DNA triplets. Note however that the large h

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length scale would be L ∼ nL(151 + 22 = 173) ' n× 20 µm. The decay ofthe BE condensate of dark W (151) bosons (with large value of h) to non-dark W (151) bosons could allowing the control of k = 151 length scale byk = 173 length scale.

In this case one would have ∆E ∼ 5 keV so that chiral selection would behighly stable. This option could be realized for linear bio-molecules. Hencethe Bose-Einstein condensate of screening k = 151 W+ bosons possessingnet spin must be considered as a candidate for a mechanism inducing chiralselection of bio-polymers. The positive charge of the W+ condensate couldrelate to the negative charge characterizing bio-polymers.

If the order parameter of W+ condensate around the molecule is spher-ically symmetric, the average interaction energy vanishes so that W bosonsshould possess also orbital angular momentum: the simplest option is thatnet angular momentum vanishes. The geometric breaking of spherical andreflection symmetries of the molecule would naturally induce the neededasymmetry of the order parameter.

6.8 Hydrogen bond revisited

Hydrogen bond is fundamental for the physics of water and believed to relateto its anomalous expansion at freezing point and anomalous contraction inheating above freezing point. Hydrogen bond plays also a key role in theliving matter. Against this background it is perhaps somewhat surprisinghow poorly understood the physics of the hydrogen bond is.

The special role of hydrogen bond is consistent with the suggested roleof dark Z0 force. Hydrogen bond is believed to reflect ordinary Coulombinteraction between hydrogen bound to molecule and lost its electron par-tially to the molecule and electronegative atom (N, O, Cl,...) which hascaptured partially the electron of the atom with which its bonds, say C, andwhich therefore looks like having positive charge. Hydrogen bonds are ina key role in the binding of DNA strands, in the generation of geometricstructure of proteins and RNA molecules, and also the molecular motors areconstructed from their building blocks by hydrogen bonds. The reason whycould be very simple: hydrogen bonds unlike valence and ionic bonds arerelatively immune to the bio-control based on the variation of the classicalZ0 force by varying the Z0 screening.

An interesting question is whether the hydrogen bonded state A+B ofatoms A and B could be in a superposition of states with A and B inthe ordinary state and a state in which A/B contains positively/negativelycharged color bond changing the charge numbers A and B and effectively

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creating ionic bond.If the hydrogen bond corresponds to a non-vacuum extremal in necessar-

ily carries color gauge flux. Quantum classical correspondence together withthe picture about nuclei as nuclear strings with nucleons connected by longcolor bonds forces to ask whether the nuclear strings of hydrogen bondedatoms fuse to form single nuclear string containing long straight section con-necting the nuclei. Hydrogen bonded nuclei would become both colored andweakly charged in this kind of situation and would posses also a fractionalelectromagnetic charge not explainable in terms of fractional quantum Halleffect. In this kind of situation the first guess is that the exotic quark pairsassociated with the color bond could play the role of valence electrons andcharacterize both the binding energy and parity breaking possibly associatedwith the bond.

7 Long ranged weak and color forces, phonons,and sensory qualia

Phase conjugate electromagnetic waves [41, 42] correspond in TGD frame-work negative energy topological light rays representing signals propagatingto the geometric future [B9]. Phase conjugation is known to make senseeven for sound waves [42]. Since phase conjugation means time reversal andnegative energies in TGD framework, the only possible conclusion seems tobe that classical sound waves and photons must correspond to their ownspace-time sheets. Depending on the time orientation of these space-timesheets, sound waves or their phase conjugates result in the interaction ofthese space-time sheets with matter.

If condensed matter is partially dark in the sense that nuclei tend tocombine to form super-nuclei, the question arises whether dark weak forceand dark nuclear strong force are involved with the sound waves besidesem forces. Topological light rays (”massless extremals”, briefly MEs) car-rying classical gauge fields corresponding to an Abelian subgroup of thegauge group, be it color or electro-weak gauge group, and drifting quantumjump by quantum jump in the direction of sound wave define candidates forthe space-time correlates of sound waves. Also the deformations of warpedimbeddings of M4 to M4 × CP2 with maximal signal velocity reduced tosound velocity using M4 as standard define candidate for the space-timesheets associated with sound waves.

In plasma phase classical electric field can cause plasma waves as lon-gitudinal oscillations of charge density. Also the notion of Z0 plasma wave

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makes sense if nuclei carry anomalous Z0 charges due to charged color bonds.Entire dark hierarchy of these waves is possible. Even the counterparts ofQCD plasma waves are possible.

7.1 Slowly varying periodic external force as the inducer ofsound waves

The basic idea is that an external force, which is constant in the lengthscale of atomic nuclei or molecules sets them in a harmonic motion aroundequilibrium point. This slowly varying force is associated with the space-time sheet serving as the space-time correlate of phonon.

The basic fact about quantum physics of harmonic oscillator is that theresulting new ground state represents a coherent state having interpretationas a classical state of harmonic oscillator. If the external force dependsperiodically on time and spatial coordinates the intensity of the parametercharacterizing coherent state varies in oscillatory manner and classical soundwave results as a consequence.

7.2 About space-time correlates of sound waves

Z0 MEs (”massless extremals”) represent transversal classical Z0 fields prop-agating with light velocity. These transversal fields are candidates for theexternal force generating the coherent states giving rise to sound waves.There are however two problems.

a) How it is possible that sound velocity v is below light velocity?b) How the Lorentz force orthogonal to the direction of propagation of

classical fields inside ME can give rise to longitudinal sound waves.One can imagine two solutions to these problems.Option I: The first solution to both problems could be as follows. Let

Z0 ME represent a wave moving in z-direction with light velocity and letsound wave propagate in the direction of x-axis with sound velocity vs.Assume that Z0 electric field of linearly polarized ME is in x-direction, andthus defines a longitudinal force field inducing the coherent state. Also Z0

magnetic field is present but for non-relativistic particles it is by a factorv/c weaker than Z0 electric force and can be thus neglected.

Z0 ME suffers in each quantum jump a shift consisting of a shift inz-direction and a shift in x-direction. The shift in the z-direction causesan effective reduction of the phase velocity of the field pattern inside ME.The shift in the x-direction means that the Z0 electric field of ME moves is

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in x-direction and causes a longitudinal force. The velocity of the shiftingmotion in the x-direction must be sound velocity.

