Monday, May 30, 2011

150 GeV bump seems to be real!

The 150 GeV bump reported by CDF seems to be real! Jester reports in Resonaances that new data conform with 4.8 confidence level that the bump is really there. Only .2 sigmas to a discovery. The slides are here.

I have told in two postings about the TGD based interpretation of 150 GeV bump in terms of a neutral pion of M89 hadron physics: see this and this. Wjj would result from a decay of charged ρ or p-adic octave of ground state pion with mass slightly below 300 GeV and the mechanism would be the same as in the case of tau-pion production explaining the two and half year CDF anomaly and also the DAMA observations and DAMA-Xenon100 discrepancy. If charged pion rather than ρ is in question, it would appear with mass twice the ground state mass of pion: this would be allowed by p-adic length scale hypothesis. A competing standard explanation is in terms of Z' boson which is leptophobic meaning that it prefers to decay to quarks. This is of course extremely un-natural for a particle analogous to weak boson. For the hadrons of a scaled up variant of hadron physics leptophoby is however the tell-tale signature.

TGD predicts entire hadron spectroscopy in TeV scale and also other bumps have been already detected. The bumps found by CDF and D0 at 325 GeV (see the earlier posting) might have interpretation in terms of kaon of M89 hadron physics (see this). This would require large CP or even CPT breaking about .2 GeV and a large 4 GeV breaking for top mass meaning CPT breaking has been reported (see the earlier posting).

The puzzling situation in dark matter searches and the two and half year old CDF anomaly find common explanation in terms of tau-hadron hypothesis (leptons are predicted to have colored excitations in TGD Universe), which is dark in TGD sense meaning that it has non-standard value of Planck constant. Furthermore, the Pamela anomaly suggests an interpretation in terms of electro-pion with standard value of Planck constant and having mass of about 500 GeV. Essentially the same explanation would apply also to the M89 pion if the recent slight evidence about mother resonance slightly below 300 GeV decaying to W boson and resonance slightly below 150 GeV turns out to solid.

To explain this anomaly in standard approaches one must introduce leptophilic particles preferring to decay to leptons with constituents having long range interaction most naturally identifiable as electromagnetic interaction: the irony is that this interaction is in conflict with the very notion of darkness. Also the emergence of particles suffering all kinds of -philies and -phobies is a clear signature that something is badly wrong in the standard theoretical framework. The ability of the theory to cure these -phobies and -philies by assuming p-adic length scale hypothesis and new view about color predicting hierarchies of new physics is also a signature of something.

It seems that several hadron and leptohadron physics are revealing themselves for those who are mature to see them;-)! Maturity requires the readiness to accept p-adic fractality instead of stubborn sticking to the belief that physics involves only single fundamental length scale. A profound modification of the naive reductionism as a reduction of everything to shorter and shorter length scales to fractal reductionism is necessary.

This is really great time for TGD. A long list of anomalies finds a beautiful explanation in TGD framework and LHC will produce new fascinating findings allowing to test the predictions. Of course, it still takes a years before we can open a bottle of champaigne. And this only within family. Decades will be required before we can invite colleagues to celebrate TGD;-).

Addition: ATLAS and CMS have not seen the CDF bump but according to Sean Carroll the probable reason is that they do not have enough data yet. D0 will probably also tell something in the near future.

For background see the chapter New Particle Physics Predicted by TGD: I of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Sunday, May 29, 2011

Motives and twistors in TGD

Motivic cohomology has turned out to pop up in the calculations of the twistorial amplitudes using Grassmannian approach (see this and this). The amplitudes reduce to multiple residue integrals over smooth projective sub-varieties of projective spaces. Therefore they represent the simplest kind of algebraic geometry for which cohomology theory exists.

Also in Grothendieck's vision about motivic cohomology projective spaces are fundamental as spaces to which more general spaces can be mapped in the construction of the cohomology groups (factorization). In the previous posting I gave an abstract of a chapter about motives and TGD explaining a proposal for a non-commutative variant of homology theory based on a hierarchy of Galois groups assigned with the zero locus of polynomial and its restrictions to lover dimension planes obtained by putting variables appearing in it to zero one by one: the basic idea is simple but I would have never discovered it without infinite primes.

The basic problem is to define boundary homomorphism for the hierarchy of Galois groups Gk satisfying the non-abelian generalization of δ2=0 stating that the image under δ2 belongs to the commutator subgroup of Gk-2 and therefore is mapped to zero in abelianization, which means division by commutator sub-group.

  1. The proposal is also that the roots can be represented as points of 2-D surface (partonic 2-surface) and that Galois groups can be lifted to braid groups acting on a braid of braids of .... to which infinite primes can be mapped. Infinite primes at n:th level of hierarchy describe a states of n times quantized arithmetic SUSY for which the many particles states of the previous level take the role of elementary particles.

  2. The basic idea is very physical: the braiding for a braid of braids induces braiding of sub-braids and this is represented as a homomorphism of the Galois group lifted to braid group of the braid to the corresponding groups of sub-braids. This nothing but a representation of symmetries and braiding as a isotopic flow gives excellent hopes about a unique realization of the boundary homomorphism.

  3. This SUSY is physically extremely interesting since irreducible polynomials of degree n> 1 have interpretation as bound states. Therefore bound states, which are the basic problem of perturbative quantum field theory, would have purely number theoretic meaning. As a matter fact, infinite rationals reducing to real units in real sense represent zero energy states in zero energy ontology and it is natural to assign Galois group hierarchies also to the poles of this rational function.

Summarizing, the infinite prime - irreducible polynomial - braid - quantum state connection suggests very deep connections between number theory, algebraic geometry, topological quantum field theories, and super-symmetric quantum field theories. The article Motives and Infinite Primes gives a more detailed discussion.

Defining integration in p-adic context is one of the basic challenges of quantum TGD in which real and various p-adic physics ought to be unified to a larger theory by realizing what I have called number theoretical Universality. Grothendieck's motivic comology can be seen as a program for the realization of integration of forms making sense also in p-adic context. In the following I shall discuss some aspects of the problem in TGD framework. The discussion of course fails to satisfy all standards of mathematical rigor but it relies of extremely deep and general physical principles and my conviction is that good physics is the best guideline for developing good mathematics.

Number theoretic universality, residue integrals, and symplectic symmetry

A key challenge in the realization of the number theoretic universality is the definition of p-adic definite integral. In twistor approach integration reduces to the calculation of multiple residue integrals over closed varieties. These could exist also for p-adic number fields. Even more general integrals identifiable as integrals of forms can be defined in terms of motivic cohomology.

Yangian symmetry (see this and this) is the symmetry behind the successes of twistor Grassmannian approach and has a very natural realization in zero energy ontology (see this). Also the basic prerequisites for twistorialization are satisfied. Even more, it is possible to have massive states as bound states of massless ones and one can circumvent the IR difficulties of massless gauge theories. Even UV divergences are tamed since virtual particles consist of massless wormhole throats without bound state condition on masses. Space-like momentum exchanges correspond to pairs of throats with opposite sign of energy.

Algebraic universality could be realized if the calculation of the scattering amplitudes reduces to multiple residue integrals just as in twistor Grassmannian approach. This is because also p-adic integrals could be defined as residue integrals. For rational functions with rational coefficients field the outcome would be an algebraic number apart from power of 2π, which in p-adic framework is a nuisance unless it is possible to get rid of it by a proper normalization or unless one can accepts the infinite-dimensional transcendental extension defined by 2π. It could also happen that physical predictions do not contain the power of 2π.

Motivic cohomology defines much more general approach allowing to calculate analogs of integrals of forms over closed varieties for arbitrary number fields. In motivic integration - to be discussed below - the basic idea is to replace integrals as real numbers with elements of so called scissor group whose elements are geometric objects. In the recent case one could consider the possibility that (2π)n is interpreted as torus (S1)n regarded as an element of scissor group which is free group formed by formal sums of varieties modulo certain natural relations meaning.

Motivic cohomology allows to realize integrals of forms over cycles also in p-adic context. Symplectic transformations are transformation leaving areas invariant. Symplectic form and its exterior powers define natural volume measures as elements of cohomology and p-adic variant of integrals over closed and even surfaces with boundary might make sense. In TGD framework symplectic transformations indeed define a fundamental symmetry and quantum fluctuating degrees of freedom reduce to a symplectic group assignable to δ M4+/-× CP2 in well-defined sense (see this). One might hope that they could allow to define scissor group with very simple canonical representatives- perhaps even polygons- so that integrals could be defined purely algebraically using elementary area (volume) formulas and allowing continuation to real and p-adic number fields. The basic argument could be that varieties with rational symplectic volumes form a dense set of all varieties involved.

How to define the p-adic variant for the exponent of Kähler action?

The exponent of Kähler function defined by the Kähler action (integral of Maxwell action for induced Kähler form) is central for quantum at least in the real sector of WCW. The question is whether this exponent could have p-adic counterpart and if so, how it should be defined.