The classical force field is in a correct phase if Z0 ME shifts in z-directionwith such an average velocity that the phase ωt − kz along ME at point(t, x, y, z) changes to ωt − kz + ω∆t − k1∆x in the shift x → x + ∆x ofthe position of ME resulting in quantum jump sequence corresponding tot → t + ∆t. This requires ∆z = (k1/k)∆x giving dz/dx = c/vs. Hence therays x = vst of constant phase for sound wave correspond to the rays ofconstant phase z = ct along ME.

In the case of transversal sound oscillations possible in solid state Z0

MEs shift in each quantum jump in z-direction in such a manner that ef-fective phase velocity becomes sound velocity. Z0 MEs generate oscillatingtransverse electric field inducing a coherent state of phonons. I have alreadyearlier proposed that nerve pulse propagation corresponds to a propagationof Z0 ME in an analogous manner [D7].

Option II: By quantum classical correspondence one might argue thatsound propagation should have a direct space-time correlate. There ex-ists an infinite variety of vacuum extremals with D = 1-dimensional CP2

projection having a flat induced metric. These extremals correspond towarped imbeddings m0 = t, sk = sk(t) of M4 with the induced metricgtt = 1 − R2skl∂ts

k∂tsl, gij = −δij . The maximal signal velocity using

the canonical imbedding of M4 as a reference is reduced to c# =√

gtt.D = 2 vacuum deformations for this kind of space-time sheets exist but

the great question mark are there non-vacuum deformations which corre-spond to solutions of field equations. Do they represent waves propagatingwith c#? This could be the case since the field equations for these de-formations contain a term proportional to linearized d’Alembert equationin the background metric. Could phonon space-time sheets correspond todeformations of vacuum extremals of this kind analogous to MEs with c#

identifiable as sound velocity? Could phonons correspond to 3-D light-likesurfaces representing wave fronts inside deformed vacuum extremals of thiskind? Could the drifting of MEs have this kind of space-time sheets as aspace-time correlate?

7.3 A more detailed description of classical sound waves interms of Z0 force

The proposed rough model is the simplest description in the case of con-densed matter as long as the positions of particles vary slowly in the timescale of the oscillations associated with the sound wave.

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A modified description applies when harmonic forces are between neigh-boring atoms. In this case the modification of standard wave equation wouldintroduce a term representing external force to the wave equation. In one-dimensional case of one-dimensional periodic lattice with lattice constant a,elastic constant k for the elastic force between nearest neighbors, and atommass m, one would have in the continuum approximation

(∂2t − v2

s∇2)A =QZEZ

ma,

v2s =

ka2

m. (15)

Here a denotes lattice constant.Temporally slowing varying Z0 force to an harmonic external force yield-

ing coherent states of the quantized system. Velocity resonance results whenthe external Z0 field pattern has effective phase velocity equal to sound ve-locity EZ = f(u+), u± = x± vst. Writing the equations in the form

∂+∂−A =QZf(u+)

ma, (16)

one finds that the the general solution is of form

A = A+(u+) + A−(u−) + u+QZ

ma

∫du−f(u−) . (17)

A+ and A− are arbitrary functions of their argument. In the absence ofdissipative effects the amplitude increases without bound.

The quantization of the model is straightforward since a one-dimensional”massless” field coupled to an external source is in question with sound ve-locity taking the role of light velocity. The resulting asymptotic ground stateis a product of coherent states for the frequencies present in the externalforce term. In quantum field theory this kind of state is interpreted as amaximally classical state and thus classical sound wave.

The intensity of the sound wave would be proportional to the modulussquared of the order parameter of the coherent state proportional to theFourier transform of the classical Z0 force. The standard classical modelfor sound waves would thus be only apparently correct. In TGD frameworkthe screened dark Z0 force gives a contribution also to the elastic forcesbetween atoms and explains the strong repulsive potential below atomic

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distances implying incompressibility of condensed matter and needed in vander Waals equation of state.

Also in the hydrodynamics dark Z0 force would take the role of an ex-ternal force. Although the quantization of the Euler’s equations is far frombeing a trivial task and perhaps not even sensible, the proposed picture isexpected to be the same also in this case for small oscillations for which waveequation holds true. In TGD framework incompressible hydrodynamic flowis interpreted from the beginning in terms of dark Z0 magnetic force [A5],and this should make possible a first principle quantization of sound wavesin the case of liquid and gas phases.

a) The hydrodynamic flow occurs along the flux tubes of Z0 magneticfield and it is quite possible that Z0 superconductivity equivalent with super-fluidity along flux tube occurs in sufficiently short length scales. The pres-ence of Z0 magnetic flux tubes parallel to the flow lines is what makespossible to apply hydrodynamic description. The incompressibility insideZ0 magnetic flux tubes is due to the fact that Z0 magnetic field has avanishing divergence. Alfwen waves, identifiable as transverse oscillationsof magnetic flux tubes and propagating with light velocity along the flowlines should have Z0 counterparts and might have detectable effects on thehydrodynamic flow.

b) The Beltrami condition ∇ × v = αv guarantees that a coordinatevarying along flow lines is globally defined and means that super-conductingorder parameter defined along the flow lines can be continued to a functiondefined everywhere so that there is Z0 superconductivity also in the globalsense. The complex patterns of flow reduce to the generalized Beltramiproperty of the topologically quantized flow. Also in the case of gas phaseone expects incompressibility inside the flux tubes at least.

7.4 Does the physics of sound provide an operational defini-tion of the dark Z0 force?

The somewhat surprising conclusion supported by the existence of phaseconjugate sound waves is that coherent sound waves could be a direct man-ifestation of the dark Z0 force directly determining the amplitude of thesound wave understood as a coherent state. Therefore the problem of defin-ing the notion of dark Z0 force operationally would become trivial.

The hypothesis would predict that sound intensity for a given strengthof the dark Z0 field proportional to amplitude squared is proportional to(N/k)2, where N is the anomalous color charge of the oscillating nucleus,and k elastic constant for the harmonic oscillations around the equilibrium

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position of (say) atom.

7.5 Weak plasma waves and the physics of living matter

In plasma phase electromagnetic MEs, and even more so scalar wave pulses,can generate plasma waves accompanied by longitudinal electric fields. Inthe case of scalar wave pulses the mechanism is simple: the longitudinalelectric field of the scalar wave pulse kicks electrons so that a gradient ofelectron density results and oscillation starts at plasma frequency ωp =e√

n/me in case of electron. The frequencies of transversal plasma wavesare above the plasma frequency.

The notion of weak plasma frequency makes sense if condensed mattercan be regarded as Z0 plasma below the weak length scale Lw with nucleicarrying anomalous weak isospin I3,L. Let I3,L be equal to N using neutron’sisospin I3,L = 1/2 as a unit so that single charged color bond correspondsto N = ±2.

For a hydrodynamic flow of water of density ρ = 1 kg/dm3 giving18n(H2O) ' 1030/m3 and m(H2O) = 18mp, W and Z0 plasma frequen-cies are given by

ωp(W ) = gW N√

n/m ,

ωp(Z0) = gZN√

12 − sin2(θW )

√n/m =

√12−sin2(θW )

sin2(θW )× ωp(W ) ,

g2W = e2tan(θW ) , g2

Z = e2

sin(θW )cos(θW ) , sin2(θw) ' .23 .

(18)

For N = 2 corresponding to single color bond Z0 plasma frequency corre-sponds to an energy E ' .062 eV. Note that ωp(W ) = 1.08ωp(Z0) is verynear to ωp(Z0). The two plasma frequencies are identical for p = 1/4.

ωp(W ) is very nearly the frequency associated with the resting potential0.065 eV of the cell membrane [D7]. Although this result could be a sheerco-incidence, it supports the idea that Z0 plasma vacuum-screened in atomiclength scale has a fundamental role in living matter. Of course, entire hier-archy of weak plasmas are possible and more or less forced by the fact thatvacuum weak fields appear in all length scales. Weak scalar wave pulseswould be an ideal tool for generating plasma oscillations whereas weak MEswould generate sound and transversal plasma waves.

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7.6 Sensory qualia and dark forces

The TGD based model of sensory qualia relies on universality hypothesisstating that the increments of various quantum numbers in quantum jumpdefine qualia at fundamental level in all p-adic length scales. The hierarchyof dark matters would allows to realize similar qualia in all length and timescales.

Quantum classical correspondence suggest that qualia identified as theincrements of quantum numbers should have space-time correlates and chargedcomponents of weak and color gauge fields are natural candidates in thisrespect. If this interpretation is correct, sensations of qualia would beassignable to those space-time regions for which space-time sheet has D > 2-dimensional CP2 projection. MEs would not thus serve as space-time cor-relates for qualia but only as communication and control tools.

D = 3 extremals allow interpretation them as analogs of spin glassphase possible in the vicinity of magnetization-demagnetization tempera-ture whereas D = 2 phase would be analogous to ferromagnetic phase andD = 4 phase to de-magnetized phase [A5]. Spin glass property suggests theidentification of D = 3 extremals as fundamental building bricks of livingsystems. D = 3 extremals have also extremely rich hidden order related tothe topology of the field lines of the induced magnetic field lines. Thereforethe interpretation of D = 3 extremals as space-time correlates of qualia isnatural.

A couple of examples are in order.a) Hearing could correspond to the increment of weak isospin or em

charge (or both of them in fixed proportion) and to D ≥ 3 weak space-timesheets. Classical W fields would serve as a space-time correlate for the basicquale associated with hearing.

b) The increments of color quantum numbers would correspond to thevisual colors. The 3+3 charged components of classical gluon field wouldcorrespond to basic color-conjugate color pairs. The reduction to U(2) sub-group of color group (for instance, CP2 projection in r = constant 3-sphereof CP2) would correspond to the restriction of color vision to black-whitevision. Non-vacuum extremals having D > 2 (also those having D = 2)carrying classical em fields are always accompanied by classical color fieldsso that the identification is not in conflict with the existing wisdom. Space-time sheets serving as correlates for color qualia would correspond to p-adiclength scales associated with multiply dark gluons.

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8 Mechanisms of Z0 screening

8.1 General view about dark hierarchy

Classical color gauge fields are always present for non-vacuum extremalsand non-Abelian classical weak fields always when the dimension D of theCP2 projection of the space-time sheet satisfies D > 2. Quantum classicalcorrespondence forces the conclusion that there must be a p-adic hierarchyof dark matters creating these fields in all length scales. At the level ofquantum TGD the p-adic hierarchy of dark matters relates closely with thehierarchy of space-time sheets, hierarchy of infinite primes, and hierarchy ofJones inclusions for hyper-finite type II1 factors. In TGD inspired theory ofconsciousness the hierarchy corresponds to the self hierarchy and hierarchyof moments of consciousness with increasing averages duration.

There already exists some guidelines about the physical realization ofthis hierarchy.

a) Already the p-adic mass calculations of hadron masses led to theconclusion that quarks can appear as several p-adically scaled up variantswith masses of variants differing by a multiple of half-octave. There is alsoexperimental support for the view that ordinary neutrinos can appear asseveral p-adically scaled up variants [61]. This forces to ask whether alsoelectrons could appear as scaled up of scaled down variants even in theordinary condensed matter, and whether the notion of effective mass ofelectron varying in wide limits could be replaced by p-adically scaled upmass. A testable prediction is atomic spectra scaled by a power of

√2.

b) In the TGD based model for atomic nuclei as color bonded nucleonswith the quarks/antiquarks at the ends of bonds are identified as p-adicallyscaled down quarks with electromagnetic space-time sheet having k = 127rather than k = 113. Quite generally, exotic quarks and perhaps also leptons(possibly also their color excitations) with p-adically scaled down masseswould be associated with the ends of join along boundaries bonds servingas correlates for the bound state formation.

c) The decay width of ordinary weak bosons force the conclusion that theweak space-time sheets associated with exotic quarks have k 6= 89 k = 113is a good guess in this respect and would in large h phase correspond toa length scale of order atomic size. The model for tetra-neutron identifiestetra-neutron as alpha particle with two charge color bonds. There is noreason to assume that charged bonds could not appear also in heavier nuclei.