In the real context the replacement of the exponent with power of p changes nothing but in the p-adic context the interpretation is affected in a dramatic manner. Physical intuition provided by p-adic thermodynamics (see this) suggests that the exponent of Kähler function is analogous to Bolzmann weight replaced in the p-adic context with non-negative power of p in order to achieve convergence of the series defining the partition function not possible for the exponent function in p-adic context.

  1. The quantization of Kähler function as K= rlog(m/n), where r is integer, m>n is divisible by a positive power of p and n is indivisible by a power of p, implies that the exponent of Kähler function is of form (m/n)r and therefore exists also p-adically. This would guarantee the p-adic existence of the vacuum functional for any prime dividing m and for a given prime p would select a restricted set of p-adic space-time sheets (or partonic 2-surfaces) in the intersection of real and p-adic worlds. It would be possible to assign several p-adic primes to a given space-time sheet (or partonic 2-surface). In elementary particle physics a possible interpretation is that elementary particle can correspond to several p-adic mass scales differing by a power of two (see this). One could also consider a more general quantization of Kähler action as sum K=K1+K2 where K1=rlog(m/n) and K2=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e1/n can be considered.

  2. The natural continuation to p-adic sector would be the replacement of integer coefficient r with a p-adic integer. For p-adic integers not reducing to finite integers the p-adic norm of the vacuum functional would however vanish and their contribution to the transition amplitude vanish unless the number of these space-time sheets increases with an exponential rate making the net contribution proportional to a finite positive power of p. This situation would correspond to a critical situation analogous to that encounted in string models as the temperature approaches Hagedorn temperature and the number states with given energy increases as fast as the Boltzmann weight. Hagedorn temperature is essentially due to the extended nature of particles identified as strings. Therefore this kind of non-perturbative situation might be encountered also now.

  3. Rational numbers m/n with n not divisible by p are also infinite as real integers. They are somewhat problematic. Does it make sense to speak about algebraic extensions of p-adic numbers generated by p1/n and giving n-1 fractional powers of p in the extension or does this extension reduce to something equivalent with the original p-adic number field when one redefines the p-adic norm as |x|p → |vert x|1/n? Physically this kind of extension could have a well defined meaning. If this does not make sense, it seems that one must treat p-adic rationals as infinite real integers so that the exponent would vanish p-adically.

  4. If one wants that Kähler action exists p-adically a transcendental extension of rational numbers allowing all powers of log(p) and log(k), where k<p is primitive p-1:th root of unity in G(p). A weaker condition would be an extension to a ring with containing only log(p) and log(k) but not their powers. That only single k<p is needed is clear from the identity log(kr)=rlog(k), from primitive root property, and from the possibility to expand log(kr+pn), where n is p-adic integer, to powers series with respect to p. If the exponent of Kähler function is the quantity coding for physics and naturally required to be ordinary p-adic number, one could allow log(p) and log(k) to exists only in symbolic sense or in the extension of p-adic numbers to a ring with minimal dimension.

    Remark: One can get rid of the extension by log(p) and log(k) if one accepts the definition of p-adic logarithm as log(x)=log(p-kx/x0) for x=pk(x0+ py), |y|p<1. To me this definition looks somewhat artificial since this function is not strictly speaking the inverse of exponent function but it might have a deeper justification.

  5. What happens in the real sector? The quantization of Kähler action cannot take place for all real surfaces since a discrete value set for Kähler function would mean that WCW metric is not defined. Hence the most natural interpretation is that the quantization takes place only in the intersection of real and p-adic worlds, that is for surfaces which are algebraic surfaces in some sense. What this actually means is not quite clear. Are partonic 2-surfaces and their tangent space data algebraic in some preferred coordinates? Can one find a universal identification for the preferred coordinates- say as subset of imbedding space coordinates selected by isometries?

If this picture inspired by p-adic thermodynamics holds true, p-adic integration at the level of WCW would give analog of partition function with Boltzman weight replaced by a power of p reducing a sum over contributions corresponding to different powers of p with WCW integra.l over space-time sheets with this value of Kähler action defining the analog for the degeneracy of states with a given value of energy. The integral over space-time sheets corresponding to fixed value of Kähler action should allow definition in terms of a symplectic form defined in the p-adic variant of WCW. In finite-dimensional case one could worry about odd dimension of this sub-manifold but in infinite-dimensional case this need not be a problem. Kähler function could defines one particular zero mode of WCW Kähler metric possessing an infinite number of zero modes.

One should also give a meaning to the p-adic integral of Kähler action over space-time surface assumed to be quantized as multiples of log(m/n).

  1. The key observation is that Kähler action for preferred extrememals reduces to 3-D Chern-Simons form by the weak form of electric-magnetic duality. Therefore the reduction to cohomology takes place and the existing p-adic cohomology gives excellent hopes about the existence of the p-adic variant of Kähler action. Therefore the reduction of TGD to almost topological QFT would be an essential aspect of number theoretical universality.

  2. This integral should have a clear meaning also in the intersection of real and p-adic world. Why the integrals in the intersection would be quantized as multiple of log(m/n), m/n divisible by a positive power of p? Could log(m/n) relate to the integral of ∫1p dx/x, which brings in mind ∮ dz/z in residue calculus. Could the integration range [1,m/n] be analogous to the integration range [0,2π]. Both multiples of 2π and logarithms of rationals indeed emerge from definite integrals of rational functions with rational coefficients and allowing rational valued limits and in both cases 1/z is the rational function responsible for this.

  3. log(m/n) would play a role similar to 2π in the approach based on motivic integration where integral has geometric objects as its values. In the case of 2π the value would be circle. In the case of log(m/n) the value could be the arc between the points r=m/n>1 and r=1 with r identified the radial coordinate of light-cone boundary with conformally invariant length measures dr/r. One can also consider the idea that log(m/n) is the hyperbolic angle analogous to 2π so that these two integrals could correspond to hyper-complex and complex residue calculus respectively.

  4. TGD as almost topological QFT means that for preferred extremals the Kähler action reduces to 3-D Chern-Simons action, which is indeed 3-form as cohomology interpretation requires, and one could consider the possibility that the integration giving log(m/n) factor to Kähler action is associated with the integral of Chern-Simons action density in time direction along light-like 3-surface and that the integral over the transversal degrees of freedom could be reduced to the flux of the induced CP2 Kähler form. The logarithmic quantization of the effective distance between the braid end points the in metric defined by modified gamma matrices has been proposed earlier.

Since p-adic objects do not possess boundaries, one could argue that only the integrals over closed varieties make sense. Hence the basic premise of cohomology would fail when one has p-adic integral over braid strand since it does not represent closed curve. The question is whether one could identify the end points of braid in some sense so that one would have a closed curve effectively or alternatively relative cohomology. Periodic boundary conditions is certainly one prerequisite for this kind of identification.

  1. In one of the many cohomologies known as quantum cohomology one indeed assumes that the intersection of varieties is fuzzy in the sense that two surfaces for points are connected by a curve of certain kind known as pseudo-holomorphic curve can be said to intersect at these points.

  2. In the construction of the solutions of the modified Dirac equation one assumes periodic boundary conditions so that in physical sense these points are identified (see this). This assumption actually reduces the locus of solutions of the modified Dirac equation to a union of braids at light-like 3-surfaces so that finite measurement resolution for which discretization defines space-time correlates becomes an inherent property of the dynamics. The coordinate varying along the braid strands is light-like so that the distance in the induced metric vanishes between its end points (unlike the distance in the effective metric defined by the modified gamma matrices): therefore also in metric sense the end points represent intersection point. Also the effective 2-dimensionality means are effectively one and same point.

  3. The effective metric 2-dimensionality of the light-like 2-surfaces implies the counterpart of conformal invariance with the light-like coordinate varying along braid strands so that it might make sense to say that braid strands are pseudo-holomorphic curves. Note also that the end points of a braid along light-like 3-surface are not causally independent: this is why M-matrix in zero energy ontology is non-trivial. Maybe the causal dependence together with periodic boundary conditions, light-likeness, and pseudo-holomorphy could imply a variant of quantum cohomology and justify the p-adic integration over the braid strands.

Motivic integration

While doing web searches related to motivic cohomology I encountered also the notion of motivic measure proposed first by Kontsevich. Motivic integration is a purely algebraic procedure in the sense that assigns to the symbol defining the variety for which one wants to calculate measure. The measure is not real valued but takes values in so called scissor group, which is a free group with group operation defined by a formal sum of varieties subject to relations. Motivic measure is number theoretical universal in the sense that it is independent of number field but can be given a value in particular number field via a homomorphism of motivic group to the number field with respect to sum operation.

Some examples are in order.

  1. A simple example about scissor group is scissor group consisting operations needed in the algorithm transforming plane polygon to a rectangle with unit edge. Polygon is triangulated; triangles are transformed to rectangle using scissors; long rectangles are folded in one half; rectangles are rescaled to give an unit edge (say in horizontal direction); finally the resulting rectangles with unit edge are stacked over each other so that the height of the stack gives the area of the polygon. Polygons which can be transformed to each other using the basic area preserving building bricks of this algorithm are said to be congruent.