Their presence would mean also that nucleus has anomalous em and weakcharges. One can even consider the possibility that the nuclear strings of

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neighboring atoms fuse to single nuclear string with long straight portion sothat nuclei become colored and possess fractional em charges. Also linkingof the nuclear strings might occur.

If this general picture forced by quantum classical correspondence istaken seriously, one begins to wonder whether even chemical bonds couldinvolve light dark elementary fermions. These dark particles could coupleto scaled down copies of both weak bosons and colored gluons.

Chiral selection in living matter could be due to the axial part of weakinteractions between exotic quarks of different nuclei. Even the low com-pressibility of liquid phase could be due to the Z0 repulsion between nucleihaving anomalous weak charges in condensed phase: note that no isotopicdependence is predicted as in the earlier proposal based on the assumptionthat ordinary quarks are Z0 charged.

c) Besides color and electro-weak numbers dark particles can carry com-plex conformal weights expressible in terms of zeros of Riemann Zeta. If theconformal weight is conserved in particle reactions and given particle cancorrespond to only single complex conformal weight, it must be expressiblein terms of conserved quantum numbers so that neutral particles have realconformal weights. In the transition to the next level of darkness the parti-cles of previous level could receive complex conformal weights and color andweak quantum numbers.

d) Dark ↔ visible phase transitions are describable as ordinary verticesin which also a scaling of h occurs and scales the size of the space-time sheetrepresenting the particle.

8.2 Vacuum screening and screening by particles

Suppose that phase transitions generating charged color bonds and makingmolecules of condensed matter Z0 charged with the same value of Z0 chargeare possible. This transition need not generate em charge since ordinarynuclear charge can be reduced in the transition. Weak charge is howevergenerated. This kind of transition could proceed spontaneously as a two-nucleon process if the nuclei are close enough.

This raises the question about the basic mechanisms of screening of weakcharges, in particular Z0 charge. There are two basic mechanism of screen-ing. Vacuum screening occurs automatically above weak length scale Lw

and is responsible for the massivation of weak bosons. The screening byZ0 charges of particles occurs in length scales L ≤ Lw in a dense weak(lycharged) plasma containing a large number of charged particles in the vol-ume defined by Lw.

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8.2.1 Vacuum screening

Vacuum screening occurs automatically and is based on the generation ofvacuum charges which reduces the value of weak charge of particle at theweak space-time sheet associated with particle so that the flux feeded tothe next sheet is reduced. This mechanism implies massivation of gaugebosons which at each space-time sheet behave classically like massless fields.It is basically the loss of coherence and correlations due to the finiteness ofparticle space-time sheet which implies the massivation and screening. Thescreening by vacuum charges makes sense only above the length scale Lw

defined by the mass scale of weak bosons.

8.2.2 Screening by weakly charged dark particles

The screening by dark particles carrying weak charges is appropriate in weakplasma. In situation when the density of Z0 charge is so high that Lw sizedregion contains large number of Z0 charges, screening must be due to darkparticles, such as dark electrons and neutrinos.

a) If ordinary atomic nuclei can make a transition to a phase in whichk = 113 defines the weak length scale followed by a transition to dark phasewith hs = nh/v0. For n = 3, the length scale Lw above which vacuumscreening occurs is about nx Angstrom, where x varies in the range [1, 2]and n = 1, 2, 3, ... and screening by dark particles is not necessary in thedensities typical to condensed matter. For n = 3 the Lw is in the range3-6 A. The fact that the screening length is of the order of atomic size andlength of a typical chemical bond means that dark weak force could play animportant role in bio-catalysis as already discussed.

The situation is quite different from that for Z0 charge localized in nu-clear volume. A complete screening by particles is achieved by constantdensity of Z0 charge for the screening particles equal to the average Z0

charge density of nuclei since the charge density to be screened is constantbelow Lw. By varying the density of screening particles the degree of Z0

screening can be varied.b) The hypothesis that weak bosons with complex conformal weights cor-

respond to Gaussian Mersennes, such as the biologically highly interestinglength scales k = 151, 157, 163, 167 varying in the biologically most inter-esting length scale range 10 nm-2.5 µm is worth of studying. This kind ofdark particles could have ordinary value of h but would possess large weaksize Lw. In condensed matter weak plasma phase would appear below thelength scale L(k) and the weak nuclear charges would be screened by dark

76

electrons.Since the Z0 charge density is constant below L(k) screening by constant

charge density of dark neutrinos is possible. Experimentally one cannotexclude the possibility that scaled up variants of ordinary neutrinos andtheir dark counterparts could appear at p-adic length scales k = 151, ..., 167.For instance, the model of nerve pulse relies crucially on the assumption thatk = 151 cell membrane space-time sheet carries neutrinos [D7].

In the sequel a classical model of Z0 screening by dark neutrinos gener-alizing the Debye model of ionic screening and a genuinely quantum modelof screening based on the Bose-Einstein condensate of dark neutrino Cooperpairs are discussed. The Bose-Einstein condensate of sneutrinos predictedby space-time super-symmetry would be ideal for screening purposes. Super-conformal symmetries are basic symmetries of quantum TGD at the level ofthe ”world of classical worlds” but it seems that sparticles are not predictedby quantum TGD if its recent interpretation is correct.

8.2.3 Different variants of Z0 screening by particles

The model for the Z0 screening allows to consider at least the followingoptions.

1. Screening by a Bose-Einstein condensate

Some particles which are bosons would Bose-Einstein condense to theground state. One can consider several options.

a) Sneutrinos, which are predicted by theories allowing space-time super-symmetry, would be nice option but there are reasons to believe that TGDdoes not predict them: super-symmetry would be realized only at the levelof configuration space of 3-surfaces.

b) Cooper pairs of dark neutrinos is second candidate. A phonon ex-change mechanism based on classical Z0 force could allow the formation ofCooper pairs making possible neutrino super conductivity. This mechanismis discussed in some detail in [C2]. The neutrinos in question would be atthe top of the Fermi sphere and highly relativistic and it is not clear whetherthe formation of Cooper pairs is possible.

The questionable feature of the Cooper pair option is that the densityof neutrinos is so high as compared to the Compton length defined by therest mass of the neutrino. One can ask whether it makes to sense to regardmulti-neutrino state as consisting of Cooper pairs in this kind of situation.

c) The Bose-Einstein condensate of W bosons giving rise to W super-conductivity would define the third option. The simplest option is that the

77

very process generating the charged color bonds in nuclei occurs via emissionof W bosons taking also care of screening.

For k = 113 dark W bosons this option is energetically problematic sincethe rest mass of dark W bosons with k = 113 is about 25 MeV and ratherhigh and these bosons are also highly unstable. Note however that com-plete screening is not needed since vacuum screening occurs automaticallyabove Lw, and W Bose-Einstein condensate could control the degree of Z0

screening.For k = 151 W mass is ∼ 50 eV and these bosons could be stable (if

the masses of exotic leptons are small enough). The negative Z0 Coulombicinteraction energy with exotic quark, given roughly by ∼ 2αZQ2

Z(ν)/a, aatomic radius, is of same order of magnitude as the rest mass. Therefore thegeneration of k = 151 W Bose-Einstein BE condensate would require rathersmall net energy and would lead to a gain of energy for k = 157, 163, 167.

2. Dark neutrinos screen the Z0 charge

For this option dark neutrinos do not form Cooper pairs and thus fill thewhole Fermi sphere. For a complete screening the Fermi energy is extremelyrelativistic, of the order πhs/a, a atomic radius so that this option is notenergetically favored despite the fact that the ground state energy is negativedue to the large Z0 interaction energy having magnitude larger than neutrinomass.

For full screening the value of the Fermi energy for dark neutrinos atlevel k = kZ is determined essentially by the density of anomalous isospinper nucleon. This implies that neutrinos at the top of Fermi surface arerelativistic: the Fermi energy for N units of weak isospin per nucleon isgiven by

EF ' N1/3hsπ

a,

a ' 10−10 m (19)

and does not depend on condensate level. The order of magnitude is 104

eV for ordinary value of h but n × 20 MeV for hs = nh/v0 and of thesame order of magnitude as the rest mass of dark W boson. Hence thisoption is not energetically much better than W boson option. As noticed,complete screening is not needed so that neutrino screening could servecontrol purposes.

78

8.3 A quantum model for the screening of the dark nuclearZ0 charge

In the sequel a quantum model for the screening of dark Z0 charge is dis-cussed. There are several options corresponding to a screening by neutrinos,by their Cooper pairs, or by light variants of W bosons. The screening bysneutrinos predicted if the theory allows space-time super-symmetry butthis does not seem to be the case in TGD.

8.3.1 Some relevant observations about dark neutrinos

The experimental data about neutrino mass differences suggests that neu-trinos correspond to the p-adic length scale k = 169 and possibly also somelarger p-adic primes such as k = 173 [61]. k = 169 neutrinos would haveCompton length of about L(169), cell size.

Neutrinos with dark k = 113 weak space-time sheet need of course notcorrespond to the same p-adic length scale as ordinary neutrinos but onecan make this assumption as a convenient working hypothesis in order toget some acquaintance with the numbers involved.

A constant Z0 charge density of dark neutrino background can in prin-ciple cancel k = 113 dark Z0 charge density which is constant in lengthscales L < Lw(keff = 137) of order atomic size. The degree of screening isthe proper parameter and cannot vary considerably in length scales smallerthan L(169) since this would require highly energetic neutrinos.

The Fermi sea of dark neutrinos screening completely the anomalous Z0

charge of nuclei gives rise to Fermi momentum equal to EF = pF = hsn1/3 '

N1/3h/L(137) ' N1/3(hs/h) × 104 eV but this requires energy. Here N isthe number of Z0 charges per nucleus.

8.3.2 The model of Z0 screening based on harmonic oscillatorpotential does not work

The density of the nuclei is so high that there is large number of nucleiwithin the Bohr radius, which increases by a factor n/v0 in large h phase.Also the fact that Z0 charge density is constant within Lw favors a differenttreatment.

The first guess is that the presence of the anomalous nuclear Z0 chargecould be treated as a harmonic oscillator potential with origin at the center ofthe region containing the dark phase. One might hope that this treatmentmakes sense if the nuclei can be regarded as forming a fixed backgroundstabilized by electromagnetic interactions and by screening. The objection

79

is that translational invariance is lost. It is easy to see that the treatmentfails also for other reasons.

The effective potential is given by

Veff =E

mVZ −

V 2Z

2mν,

VZ =kr2

2,

k =13Q2

Z(ν)hsαZNρn , (20)

where ρn ≡ 1/a3 is the number density of nuclei. N is the Z0 charge pernucleus due to the charged color bonds using QZ(ν) as a unit.

The presence of the relativistic correction in-stabilizes the system abovesome critical value of r. The maximum V = E2/2mν of the effective poten-tial at V = E corresponds to

r =

√6Ea3

hs×

√1

αZNQ2Z(ν)

. (21)

For non-relativistic energies the order of magnitude for r is

r ∼√

v0mνa/√

NαZQ2Z(ν)

and smaller than the atomic radius. Thus it would seem that the potentialis in practice repulsive in the non-relativistic case. For negative energiesthe potential is repulsive everywhere. Even for relativistic energies of oforder hs/a at the Fermi surface one has r ∼ a/

√NαZQ2

Z(ν) and not muchlarger than atomic radius. Obviously the treatment of nuclei in the proposedmanner does not work.

8.3.3 The model for Z0 screening based on constant potentialwell

Since Z0 charge density is constant within Lw, the safest manner to describethe system is as free dark neutrinos or neutrino Cooper pairs in a potentialwell characterized by the average Z0 interaction energy of neutrino withnucleus, both idealized as balls of radius Lw carrying a constant Z0 chargedensity.

By performing a time dependent gauge transformation

80

Z0µ → Z0

µ + ∂µΦ , Φ = VZt× χ ,

where χ equals to unity inside the potential well and vanishes outside, freed’Alembert equation inside potential well results and solutions can be writ-ten as standing waves, which must vanish at the boundary of the well tominimize the singularity resulting from the fact that AµAµ term gives squareof delta function at boundary. The energy identified from the time depen-dence of the phase factor of solution is E0 + VZ =

√p2 + m2 + VZ as the

non-relativistic treatment would suggest. Negative energy states obviouslyresult if Z0 Coulomb interaction energy E ∼ αZQ2

Z(ν)N/a is larger thanneutrino mass.