    The basic object is the free abelian group of polygons subject to two relations analogous to second homology group. If P is polygon which can be cut to two polygons P1 and P2 one has [P]=[P1]+[P2]. If P and P' are congruent polygons, one has [P]=[P']. For plane polygons the scissor group turns out to be the group of real numbers and the area of polygon is the area of the resulting rectangle. The value of the integral is obtained by mapping the element of scissor group to a real number by group homomorphism.

  2. One can also consider symplectic transformations leaving areas invariant as allowed congruences besides the slicing to pieces as congruences appearing as parts of the algorithm leading to a standard representation. In this framework polygons would be replaced by a much larger space of varieties so that the outcome of the integral is variety and integration means finding a simple representative for this variety using the relations of the scissor group. One might hope that a symplectic transformations singular at the vertices of polygon combined with with scissor transformations could reduce arbitrary area bounded by a curve into polygon.

  3. One can identify also for discrete sets the analog of scissor group. In this case the integral could be simply the number of points. Even more abstractly: one can consider algebraic formulas defining algebraic varieties and define scissor operations defining scissor congruences and scissor group as sums of the formulas modulo scissor relations. This would obviously abstract the analytic calculation algorithm for integral. Integration would mean that transformation of the formula to a formula stating the outcome of the integral. Free group for formulas with disjunction of formulas is the additive operation(see this). Congruence must correspond to equivalence of some kind. For finite fields it could be bijection between solutions of the formulas. The outcome of the integration is the scissor group element associated with the formula defining the variety.

  4. For residue integrals the free group would be generated as formal sums of even-dimensional complex integration contours. Two contours would be equivalent if they can be deformed to each other without going through poles. The standard form of variety consists of arbitrary small circles surrounding the poles of the integrand multiplied by the residues which are algebraic numbers for rational functions. This generalizes to rational functions with both real and p-adic coefficients if one accepts the identification of integral as a variety modulo the described equivalence so that (2π)n corresponds to torus (S1)n. One can replace torus with 2π if one accepts an infinite-dimensional algebraic extension of p-adic numbers by powers of 2π. A weaker condition is that one allows ring containing only the positive powers of 2π.

  5. The Grassmannian twistor approach for two-loop hexagon Wilson loop gives classical polylogarithms Lk(s) (see this). General polylogarithm is defined by obey the recursion formula:

    Lis+1(z)= ∫0zLis(t)dt/t .

    Ordinary logarithm Li1(s) = -log(1-s) exists p-adically and generates a hierarchy containing dilogarithm, trilogarithm, and so on, which each exist p-adically for |x|p < 1 as is easy to see. If one accepts the general definition of p-adic logariths one finds that the entire function series exists p-adically for integer values of s. An interesting question is how strong constraints p-adic existence gives to the thetwistor loop integrals and to the underlying QFT.

  6. The ring having p-adic numbers as coefficients and spanned by transcendentals log(k) and log(p), where k is primitive root of unity in G(p) emerges in the proposed p-adicization of vacuum functional as exponent of Kähler action. The action for the preferred extremals reducing to 3-D Chern-Simons action for space-time surfaces in the intersection of real and p-adic worlds would be expressible p-adically as a linear combination of log(p) and log(k). log(m/n) expressible in this manner p-adically would be the symbolic outcome of p-adic integral ∫ dx/x between rational points. x could be identified as a preferred coordinate along braid strand. A possible identification for x earlier would be as the length in the effective metric defined by modified gamma matrices appearing in the modified Dirac equation (see this).

Infinite rationals and multiple residue integrals as Galois invariants and Galois groups as symmetry groups of quantum physics

In TGD framework one could consider also another kind of cohomological interpretation. The basic structures are braids at light-like 3-surfaces and space-like 3-surfaces at the ends of space-time surfaces. Braids intersects have common ends points at the partonic 2-surfaces at the light-like boundaries of a causal diamond. String world sheets define braid cobordism and in more general case 2-knot (see this)). Strong form of holography with finite measurement resolution would suggest that physics is coded by the data associated with the discrete set of points at partonic 2-surfaces. Cohomological interpretation would in turn would suggest that these points could be identified as intersections of string world sheets and partonic 2-surface defining dual descriptions of physics and would represent intersection form for string world sheets and partonic 2-surfaces.

Infinite rationals define rational functions and one can assign to them residue integrals if the variables xn are interpreted as complex variables. These rational functions could be replaced with a hierarchy of sub-varieties defined by their poles of various dimensions. Just as the zeros allow realization as braids or braids also poles would allow a realization as braids of braids. Hence the n-fold residue integral could have a representation in terms of braids. Given level of the braid hierarchy with n levels would correspond to a level in the hierarchy of complex varieties with decreasing complex dimension.

One can assign also to the poles (zeros of polynomial in the denominator of rational function) Galois group and obtains a hierarchy of Galois groups in this manner. Also the braid representation would exists for these Galois groups and define even cohomology and homology if they do so for the zeros. The intersections of braids with of the partonic 2-surfaces would represent the poles in the preferred coordinates and various residue integrals would have representation in terms of products of complex points of partonic 2-surface in preferred coordinates. The interpretation would be in terms of quantum classical correspondence.

Galois groups transform the poles to each other and one can ask how much information they give about the residue integral. One would expect that the n-fold residue integral as a sum over residues expressible in terms of the poles is invariant under Galois group. This is the case for the simplest integrals in plane with n poles and probably quite generally. Physically the invariance under the hierarchy of Galois group would mean that Galois groups act as the symmetry group of quantum physics. This conforms with the number theoretic vision and one could justify the formula for the residue integral also as a definition motivated by the condition of Galois invariance. Of course, all symmetric functions of roots would be Galois invariants and would be expected to appear in the expressions for scattering amplitudes.

The Galois groups associated with zeros and poles of the infinite rational seem to have a clear physical significance. This can be understood in zero energy ontology if positive (negative) physical states are indeed identifiable as infinite integers and if zero energy states can be mapped to infinite rationals which as real numbers reduce to real units. The positive/negative energy part of the zero energy state would correspond to zeros/poles in this correspondence. An interesting question is how strong correlations the real unit property poses on the two Galois group hierarchies. The asymmetry between positive and negative energy states would have interpretation in terms of the thermodynamic arrow of geometric time (see this) implied by the condition that either positive or negative energy states correspond to state function reduced/prepared states with well defined particle numbers and minimum amount of entanglement.

For more details see the new chapter Infinite Primes and Motives of "TGD as Generalized Number Theory" or the article with same title.

Tuesday, May 24, 2011

p-Adic physics as a correlate for Boolean cognition

I have had some discussions with Stephen King and Hitoshi Kitada in a closed discussion group about the idea that the duality between Boolean algebras and Stone spaces could be important for the understanding of consciousness, at least cognition. In this vision Boolean algebras would represent conscious mind and Stone spaces would represent the matter: space-time would emerge.

I am personally somewhat skeptic because I see consciousness and matter as totally different levels of existence. Consciousness (and information) is about something, matter just is. Consciousness involves always a change as we no from basic laws about perception. There is of course also the experience of free will and the associated non-determinism. Boolean algebra is a model for logic, not for conscious logical reasoning. There are also many other aspects of consciousness making it very difficult to take this kind of duality seriously.

I am also skeptic about the emergence of space-time say in the extremely foggy form as it used in entropic gravity arguments. Recent day physics poses really strong constraints on our view about space-time and one must take them very seriously.

This does not however mean that Stone spaces could not serve as geometrical correlates for Boolean consciousness. In fact, p-adic integers can be seen as a Stone space naturally assignable to Boolean algebra with infinite number of bits.

1. Innocent questions

I was asked to act as some kind of mathematical consultant and explain what Stone spaces actually are and whether they could have a connection to p-adic numbers. Anyone can of course go to Wikipedia and read the article Stone's representation theorem for Boolean algebras. For a layman this article does not however tell too much.

Intuitively the content of the representation theorem looks rather obvious, at least at the first sight. As a matter fact, the connection looks so obvious that physicists often identify the Boolean algebra and its geometric representation without even realizing that two different things are in question. The subsets of given space- say Euclidian 3-space- with union and intersection as basic algebraic operations and inclusion of sets as ordering relation defined a Boolean algebra for the purposes of physicist. One can assign to each point of space a bit. The points for which the value of bit equals to one define the subset. Union of subsets corresponds to logical OR and intersection to AND. Logical implication B→ A corresponds to A contains B.

When one goes to details problems begin to appear. One would like to have some non-trivial form of continuity.

  1. For instance, if the sets are form open sets in real topology their complements representing negations of statements are closed, not open. This breaks the symmetry between statement and it negation unless the topology is such that closed sets are open. Stone's view about Boolean algebra assumes this. This would lead to discrete topology for which all sets would be open sets and one would lose connection with physics where continuity and differential structure are in key role.

  2. Could one then dare to disagree with Stone;-) and allow both closed and open sets of E3 in real topology and thus give up clopen assumption? Or could one tolerate the asymmetry between statements and their negations and give some special meaning for open or closet sets- say as kind of axiomatic statements holding true automatically. If so, one an also consider algebraic varieties of lower dimension as collections of bits which are equal to one. In Zariski topology used in algebraic geometry these sets are closed. Again the complements would be open. Could one regard the lower dimensional varieties as identically true statements so that the set of identically true statements would be rather scarce as compared to falsities? If one tolerates some quantum TGD, one could ask whether the 4-D quaternionic/associative varieties defining classical space-times and thus classical physics could be identified as the axiomatic truths. Associativity would be the basic truth inducing the identically true collections of bits.

2. Stone theorem and Stone spaces

For reasons which should be clear it is perhaps a good idea to consider in more detail what Stone duality says. Stone theorem states that Boolean algebras are dual with their Stone spaces. Logic and certain kind of geometry are dual. More precisely, any Boolean algebra is isomorphic to closed open subsets of some Stone space and vice versa. Stone theorem respects category theory. The homomorphisms between Boolean algebras A and B corresponds to homomorphism between Stone spaces S(B) and S(A): one has contravariant functor between categories of Boolean algebras and Stone spaces. In the following set theoretic realization of Boolean algebra provides the intuitive guidelines but one can of course forget the set theoretic picture altogether and consider just abstract Boolean algebra.

  1. Stone space is defined as the space of homomorphisms from Boolean algebra to 2-element Boolean algebra. More general spaces are spaces of homomorphisms between two Boolean algebras. The analogy in the category of linear spaces would be the space of linear maps between two linear spaces. Homomorphism is in this case truth preserving map: h(a AND B) = h(a) AND h(B), h(a OR B) = h(a) OR h(B) and so on. These homomorphisms are like always-the-truth-tellers, which are of course social catastrophes;-).

  2. For any Boolean algebra Stone space is compact, totally disconnected Hausdorff space. Conversely, for any topological space, the subsets, which are both closed and open define Boolean algebra. Note that for a real line this would give 2-element Boolean algebra. Set is closed and open simultaneously only if its boundary is empty and in p-adic context there are no boundaries. Therefore for p-adic numbers closed sets are open and the sets of p-adic numbers with p-adic norm above some lower bound and having some set of fixed pinary digits define closed-open subsets.

  3. Stone space dual to the Boolean algebra does not conform with the physicist's ideas about space-time. Stone space is a compact totally disconnected Hausdorff space. Disconnected space is representable as a union of two or more disjoint open sets. For totally disconnected space this is true for every subset. Path connectedness is stronger notion than connected and says that two points of the space can be always connected by a curve defined as a mapping of real unit interval to the space. Our physical space-time seems to be however connected in real sense.

  4. The points of the Stone space S(B) can be identified ultrafilters. Ultrafilter defines homomorphism of B to 2-element of Boolean algebra Boolean algebra. Set theoretic realization allows to understand what this means. Ultrafilter is a set of subsets with the property that intersections belong to it and if set belongs to it also sets containing it belong to it: this corresponds to the fact that set inclusion A ⊃ B corresponds to logical implication. Either set or its complement belongs to ultrafilter (either statement or its negation is true). Empty set does not. Ultrafilter obviously corresponds to a collection of statements which are simultaneously true without contradictions. The sets of ultrafilter correspond to the statements interpreted as collections of bits for which each bit equals to 1.

  5. The subsets of B containing a fixed point b of Boolean algebra define an ultrafilter and imbedding of b to the Stone space by assigning to it this particular principal ultrafilter. b represents a statement which is always true, kind of axiom for this principal ultrafilter and ultrafilter is the set of all statements consistent with b.

    Actually any finite set in the Boolean algebra consisting of a collection of fixed bits bi defines an ultrafilter as the set all subsets of Boolean algebra containing this subset. Therefore the space of all ultra-filters is in one-one correspondence with the space of subsets of Boolean statements. This set corresponds to the set of statements consistent with the truthness of bi analogous to axioms.

3. 2-adic integers and 2-adic numbers as Stone spaces

I was surprised to find that p-adic numbers are regarded as a totally disconnected space. The intuitive notion of connected is that one can have a continuous curve connecting two points and this is certainly true for p-adic numbers with curve parameter which is p-adic number but not for curves with real parameter which became obvious when I started to work with p-adic numbers and invented the notion of p-adic fractal. In other words, p-adic integers form a continuum in p-adic but not in real sense. This example shows how careful one must be with definitions. In any case, to my opinion the notion of path based on p-adic parameter is much more natural in p-adic case. For given p-adic integers one can find p-adic integers arbitrary near to it since at the limit n→∞ the p-adic norm of pn approaches zero. Note also that most p-adic integers are infinite as real integers.

Disconnectedness in real sense means that 2-adic integers define an excellent candidate for a Stone space and the inverse of the Stone theorem allows indeed to realize this expectation. Also 2-adic numbers define this kind of candidate since 2-adic numbers with norm smaller than 2n for any n can be mapped to 2-adic integers. One would have union of Boolean algebras labelled by the 2-adic norm of the 2-adic number. p-Adic integers for a general prime p define obviously a generalization of Stone space making sense for effectively p-valued logic: the interpretation of this will be discussed below.

Consider now a Boolean algebra consisting of all possible infinitely long bit sequences. This algebra corresponds naturally to 2-adic integers. The generating Boolean statements correspond to sequences with single non-vanishing bit: by taking the unions of these points one obtains all sets. The natural topology is that for which the lowest bits are the most significant. 2-adic topology realizes this idea since n:th bit has norm 2-n. 2-adic integers as an p-adic integers are as spaces totally disconnected.

That 2-adic integers and more generally, 2-adic variants of n-dimensional manifolds would define Stone bases assignable to Boolean algebras is consistent with the identification of p-adic space-time sheets as correlates of cognition. Each point of 2-adic space-time sheet would represent 8 bits as a point of 8-D imbedding space. In TGD framework WCW ("world of classical worlds") spinors correspond to Fock space for fermions and fermionic Fock space has natural identification as a Boolean algebra. Fermion present/not present in given mode would correspond to true/false. Spinors decompose to a tensor product of 2-spinors so that the labels for Boolean statements form a Boolean algebra two.

In TGD Universe life and thus cognition reside in the intersection of real and p-adic worlds. Therefore the intersections of real and p-adic partonic 2-surfaces represent the intersection of real and p-adic worlds, those Boolean statements which are expected to be accessible for conscious cognition. They correspond to rational numbers or possibly numbers in n algebraic estension of rationals. For rationals pinary expansion starts to repeat itself so that the number of bits is finite. This intersection is also always discrete and for finite real space-time regions finite so that the identification looks a very natural since our cognitive abilities seem to be rather limited. In TGD inspired physics magnetic bodies are the key players and have much larger size than the biological body so that their intersection with their p-adic counterparts can contain much more bits. This conforms with the interpretation that the evolution of cognition means the emergence of increasingly longer time scales. Dark matter hierarchy realized in terms of hierarchy of Planck constants realizes this.

3. What about p-adic integers with p>2?

The natural generalization of Stone space would be to a geometric counterpart of p-adic logic which I discussed for some years ago. The representation of the statements of p-valued logic as sequences of pinary digits makes the correspondence trivial if one accepts the above represented arguments. The generalization of Stone space would consist of p-adic integers and imbedding of a p-valued Boolean algebra would map the number with only n:th digit equal to 1,...,p-1 to corresponding p-adic number.

One should however understand what p-valued statements mean and why p-adic numbers near powers of 2 are important. What is clear that p-valued logic is too romantic to survive. At least our every-day cognition is firmly anchored to a reality where everything is experience to be true or false.

  1. The most natural explanation for p> 2 adic logic is that all Boolean statements do not allow a physical representation and that this forces reduction of 2n valued logic to p< 2n valued one. For instance, empty set in the set theoretical representation of Boolean logic has no physical representation. In the same manner, the state containing no fermions fails to represent anything physically. One can represent physically at most 2n-1 one statements of n-bit Boolean algebra and one must be happy with n-1 completely represented digits. The remaining statements containing at least one non-vanishing digit would have some meaning, perhaps the last digit allowed could serve as a kind of parity check.

  2. If this is accepted then p-adic primes near to power 2n of 2 but below it and larger than the previous power 2n-1 can be accepted and provide a natural topology for the Boolean statements grouping the binary digits to p-valued digit which represents the allowed statements in 2n valued Boolean algebra. Bit sequence as a unit would be represented as a sequence of physically realizable bits. This would represent evolution of cognition in which simple yes or not statements are replaced with sequences of this kind of statements just as working computer programs are fused as modules to give larger computer programs. Note that also for computers similar evolution is taking place: the earliest processors used byte length 8 and now 32, 64 and maybe even 128 are used.

  3. Mersenne primes Mn=2n-1 would be ideal for logic purposes and they indeed play a key role in quantum TGD. Mersenne primes define p-adic length scales characterize many elementary particles and also hadron physics. There is also evidence for p-adically scaled up variants of hadron physics (also leptohadron physics allowed by the TGD based notion of color predicting colored excitations of leptons). LHC will certainly show whether M89 hadron physics at TeV energy scale is realized and whether also leptons might have scaled up variants.