8.3.4 Is Bose-Einstein condensate generated spontaneously?

The formation of neutrino Cooper pairs would correspond to the pairing ofneutrinos of opposite spin and would be analogous to the pairing of valenceelectrons and nucleon pairs inside nuclei. The Bose-Einstein condensationwould result basically from the energy gap between the states at the topof Fermi sphere and bound states formed via the scattering possible at thetop of Fermi sphere. If the Z0 interaction energy of neutrinos is negativeand has larger magnitude than the rest mass at the bottom of Fermi sphere,it is energetically favorable to generate Fermi sea up to a positive energyfor which the neutrino system vanishes. Zero energy neutrino-antineutrinopairs for which neutrino has negative energy could be created spontaneouslyfrom vacuum and the condensate could thus be generated spontaneously.

k = 151 W bosons could form automatically Bose-Einstein condensate.The fact that Z0 interaction energy has larger magnitude than W bosonmass favors the spontaneous occurrence of the process. If W bosons arecreated by the phase transition generating charged color bonds in nucleitheir charge is automatically screened.

It is illustrative to recall the basic aspects of the model for Bose-Einsteincondensation in case of ordinary ideal Boson gas.

a) In the absence of the classical Z0 force the energy spectrum of non-relativistic neutrino Cooper pairs is that for a particle in box: En = k

∑i n

2i×

π2/mL2(169), where k is a numerical factor k characterizing the geometry.The natural unit of energy is π2h2/2mL2(169) ' .05 eV.

b) The critical temperature for Bose-Einstein condensation is in recentcase obtained by applying the general formula applying in case of free bosongas with fixed particle number N in volume V :

81

Tc =2πh2

s

m(

n

2.61)2/3 = 2πh2

s × (A− Z

2.61)2/3 × a2

m. (22)

Tc is of order .1 GeV so that Bose-Einstein condensation certainly occurs.The fraction of Bose-Einstein condensed particles is given by

NBE

N= 1− (

T

Tc)3/2 . (23)

From these estimates it should be obvious that also in the recent case BoseEinstein condensation indeed can occur and that most of the bosons are inthe negative energy state.

9 Appendix: Dark neutrino atoms

Dark neutrinos provide a possible screening mechanism for classical Z0 forcepresent in dark condensed matter with weak bosons in dark k = 113 phase.If one takes seriously recent experimental evidence [35] and the explanationof the anomalous atmospheric µ/e ratio [36] in terms of neutrino mixing onemust conclude that νµ and ντ are condensed on k = kZ level and that muonand τ neutrino have suffered large mixing whereas the mixing of νe withremaining neutrinos is much small.

The discussion of [B2] led to the predictions for neutrino masses as afunction of common condensation level. In the following table also the k =132 = 169 level is included since it predicts exactly the best fit value forντ − νµ mass squared difference whereas k = 167 predicts it within 90 percent confidence limits. k = 169 = 132 would be allowed if the physicallyinteresting k:s are powers of primes instead of primes: this introduces onlyfew new p-adic length scales below one meter.

k m(νe)/eV m(νµ)/eV m(ντ/eV

163 2.16 5.28 5.36167 .54 1.32 1.34169 = 132 .27 .66 .67

Table 2. The table gives the masses of neutrinos as predicted by p-adicmass calculations for three condensate levels.Only k = 167 is allowed by the experimental constraints and p-adic lengthscale hypothesis in its most stringent form. It must be however emphasized

82

that the elementary particle black hole analogy, discussed in the third part ofthe book, allows also k = 169 = 132 giving the best fit to the neutrino masssquared differences. Since the experimental results about electron neutrino-muon neutrino mass difference are preliminary one cannot however excludethe existence of heavy τ neutrino effecting screening of classical Z0 force inatomic length scales. The upper bound .3 MeV of neutrino mass almostallows k = 131 τ neutrino with mass of .4 MeV and it is interesting to findwhether k = 131 τ is physically acceptable alternative. It turns out thatthis is not the case.

9.1 Dark neutrino atoms in non-relativistic approximation

To get order of magnitude picture it is useful to look first the Bohr radiiand ground state energies for dark neutrino atoms assuming that the non-relativistic approximation makes sense. The Bohr radius aν = 1

mναZQ2z(ν)(A−Z)

and ground state energy of the neutrino atom read in terms of the ordinaryBohr radius a0 ' 0.5 · 10−10 m and hydrogen atom ground state energyEH ' 13.6 eV

aν =me

mν

αem

αZQ2Z(ν)

a0

(A− Z)

' me

mνX

a0

(A− Z),

Eν = X−2 mν

me(A− Z)2EH ,

X =sin(θW )cos(θW )

Q2Z(ν)

' 1.68 . (24)

For ντ (131) (see the table below) Bohr radius is a(ν) = 1.95a0 = 1.05L(137)and quite near to the typical size of lattice cell in condensed matter systems.

ν m aν E0/eV TI/K

ντ (131) 0.45 MeV 7.5E − 10 m 4.3 .5E + 4νµ,τ (167) 1.32 eV 12.8 µm 1.32E − 5 .13νe(167) .45 eV 49.8µm .40E − 5 .04

Table 3. Table gives Bohr radius, energy of ground state and ionizationtemperature for ground state of neutrino atom for different neutrino species.Data are also given for k = 131 τ neutrino.

83

For dark matter densities which are of order condensed matter densitiesneutrino atoms are not possible. One can however consider the possibilitythat a block of dark matter takes the role of ”super nucleus” creating aneutrino ”super-atom” with Bohr radius ∝ 1/N(A−Z) and binding energy∝ N2(A− Z)2, where N is the number of nuclei involved.