  4. For instance, M127 assignable to electron secondary p-adic time scale is .1 seconds, the fundamental time scale of sensory perception. Thus cognition in .1 second time scale single pinary statement would contain 126 digits as I have proposed in the model of memetic code. Memetic codons would correspond to 126 digit patterns with duration of .1 seconds giving 126 bits of information about percept.

If this picture is correct, the interpretation of p-adic space-time sheets- or rather their intersections with real ones- would represent space-time correlates for Boolean algebra represented at quantum level by fermionic many particle states. In quantum TGD one assigns with these intersections braids- or number theoretic braids- and this would give a connection with topological quantum field theories (TGD can be regarded as almost topological quantum field theory).

4. One more road to TGD

The following arguments suggests one more manner to end up with TGD by requiring that fermionic Fock states identified as a Boolean algebra have their Stone space as space-time correlate required by quantum classical correspondence. Second idea is that space-time surfaces define the collections of binary digits which can be equal to one: kind of eternal truths.In number theoretical vision associativity condition in some sense would define these divine truths. Standard model symmetries are a must- at least as their p-adic variants -and simple arguments forces the completion of discrete lattice counterpart of M4 to a continuum.

  1. If one wants Poincare symmetries at least in p-adic sense then a 4-D lattice in M4 with SL(2,Z)×T4, where T4 is discrete translation group is a natural choice. SL(2,Z) acts in discrete Minkowski space T4 which is lattice. Poincare invariance would be discretized. Angles and relative velocities would be discretized, etc..

  2. The p-adic variant of this group is obtained by replacing Z and T4 by their p-adic counterparts: in other words Z is replaced with the group Zpof p-adic integers. This group is p-adically continuous group and acts continuously in T4 defining p-adic variant of Minkowski space consisting of all bit sequences consisting of 4-tuples of bits. Only in real sense one would have discreteness: note also that most points would be at infinity in real sense. Therefore it is possible to speak about analytic functions, differential calculus, and to write partial differential equations and to solve them. One can construct group representations and talk about angular momentum, spin and 4-momentum as labels of quantum states.

  3. If one wants standard model symmetries p-adically one must replace T4 with T4 × CP2. CP2 would be now discrete version of CP2 obtained from discrete complex space C3 by identifying points different by a scaling by complex integer. Discrete versions of color and electroweak groups would be obtained.

The next step is to ask what are the laws of physics. TGD fan would answer immediately: they are of course logical statements which can be true identified as subsets of T4× CP2 just as subset in Boolean algebra of sets corresponds to bits which are true.

  1. The collections of 8-bit sequences consisting of only 1:s would define define 4-D surfaces in discrete T4× CP2 . Number theoretic vision would suggest that they are quaternionic surfaces so that one associativity be the physical law at geometric level. The conjecture is that preferred extremals of Kähler action are associative surfaces using the definition of associativity as that assignable to a 4-plane defined by modified gamma matrices at given point of space-time surface.

  2. Induced gauge field and metric make sense for p-adic integers. p-Adically the field equations for Kähler action make also sense. These p-adic surfaces would represent the analog of Boolean algebra. They would be however something more general than Stone assumes since they are not closed-open in the 8-D p-adic topology.

One however encounters a problem.

  1. Although the field equations associated with Kähler action make sense, Kähler action itself does not exists as integral nor does the genuine minimization make sense since p-adically numbers are not well ordered and one cannot in general say which of two numbers is the larger one. This is a real problem and suggests that p-adic field equations are not enough and must be accompanied by real ones. Of course, also the metric properties of p-adic space-time are in complete conflict with what we believe about them. Note however that discretized variants of symmetries might make sense but would reflect finite measurement resolution and cognitive resolution.

  2. One could argue that for preferred extremals the integral defining Kähler action is expressible as an integral of 4-form whose value could be well-defined since integrals of forms for closed surface make sense in p-adic cohomology theory pioneered by Grothendieck. The idea would be to use the definition of K\"ahler action making sense for preferred extremals as its definition in p-adic context. I have indeed proposed that space-time surfaces define representatives for homology with inspiration coming from TGD as almost topological QFT. This would give powerful constraints on the theory in accordance with the interpretation as a generalized Bohr orbit.

  3. This argument together with what we know about the topology of space-time on basis of everyday experience however more or less forces the conclusion that also real variant of M4×CP2 is there and defines the proper variational principle. The finite points (on real sense) of T4× CP2 (in discrete sense) would represent points common to real and p-adic worlds and the identification in terms of braid points makes sense if one accepts holography and restricts the consideration to partonic 2-surfaces at boundaries of causal diamond. These discrete common would represent the intersection of cognition and matter and living systems and provide a representation for Boolean cognition.

  4. Finite measurement resolution enters into the picture naturally. The proper time distance between the tips would be quantized in multiples of CP2 length. There would be several choices for the discretized imbedding space corresponding to different distance between lattice points: the interpretation is in terms of finite measurement resolution.

It should be added that discretized variant of Minkowski space and its p-adic variant emerge in TGD also in different manner in zero energy ontology.

  1. The discrete space SL(2,Z) × T4 would have also interpretation as acting in the moduli space for causal diamonds identified as intersections of future and past directed light-cones. T4 would represent lattice for possible positions of the lower tip of CD and and SL(2,C) leaving lower tip invariant would act on hyperboloid defined by the position of the upper tip obtained by discrete Lorentz transformations. This leads to cosmological predictions (quantization of red shifts). CP2 length defines a fundamental time scale and the number theoretically motivated assumption is that the proper time distances between the tips of CDs come as integer multiples of this distance. The stronger condition that they come as octaves of this scale is not in fashion anymore;-).

  2. The stronger condition explaining p-adic length scale hypothesis would be that only octaves of the basic scale are allowed. This option is not consistent with zero energy ontology. The reason is that for more general hypothesis the M-matrices of the theory for Kac-Moody type algebra with finite-dimensional Lie algebra replaced with an infinite-dimensional algebra representing hermitian square roots of density matrices and powers of the phase factor replaced with powers of S-matrix. All integer powers must be allowed to obtain generalized Kac-Moody structure, not only those which are powers of 2 and correspond naturally to integer valued proper time distance between the tips of CD. Zero energy states would define the symmetry Lie-algebra of S-matrix with generalized Yangian structure.

  3. p-Adic length scale hypothesis would be an outcome of physics and it would not be surprising that primes near power of two are favored because they optimize Boolean cognition.

The outcome is TGD as the skeptical reader already knowing my tricks might have guessed;-). Reader can of course imagine alternatives but remember the potential difficulties due to the fact that minimization in p-adic sense does not make sense and action defined as integral does not exist p-adically. Also the standard model symmetries and quantum classical correspondence are to my opinion "must":s.

5. A connection between cognition, number theory, algebraic geometry, topology, and quantum physics

Stone space is synonym for profinite space. Galois groups associated with algebraic extensions of number fields represent an extremely general class of profinite groups. Every profinite group appears in Galois theory of some field K. The most most interesting ones for algebraic extensions F/K of field K are inverse limits of Gal(F1/K), where F1 varies over all intermediate fields. Profinite groups appear also as fundamental groups in algebraic geometry. In algebraic topology fundamental groups are in general not profinite. Profiniteness means that p-adic representations are especially natural for profinite groups.

There is a fascinating connection between infinite primes and algebraic geometry summarized briefly here. This connection lead to the proposal that Galois groups - or rather their projective variants- can be represented as braid groups acting on 2-dimensional surfaces. These findings suggest a deep connection between space-time correlates of Boolean cognition, number theory, algebraic geometry, and quantum physics and TGD based vision about representations of Galois groups as groups lifted to braiding groups acting on the intersection of real and p-adic variants of partonic 2-surface conforms with this.

Fermat theorem serves as a good illustration between the connection between cognitive representations and algebraic geometry. A very general problem of algebraic geometry is to find rational points of an algebraic surface. These can be identified as common rational points of the real and p-adic variant of the surface. The interpretation in terms of consciousness theory would be as points defining cognitive representation as rational points common to real partonic 2-surface and and its p-adic variants. The mapping to polynomials given by their representation in terms of infinite primes to braids of braids of braids.... at partonic 2-surfaces would provide the mapping of n-dimensional problem to a 2-dimensional one (see this).

One considers the question whether there are integer solutions to the equation xn+yn+zn=1. This equation defines 2-surfaces in both real and p-adic spaces. In p-adic context it is easy to construct solutions but they usually represent infinite integers in real sense. Only if the expansion in powers of p contains finite number of powers of p, one obtains real solution as finite integers.

The question is whether there are any real solutions at all. If they exist they correspond to the intersections of the real and p-adic variants of these surfaces. In other words p-adic surface contains cognitively representable points. For n>2 Fermat's theorem says that only single point x=y=z=0 exists so that only single p-adic multi-bit sequence (0,0,0,...) would be cognitively representable.

This relates directly to our mathematical cognition. Linear and quadratic equations we can solve and in these cases the number in the intersection of p-adic and real surfaces is indeed very large. We learn the recipes already in school! For n>2 difficulties begin and there are no general recipes and it requires mathematician to discover the special cases: a direct reflection of the fact that the number of intersection points for real and p-adic surfaces involved contains very few points.

For details see the new chapter Motives and Infinite Primes of "Physics as a Generalized Number Theory" or the article with same title.

Sunday, May 22, 2011


Sean Carroll writes about breakdown of classical gravity in Cosmic variance. Recall that the galactic dark matter problem arose with the observation that the velocity spectrum of distance star is constant rather than behaving as 1/r as Newton's law assuming that most mass is in the galactic center predicts.

The MOND theory and its variants predict that there is a critical acceleration below which Newtonian gravity fails. This would mean that Newtonian gravitation is modified at large distances. String models and also TGD predict just the opposite since in this regime General Relativity should be a good approximation.

  1. The 1/r2 force would transform to 1/r force at some critical acceleration of about a=10-10 m/s2: this is a fraction of 10-11 about the gravitational acceleration at the Earth's surface.

  2. What Sean Carroll wrote about was the empirical study giving support for this kind of transition in the dynamics of stars at large distances and therefore breakdown of Newtonian gravity in MOND like theories.

In TGD framework critical acceleration is predicted but the recent experiment does not force to modify Newton's laws. Since Big Science is like market economy in the sense that funding is more important than truth, the attempts to communicate TGD based view about dark matter have turned out to be hopeless. Serious Scientist does not read anything not written on silk paper.

  1. One manner to produce this spectrum is to assume density of dark matter such that the mass inside sphere of radius R is proportional to R at last distances. Decay products of and ideal cosmic strings would predict this. The value of the string tension predicted correctly by TGD using the constraint that p-adic mass calculations give electron mass correctly.

  2. One could also assume that galaxies are distributed along cosmic string like pearls in necklace. The mass of the cosmic string would predict correct value for the velocity of distant stars. In the ideal case there would be no dark matter outside these cosmic strings.

    1. The difference with respect to the first mechanism is that this case gravitational acceleration would vanish along the direction of string and motion would be free motion. The prediction is that this kind of motions take place along observed linear structures formed by galaxies and also along larger structures.

    2. An attractive assumption is that dark matter corresponds to phases with large value of Planck constant is concentrated on magnetic flux tubes. Holography would suggest that the density of the magnetic energy is just the density of the matter condensed at wormhole throats associated with the topologically condensed cosmic string.
    3. Cosmic evolution modifies the ideal cosmic strings and their Minkowski space projection gets gradually thicker and thicker and their energy density - magnetic energy - characterized by string tension could be affected
TGD option differs from MOND in some respects and it is possible to test empirically which option is nearer to the truth.

  1. The transition at same critical acceleration is predicted universally by this option for all systems-now stars- with given mass scale if they are distributed along cosmic strings like like pearls in necklace. The gravitational acceleration due the necklace simply wins the gravitational acceleration due to the pearl. Fractality encourages to think like this.

  2. The critical acceleration predicted by TGDr depends on the mass scale as a ∝ GT2/M, where M is the mass of the object- now star. Since the recent study considers only stars with solar mass it does not allow to choose between MOND and TGD and Newton can continue to rest in peace in TGD Universe. Only a study using stars with different masses would allow to compare the predictions of MOND and TGD and kill either option or both. Second test distinguishing between MOND and TGD is the prediction of large scale free motions by TGD option.

TGD option explains also other strange findings of cosmology.

  1. The basic prediction is the large scale motions of dark matter along cosmic strings. The characteristic length and time scale of dynamics is scaled up by the scaling factor of hbar. This could explain the observed large scale motion of galaxy clusters -dark flow- assigned with dark matter in conflict with the expectations of standard cosmology.

  2. Cosmic strings could also relate to the strange relativistic jet like structures meaning correlations between very distant objects. Universe would be a spaghetti of cosmic strings around which matter is concentrated.

  3. The TGD based model for the final state of star actually predicts the presence of string like object defining preferred rotation axis. The beams of light emerging from supernovae would be preferentially directed along this lines- actually magnetic flux tubes. Same would apply to the gamma ray bursts from quasars, which would not be distributed evenly in all directions but would be like laser beams along cosmic strings.

For more about TGD based vision about cosmology and astrophysics see the chapters of the book Physics in Many-sheeted Space-time.

Motives and infinite primes

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.

Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.

The notion of infinite prime has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.

  1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes xn=0, xn=xn-1=0, etc...

  2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group Gk would be analogous to the relative homology group relative to the plane xk-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

  3. The algebraic counterpart of boundary map would map the elements of Gk identified as analog of homotopy group to the commutator group [Gk-2,Gk-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

  4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δikδjk-1jkδik-1 maps to the group algebra of the commutator group [Gk-2,Gk-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration.The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

  5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M2+/-× CP2 representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of fermion modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

  6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II1. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

This construction would realize thge number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as an infinite-D geometry, and TGD as a generalized number theory visions.

This picture leads also to a proposal how p-adic integrals could be defined in TGD framework.

  1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

  2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

  3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K1+K2 where K1=rlog(m/n) and K2=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e1/n can be considered.

  4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2n>p>pn-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed.

For details see the new chapter Motives and Infinite Primes of "Physics as a Generalized Number Theory" or the article with same title.

Monday, May 16, 2011

PAMELA anomaly suggests a scaled up variant of electro-hadron physics

Resonaances tells that the Fermi collaboration confirms the claim of PAMELA collaboration about anomalous e+e- pairs in cosmic ray radiation. The announcement of Pamela was my second birthday gift at October 30 for two and half years ago. The first gift was CDF anomaly which found a beautiful explanation in terms of tau-pions and the p-adically scaled up variants with color tau- lepton having mass scale by power of two. The tau-pion of mass about 14 GeV decaying in cascade like manner to lower octaves of basic tau-pion explained elegantly the observations reported by CDF. Helsinki University remembered me by giving a third and really lovely gift by banishing me from University. I understand the motivations: I had repeatedly and without careing about warnings of the authorities broken the Basic Principle of Serious Science: You ought to do nothing for the first time.

For some time ago the dilemma posed by the contradictory claims of DAMA and Cogent collaborations on one hand and XENON100 collaboration on one hand finds also nice solution in terms of 14 GeV taupion decaying to charged taupions with mass about 7 GeV. See the previous posting.

The decays of electro-pions to gamma pair can explain the observed anomalous gammas from galactic nucleus with energy very nearly to electron rest mass. Could one understand also the anomalous positrons reported by PAMELA as decay products of lepto-pion like states, say tau-pions? Intriguingly, the first figures of the article by Alessandro Strumia discussing the constraints on the possible explanations of the PAMELA anomaly show that the anomalous positron excess starts around 10 GeV, possible it starts already at 7 GeV. It is not possible to say anything certain below 10 GeV since the measurements are affected by the solar actvity below 10 GeV. What is however clear is that the excess cannot be explained by taupion decays with 14 GeV mass since the excess would be localized around energy of about 7 GeV. Higher mass is required.

The article by Alessandro Strumia summarizes various theoretical constraints on the new particle explaining positron and electron excesses. The conclusions are following.

  1. DM should result in a decay of quite a narrow particle with a mass very near to 2M, which is nearly at rest. What narrow means quantitatively is not clear to me.
  2. DM should carry a charge mediating long range interaction with the mediating boson which is must lighter than the particle itself: photon is the obvious candidate. Electromagnetically charged dark matter is however in conflict with the standard prejudices about dark matter and actually in dramatic conflict with its basic property of being invisible. Hierarchy of Planck constants is the only solution to the paradox of charged invisible dark matter.
  3. DM must prefer the decays to leptons since otherwise there would be also antiproton and proton excess which has not been observed.
  4. The mass of DM should be above 100 GeV.
These conditions encourage the idenfication of DM as a decay product of leptopion like state but with mass considerably higher than the 14 GeV mass. Tau-pions could of course be present but would not contribute to the anomaly at energies not too much above 7 GeV. Tau-pions would also give muon pair anomaly. Heavier leptopion like states are required and electropion would be the most natural candidate.
  1. If a scaled up variant of ordinary hadron physics characterized by M89 is there as the recent bumps having interpretation as mesons of this physics suggest, there is no deep reason preventing the presence of also the scale variant of leptohadron physics in this scale. Even more, one can argue that colored leptons must appear as both dark and ordinary variants. Dark variants with non-standard value of Planck constant can have masses of ordinary leptons plus possibly their octaves as in the case of tau at least. The decay widths of intermediate gauge bosons require ordinary colored leptons to have mass higher than 45 GeV.

  2. The mass of scaled up electropion would be obtained by scaling the mass of the dark electropion which for M89 electro-pion physics is in a good approximation 2me=1 MeV by a factor 2(127-89)/2= 219. This gives electropion mass equal to 500 GeV. Ordinary colored electron would therefore have mass of 250 GeV consistent with the lower bound. The conclusion would be rather ironic: we would have seen dark colored electron (in TGD sense) already at seventies and covered it carefelly under the rug and would be seeing now the ordinary colored electron and stubbornly trying to identify it as DM without caring about the fact that if dark matter is invisible in the standard sense it cannot be electromagnetically charged! Nature seems to love vulgar humour and feel no mercy towards pompous theoreticians enjoying monthly salary;-).