The observation of the spectral lines of k = kZ dark neutrino atomswould be a triumph of the theory. The transitions between different energylevels can take place via photon/phonon emission/absorption and the ob-servation of the predicted hydrogen type emission and absorption lines ortheir phonon counterparts would be a direct verification of the theory.

a) A possible signature of neutrino atoms is weak absorption of light atenergies of order 10−5 eV . In dipole approximation the transition ampli-tudes are proportional to the sum of matrix elements for electronic and nu-clear dipole moment operators so that matrix elements (m|r(nucleus)|n) and(m|r(electron)|n) are involved. The coordinate vector operators r(nucleus)and r(electron) must be expressed in terms of cm coordinates and they con-tain a small contribution proportional mν

M(nucleus) rν as is clear from r(nucleus) =rcm + mν

m(nucleus)+mν) r12 and corresponding expression for electronic coordi-nate vector. These terms proportional to rν induce transitions between dif-ferent neutrino energy levels. The transition rates are by a factor m2

νm2(nucleus)

∼10−18/A2 (!) smaller than their atomic physics counter parts. Transitionrates are also proportional to the square of the energy difference betweenthe levels in question and this gives additional factor of order 10−10 for neu-trino atoms so that reduction factor of order 10−38 results! The observationof k = 167 neutrino atoms requires temperature of order .1 K and verylow densities (fraction of order 10−12 of ordinary condensed matter density)and one can conclude that the observation of k = 167 neutrino atoms isextremely difficult by photon emission or absorption.

b) One can also consider the possibility of observing dark neutrino atomsvia phonon absorption or emission: the coupling of the neutrinos to phononswould result indirectly from the coupling of neutrinos to atomic nuclei viaclassical Z0 force and from the coupling of nuclei to phonons. A roughestimate for the relevant wavelength of sound in temperature of order .1 Kgives for the wavelength of the phonon associated with transitions λ ∼ 10−9

meters and frequency of order 1010 Hz.

9.2 A relativistic model for dark neutrino atom

The Z0 gauge potential around nucleus is very strong and the classical esti-mate for the neutrino Coulombic energy has a magnitude much larger than

84

the rest mass of neutrino. This suggests that neutrinos and their Cooperpairs could form negative energy states around nucleus.

For neutrino atoms with several neutrinos one must take into account thescreening effect of neutrinos to the Z0 Coulombic potential of the nucleon.The self consistent model is based on the relativistic counterpart of theSchrodinger equation for the order parameter describing bosons in the Z0

Coulomb potential created by the nucleus and neutrino charge density.

9.2.1 Self consistent relativistic Schrodinger equation as a modelfor Z0 screening

The Laplace equation for the self-consistent Z0 Coulomb potential reads as

∇2VZ = −g2ZQ2

Z(ν)(A− Z)δ(r) + g2ZQ2

ZΨ∂tΨ . (25)

In the lowest order approximation the solution of this equation is Coulombinteraction energy of neutrino with nucleus

V 0Z = −kZ

r,

kZ = αZQ2Z(ν)(A− Z) . (26)

The d′Alembert equation for the order parameter Ψ characterizing aBose-Einstein condensate of Cooper pairs of mass m reads as

[(−i∂t − VZ)2 +∇2

]Ψ = m2Ψ . (27)

Specializing to stationary solutions Ψ ∝ exp(iEt) corresponding to en-ergy eigenstate and assuming spherically symmetric potential, one has Ψ =R(r)Y l

m(θ, φ).If |Ψ|2 is spherically symmetric as one can assume under rather general

conditions, the model reduces to ordinary differential equations and one cansolve it numerically by iterating. By writing VZ in the form VZ = fZ/r onecan readily integrate VZ from

VZ = −kZ

r+

g2ZQ2

ZE

r

∫ r

0dr2

∫ r2

0dr1r1R

2(r1) . (28)

85

9.2.2 Bound states

It is possible to understand the general properties of this equation by trans-forming in to a form which allows to use the rather precise analogy withSchrodinger equation for hydrogen atom. There are two cases to be consid-ered: bound states and negative energy resonances.

For the bound states the appropriate representation of the equation is

[− 1

2m(∂2

r +2r∂r +

l(l + 1)r2

) +E

mVZ −

V 2Z

2m

]R =

(E2 −m2)2m

×R .(29)

When the screening is not taken into account, the equation has a closeresemblance with the Schrodinger equation for the hydrogen atom. Thecorrespondences are following:

keff = E2mk , Eeff = E2−m2

2m , leff (leff + 1) = l(l + 1)− k2Z . (30)

In the analog of Schrodinger equation Coulombic potential energy isreplaced by an effective potential energy

Veff =E

mVZ −

V 2Z

2m. (31)

Veff is negative for large values of VZ , vanishes for V = −2E, has a maxi-mum Veff (max) = E2/2m for V = E and vanishes asymptotically. There-fore Veff has an attractive infinitely deep well surrounded by a potential wallof height E2/2m so that tunnelling in principle becomes possible. Since V 2

term only modifies the effective value of the angular momentum, it is pos-sible to solve the Schrodinger equation explicitly. Bound states correspondto E < m. Bound states are non-relativistic with a very long range m/k2

Z

of about 10−4 meters and are not interesting as far as local screening of Z0

charge is considered.

9.2.3 Negative energy resonances

Relativistic negative energy resonance like solutions can be localized belowthe atomic radius and only these are appropriate for local screening of theZ0 charge. For these solutions it is natural to replace the mass of the Cooperpair with its energy |E|. With a little re-arranging the following equationanalogous to Schrodinger equation for hydrogen atom

86

[− 1

2|E|(∂2

r +2r∂r +

l(l + 1)r2

)− E

|E|VZ −

V 2Z

2|E|

]R =

(E2 −m2)2|E|

R .(32)

In the lowest order approximation one can use the unscreened Z0 Coulom-bic potential allowing very close analogy with the hydrogen atom. Theanalogy with the hydrogen atom is revealed by the replacements

meff = |E| , keff = kZ2 , Eeff = E2−m2

2|E| , leff (leff + 1) = l(l + 1)− k2Z . (33)

Note that leff can be also negative and that for negative energies theCoulombic potential term represents an attractive potential although onehas Eeff > 0. Thus the proper interpretation of the negative energy statesare as kind of resonance states.

9.2.4 An upper bound on the neutron number of nucleus

The general solution for leff allows to branches

leff = −12± 1

2

√1 + 4l(l + 1)− 4k2

Z . (34)

The second branch allows leff < 0 even when the right hand side of theequation above is positive.

The condition

l(l + 1) > k2Z − 1

4 (35)

guaranteing the reality of leff must be satisfied. This condition is automat-ically satisfied for l = 0 for nuclei satisfying kZ < 1/2: this gives

A− Z ≤ 12αZQ2

Z(ν). (36)

For biologically important nuclei the condition is satisfied since the lowerbound is very roughly A− Z = 60.