  3. By stretching one's imagination one might play with the thought that superpartners of colored leptons with mass scale of order 100 GeV could form pion like states. The superpartners decay to partner and neutrino since R-parity is not exact invariance in TGD and all depends on how fast this process occurs.

  4. Skeptic could wonder why the counterparts for colored excitations of quarks are not there and induce the increase of proton and antiproton fluxes.
To summarize, entire Zoo of not only new particles but even of new physics could be waiting for us at LHC energies if we live in TGD Universe!

Friday, May 13, 2011

Another approach to primordial cell

In the previous posting I told about the highly interesting work of the group of Martin Hanczyc in attempts to build protocell. Also the group led by Jack. W. Szostak, who was the 2009 Nobel Prize winner in physiology or medicine - has carried out beautiful experiments in which they are able to create a candidate for protocell satisfying many of the basic requirements.

One such condition is the ability of protocell to transfer various nutrient molecules through the protocell membrane. In modern cell pumps and channels consisting of proteins are believed to serve that purpose (for a different view see the remark below). Genetically coded proteins were however absent during the primordial era. Therefore the membrane is constructed of branching lipids believed to exists during prebiotic era allowing sugars which are basic building bricks of DNA to permeate to the protocell. Given the DNA template , the basic building bricks of DNA molecule assemble to a copy of DNA in this protocell.

What is still lacking is the generation of the template strand of DNA itself and also the replication of protocell. If dark DNA in the form of dark nucleon strings is really there, the template could result as the assembly of the basic bricks of DNA around it and above a proposal for the analog of this kind of process is suggested. The replication of the dark genes would have been also present from the beginning and would have preceded the replication of genes and protocell. Biological evolution could be seen as a migration from dark space-time sheets to ordinary ones and somewhat analogous to the migration of life from sea to land.

Remark: There are puzzling experimental findings about quantal currents through cell membrane even in absence of metabolic sources. In many-sheeted space-time one could interpret these currents as various kinds of Josephson currents running between cell interior and exterior along current carrying space-time shees. Pumps and channels would be more like a diagnostic tool allowing cell to measure the concentrations of various important biomolecules and ions.

At first sight the approaches of Szostak and Martin Hanczyc look very different. These approaches have however a lot of common at deeper level if one accepts TGD based view as DNA-cell membrane system or its more primitive version as a topological quantum computer like system relying on the braiding of magnetic flux tubes connecting the counterpart of DNA nucleotides to the lipids of protocell membrane and on the prebiotic realization of genetic code at the level of dark nuclear physics.

One could also argue that the protocell of Hanczyk represents oil based life as opposed to life as we know it. In TGD framework this is a mis-interpretation. The protocells of Hanczyk live in an aqueous environment. Nitrobenzene oil is an aromatic compound as also sugars and contains nitrogen taking in the proposed scenario same role as phosphorus in ordinary life. Oleic anhydride is lipid and- would provide basic building brick for a particular variant of DNA like structure half-way between dark and completely chemical realization. Oleic anhydride would provide also the building bricks of protocell membrane and serve as a nutrient just like fat molecules- also lipids- serve in "real life".

For background see the chapter Evolution in Many-Sheeted Space-time of "Genes and Memes".

Oil droplets as a primitive life form?

The origin of life is one the most fascinating problems of biology. The classic Miller-Urey experiment was carried out almost 60 years ago. In the experiment sparks were shooted through primordial atmosphere consisisting of methane, ammonia, hydrogen and water and the outcome was many of the aminoacids essential for life. The findings raised the optimism that the key to the understanding of the origins of life. After Miller's death 2007 scientists re-examined sealed test tubes from the experiment using modern methods found that well over 20 aminoacids-more than the 20 occurring in life- were produced in the experiments.

The Urey-Miller experiments have yielded also another surprise: the black tar consisting mostly of hydrogen cyanide polymer produced in the experiments has turned out to be much more interesting than originally thought and suggests a direction where the candidates for precursors of living cells might be found. In earlier experiments nitrobenzene droplets doped with oleic anhydride exhibited some signatures of life. The droplets were capable to metabolism using oleic anhydride as "fuel" making for the droplet to move. Droplets can move along chemical gradients, sense each other's presence and react to it and have also demonstrated rudimentary memory. Droplets can even "solve" a maze having "food" at its other end.

The basic objection against identification as primitive life form is that droplets have no genetic code and do not replicate. The model for dark nucleons however predicts that the states of nucleon are in one-one correspondence with DNA, RNA, tRNA, and aminoacid molecule and that vertebrate genetic code is naturally realized. The question is whether the realization of the genetic code in terms of dark nuclear strings might provide the system with genetic code and whether the replication could occur at the level of dark nucleon strings. TGD inspired quantum model of biology leads to a model for oil droplets as a primitive life form. In particular, a proposal for how dark genes could couple to chemistry of oil droplets is developed.

Intelligent oil droplets

New Scientist tells about a new twist related to the Urey-Miller experiment. Martin Hanczyc and his colleagues of University of Southern Denmark in Odense are doing research with a rather ambitious goal: the discovery of the recipe of life. The highly demanding challenge is to find candidates for the protocell that preceded the recent cell. What makes the task so difficult that it is not even clear what one should be searching for. For instance, what basic characteristics distinguishing living matter from inanimate systems protocell is expected to have before one can speak about primitive life form? And if one accepts the dogmas of standard biology, one encounters also the nasty hen-egg question which came first: metabolism or the genetic machinery.

Hanczyc and his colleagues have been experimenting with simple candidates for primitive life forms: oily nitrobenzene droplets doped with oleic anhydride immersed in alkaline aqueous solution (alkalinity is by definition an ability to reduce acidicity). They have found that these systems have some attributes generally associated with life. The recent experiments replaced oleic anhydrite with the black tar consisting of complex branched and fractal looking hydrogen cyanide (HCN) polymer produced by Urey-Miller experiments and found that also now the droplets exhibit lifelike behavior: they sense and respond their neighbors and move towards "food" sources.

The earlier experiments using nitrobenzene droplets doped with oleic anhydridge immersed in alkaline solution began immediately to move along straight lines. What happened that the oleic anhydrite at the surface of the droplet reacted with the water spilitting to two oleic acid molecules by hydration. This dropped the surface tension of the droplet and by a kind of spontaneous symmetry breaking the reaction rate had maximum at some point of the droplet and a "hot spot" was generated drawing oleic anhydride from the interior of the dropled and generating a convective flow. A pH gradient develops along the surface. The oleic acid in turn moved along the droplet surface from the hot spot to the diametrically opposite side of the droplet (see this). The net effect was a linear motion. pH gradient is claimed to be essential for the generation of motion but I must admit that I do not quite understand this point. A primitive metabolism liberating energy is obviously in question. By momentum conservation the total momentum for the convective flow and flow of oleic acid was compensated by a center of mass motion of the droplet.

One could claim that this process belongs to the same class of self-organization processes as the generation of convection patterns as one heats liquid from below. Other researchers have however discovered that the oil droplets can also travel along chemical gradients, something known as chemotaxis used by many bacteria to find food and void threats. One oil droplet managed even to "solve" a complex maze containing "food" at its other end. Whether this kind of behavior can be regarded as a mere chemistry is far from obvious to me. To me this a achievement look like a genuinely goal directed intentional behavior.

Hanczyc has also found that when the oil droplets approach each other they change course to avoid collision, or can circle each other-like partners in Viennese waltz! Oil droplets seem to have even memory. By videoing the paths of oil droplets Hanczyc found that the decision to stop or continue was not random but the behavior at any point of orbits was affected by the earlier behavior. This is by the way an elegant experimental manner to show that non-deterministic behavior is not just randomness. The experiments have been also carried using instead of oleic anhydrige mineral oil consisting of a mixture of alkanes having as building block polymeres from from CH4 by dropping two hydrogen from each C as also lipids have (methane CH4 is the simplest alkane). What distinguishes mineral oil molecules from the oleic anhydride molecules are the oxygen atoms in the middle of the reflection symmetric linear molecule. Also now the droplets move although the process takes place with a slower rate.

The basic objections against the identification of the oil droplets as a life form is that they do not replicate and there is no genetic code. One must be however very cautious with this kind of statetments. Maybe the primary life forms are not the droplets and the behavior of droplets reflects the control actions of these life forms on droplets. Perhaps also genetic code could be realized at at totally different level. The recent findings of the group of HIV Nobelist Montagnier indeed suggest a new realization of genetic code in water closely related to to water memory and TGD suggests a concrete realization of this code.

Some key ideas of TGD inspired quantum biology

Before proposing a model for intelligent oil droplets as a primitive life form its good to list some of the basic ideas of TGD inspired quantum biology.,

  1. The basic hypothesis is that the dark matter at the magnetic flux tubes of the magnetic body assignable to any physical system serves as an intentional agent controlling the behavior of the ordinary matter (see this). Dark matter can correspond to just the ordinary particles- at least electrons and protons- in a phase with non-standard large value of Planck constant forming macroscopic quantum phases. Also biologically important ions could form this kind of phases.