For l > 0 solutions the neutrino perturbation of the Coulombic potentialis not spherically symmetric. Hence only l = 0 solution allows a simplenumerical treatment based on ordinary differential equations.

87

9.2.5 The behavior of the negative energy solutions near origin

One can apply standard methods used for solving the Schrodinger equationfor hydrogen atom also in the recent case.

a) One can write the normalized order parameter R in the form

R(r) = N × rleff+1 × exp(−ir

|r0|)× w(r) . (37)

The counterpart of Bohr radius is given by

|r0| =1√

2Eeffmeff=

1√E2 −m2

. (38)

For relativistic negative energy solutions the counterpart of Bohr radius isimaginary so that the exponential represents spherical wave.

b) Negative energy solutions are slightly singular at origin as are alsothe solutions of the relativistic Dirac equation. The requirement that thesolution is square integrable at origin gives

leff > −52

, (39)

The behavior R2r2 ∝ r2δ/r for |Ψ|2 near origin is therefore the most singularoption.

A more stringent condition results if one requires that the interactionenergy between neutrinos and nucleus is finite. In the lowest order theinteraction energy density behaves as r2leff+1 so that the constraint readsas

leff > −2 . (40)

If one requires that neutrino-neutrino Coulombic interaction energy is finiteone has

leff > −54

. (41)

At large distances 1/r1−2δ even the most singular behavior of |Ψ|2 doesnot guarantee square integrability but in present case one is interested in

88

non-localized solutions analogous to those characterizing conduction elec-trons and square integrability is not needed. From the condition

leff (leff + 1) = l(l + 1)− k2Z = l(l + 1)− αZ(A− Z)Q2

Z(ν) (42)

it is clear leff can be negative only for l = 0 solution for nuclei for whichthe condition A− Z < αZQ2

Z is satisfied.

9.2.6 The condition determining the energy eigen values

In the case of bound states the function w(ρ) reduces to a polynomial. Alsofor the negative energies one can consider analogous solution ansatz as arepresentation of a negative energy resonance state.

a) The condition for the reduction to a polynomial can be deduced usingstandard power series expansion and reads as

2(k + leff + 1) = − keff

|Eeffr0|= −kZ ×

[ |E|mE2 −m2

]1/2

. (43)

b) One can write leff in the form leff = −leff (min)+∆l, where the valueof leff (min) = −7/2, 2, or −5/4 depending on the regularity conditions atthe origin so that the condition Eq. 43 gives

k < −leff (min)− 1−∆l ≥ 14−∆l . (44)

w is at most a first order polynomial in r. The most stringent conditionguaranteing the finiteness of Z0 interaction energy allows only the solutionfor which w(ρ) is constant.

c) The condition of Eq. 43 guaranteing the reduction of the series of wto a polynomial reduces to the form

1− 2δ = kZ ×[ |E|mE2 −m2

]1/2

. (45)

The solutions are

|E|m

=[b±

√b2 − 1

]1/2,

b = 1 +k2

Z

2(1 + 2δ2)2. (46)

89

Solutions are relativistic negative energy solutions but the energy is of thesame order of magnitude as the rest energy so that the total energy of theBose-Einstein condensate is relatively small. Note that the solution is scalingcovariant in the sense that in the p-adic scaling m → 2km also energy scalesin the same manner.

References

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[padTGD] M. Pitkanen (1995), Topological Geometrodynamics and p-AdicNumbers. Internal Report HU-TFT-IR-95-5 (Helsinki University).http://www.physics.helsinki.fi/∼matpitka/padtgd.html.

[cbookI] M. Pitkanen (2001), TGD inspired theory of consciousness withapplications to bio-systems.http://www.physics.helsinki.fi/∼matpitka/cbookI.html.

[cbookII] M. Pitkanen (2001) Genes, Memes, Qualia, and Semitrance,http://www.physics.helsinki.fi/∼matpitka/cbookII.html.

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[A1] The chapter Is it Possible to Understand Coupling Constant Evolutionat Space-Time Level? of [TGD].

[A2] The chapter Equivalence of Loop Diagrams with Tree Diagrams andCancellation of Infinities in Quantum TGD of [TGD].

[A3] The chapter TGD as a Generalized Number Theory III: Infinite Primesof [TGD].

[A4] The chapter Was von Neumann Right After All? of [TGD].

[A5] The chapter Basic Extremals of the Kahler Action of [TGD].

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[A6] The chapter TGD and Cosmology of [TGD].

[A7] The chapter Cosmic Strings of [TGD].

[A8] The chapter TGD and Astrophysics of [TGD].

[B1] The chapter Massless States and Particle Massivation of [padTGD].

[B2] The chapter Particle Massivation: Elementary Particle Masses of[padTGD].

[B3] The chapter Particle Massivation: Hadron Masses of [padTGD].

[B4] The chapter Particle Massivation: New Physics of [padTGD].

[B5] The chapter Topological Condensation and Evaporation of [padTGD].

[B6] The chapter TGD and Nuclear Physics of [padTGD].

[B7] The chapter Dark Nuclear Physics and Living Matter of [padTGD].

[B8] The chapter The Notion of Free Energy and Many-Sheeted Space-TimeConcept of [padTGD].

[B9] The chapter Did Tesla Discover the Mechanism Changing the Arrowof Time? of [padTGD].

[C1] The chapter ”Topological Quantum Computation in TGD Universe”of [cbookI].

[C2] The chapter Bio-systems as Super-Conductors of [cbookI].

[D1] The chapter Quantum Coherent Dark Matter and Bio-Systems asMacroscopic Quantum Systems of [cbookII].

[D2] The chapter Macro-Temporal Quantum Coherence and Spin Glass De-generacy of [cbookII].

[D3] The chapter Macroscopic Quantum Coherence and QuantumMetabolism as Different Sides of the Same Coin of [cbookII].

[D4] The chapter Bio-Systems as Conscious Holograms Bio-Systems asConscious Holograms of [cbookII].

[D5] The chapter Homeopathy in Many-sheeted Space-time of [cbookII].

[D6] The chapter General Theory of Qualia of [cbookII].

[D7] The chapter ”Quantum Model of Nerve Pulse” of [cbookII].

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