    TGD inspired nuclear physics (see this) allows also the bosonic counterparts of fermionic with same nuclear charge so that every fermionic ion could be accompanied by exotic bosonic ion so that Bose-Einstein condensates could become possible.

  2. The model for dark nucleons as entangled triplets of three quarks (see this and this) leads to the identification of the counterparts DNA, RNA, tRNA, and aminoacids as three-quark states and one can identify also vertebrate genetic code. DNA sequences correspond to dark nucleon sequences - dark nuclei - in this correspondence. The proposal is that dark proton sequences in water form dark nucleons with so large a Planck constant that nucleon size corresponds to size of singe DNA codon. There is indeed evidence that in attosecond time scale (time scale for corresponding causal diamonds) water obeys effective chemical formula H1.5O as far as scattering of electrons and neutrons is considered (see R. A. Cowley, Physics 350, 243-245, 2004 and R. Moreh et al, Phys Rev. 94, 2005)).

    This would suggest that 1/4 of protons are in dark large Planck constant phase in the experiments. This proportion is expected to depend on temperature and pressure and should explain the rich spectrum of anomalies of water by regarding it as a two phase system (see this). Perhaps these protons could form dark nucleon sequences realizing genetic code. These sequences could replicate and evolve and could define at least the analog of DNA or RNA. Maybe even DNA-mRNa-aminoacids translation processing could take place. If a translation machinery transforming exotic DNA to ordinary has developed during evolution, this fundamental realization of genetic machinery might make possible kind of Research&Development making possible to experiment with different genomes. Evolution would not be a random process anymore (see this).

  3. The proposal is that the ordered water layers associated with polar molecules dissolved in water are attached to the magnetic body of the molecule induced in water environment and that this magnetic body mimicking the original molecule is an essential element of this primitive life (see this). The self-organization processes of these layers induced by external perturbations could be the predecessor of processes like protein folding and de-folding. The mechanism of water memory could be based on dropping of the magnetic bodies of molecules as a result of repeated shaking involved with homepathic procedure inducing a sequence of catastrophes driving the evolution of these primitive life forms. One can also ask whethert these magnetic bodies could define the analog of proteins providing one realization of dark matter genetic code.

  4. If dark nucleons have been the predecessors of chemical life forms, one can circumvent the hen-egg question about whether the genetic code or metabolism came first. In zero energy ontology negative energy signals propagating in the direction of geometric past would in turn provide fundamental mechanism of intentional action, metabolism, and memory. If this is the case, evolution would have only led to a refinement of the fundamental mechanisms of life already existing: there would be no need to pull anything out of hat. The mechanisms for chemical storage and utilization of energy are needed and moving oil droplets would provide a primitive realization of these mechanisms.

  5. The notion of negentropic entanglement makes sense if one accepts the role of p-adic number fields and the vision about life as something residing in the intersection of real and p-adic worlds (see this). Entanglement probabilities for negentropic entanglement must be rational or algebraic numbers in the algebraic extension of p-adic numbers involved and there is unique prime for which this entanglement entropy is maximally negative. Negentropic entanglement makes possible new kind of many particle states which is not based on binding energy but the fact that negentropic entanglement is stable against state function reduction if Negentropy Maximization Principle determines its dynamics also in the case of negentropic entanglement. This makes it possible to have bound state like many particle states in which composite particles are in well-defined sense free and binding energy is formally negative. The proposal is that the mysterious high energy phosphate bond corresponds to negentropic entanglement and carries both metabolic energy and information (see this). In this framework ATP-ADP cycle has also information theoretic interpretation as a transfer of conscious information.

The model for DNA as topological quantum computer (see this and this) led among other things to an identification of magnetic flux tubes connecting bio-molecules as a basic building bricks of living matter.

  1. Flux tubes are assumed to connect DNA nucleotides to lipids of the nuclear and cell membranes. Flux tubes could begin from =O in the double bonds R=O or from negatively charged oxygens. In the case of DNA R would correspond to the basic unit in phosphate deoxiribose backbone consisting of aromatic 5-cycle and PO4 containing one =O and one O-. The lipid end would would contain =O and -OH and the flux tube could end to either of these or possibly -OH ionized to -O- by a transformation of proton to dark proton.

  2. The braiding of flux tubes makes topological quantum computation like processes possible (see this). The contractions and expansions of flux tubes induced by phase transitions changing the value of Planck constant would be a basic control mechanis allowing to understand how two biomolecules (say DNA and its conjugate) can find each other in the thick soup of organic molecules. The reconnections of the magnetic flux tubes would be second basic control mechanism and ATP→ADP process and its reverse would represent standardized reconnection.

  3. The flux tube ends would contain quark and antiquark (u,d and their antiquarks are involved) coding for the four DNA letters A,T,C,G so that also dark quarks and their antiquarks would provide an elementary particle level realization for the codons. Note that topological quantum computation does not necessitate genetic code and therefore also the repeating DNA sequences regarded as junk could be used for topological quantum computations.

General ideas about oil droplets as a primitive life form

It is interesting to see what one obtains if one takes the dark nucleon realization of genetic code, the mechanism of water memory realized as magnetic bodies attached to the ordered water layers associated with polar molecules, the model for DNA as topological quantum computer, and the ideas about magnetic body with dark matter as fundamental bio-control as basic ingredients of the model of intelligent oil droplets.

  1. A process resembling spontaneous symmetry breaking occurs for the droplet molecules as the hot spot is formed. The interpretation as a generation of magnetic body of approximately dipolar magnetic field is attractive. The magnetic body would control the droplet. The change of the direction of the motion of the oil droplet would correspond to the change of the orientation of the magnetic body and would thus reduce to a motor action of the magnetic body.

  2. The flux tubes of the magnetic body would be most naturally parallel to the direction of the nitrobenzene polymer strands. Oleic anhydride molecules and the hydrogen cyanid polymers would be transferred along the magnetic flux tubes of an approximately dipolar magnetic field entering to the hot spot from interior and the oleic acid molecules could move along the flux tubes continuing along the surface of the droplet to the diametrically opposite point. The migration of birds along magnetic field lines is a fractal analog for this.

  3. The dark matter at the magnetic body would give the oil drop its "intelligence". The dark nuclear genome could be realized at the magnetic body and the magnetic bodies might define the replicating life form as in the TGD based model of water memory for which the magnetic bodies represent molecules as far as low frequency electromagnetic fields characterized by cyclotron frequencies are considered. One could see intelligent oil droplets as a manifestation of control actions of a life form defined by dark matter at magnetic flux tubes and the first step in the process eventually leading to a complex control and coordination of the behavior of ordinary matter.

  4. The ability of droplets to react to the presence of other droplets would be due to the communications between magnetic bodies based on low frequency photons at cyclotron frequencies but having energy above thermal energy if the value of Planck constant is large enough.

At least oleic anhydrite, hydrogen cyanide, and mineral oil can serve as a fuel of oil droplets and this raises the question what might be the common property shared by them. Certainly this property must relate to metabolism and the model for ordinary metabolism suggests that this property is shared also by the high energy phosphate bond.

  1. Oleic anhydrite is a lipid formed by as a fusion of two oleic acids consisting of a sequence of CH2 units and the characteristic (C=O)-(O-H) group at its end. The burning of the molecule splits it to two oleic acids by hydration meaning utilizing one water molecule. The formation of oleic acid in turn involves dehydration so that the burning process is analogous to depolymerization of DNA or aminoacid sequence by hydration.

  2. Mineral oil is also a lipid and looks like oleic anhydride locally. In the ideal case however the crucial ..(C=O)-O-(C=O)-.. portions are lacking. Oxygenation could however produce this kind of defects to the mineral oil molecules so that the mechanism of burning would remain the same.

  3. Hydrogen cyanide HCN involves valence bond of valence 3 between C and N. The polymers are constructed from H-C-N sequences with single valence bond between both C:s and N:s of two subsequent horizontal H-C-N units, which one can think of as being obtained from (H-C)-(H-C)... sequence and ..N-N-N... sequences with N and C connected by horizontal bonds. This polymer replaces oleic acid as a "fuel" reacting with water and liberating metabolic energy. These polymers -which would serve as primitive analogs of proteins- would be transferred along the magnetic flux tubes and burned at the hot spot by hydration. HCN has been proposed to have been a primitive precursor of both amino acids and nuclei acids. With motivations coming from the general vision about quantum biology, tt will beproposed that also hydrogen cyanide polymers contain in their C-backbone ..(C=O)-O-(C=O)-.. portions as local defects due to oxygenation so that the burning would occur via hydration in all three cases.

What are the prerequisites for metabolism and topological quantum computation like processes?

The basic question is whether metabolism interpreted in TGD framework as negentropy transfer and thus requiring the analogs of high energy phosphate bond and ATP-ADP cycle is possible. The high energy phosphate bonds make also possible flux tube structures serving as a prerequisite for topological quantum computation like process. Both oleic anhydride, hydrogen cyanide and mineral oil can serve as a metabolic source and one should identify the common property of them making. This property should be the analog of high energy phosphate bond.

  1. High energy phosphate bond carries metabolic energy. This bond is poorly understood and I have proposed that high energy phosphate bond carries negentropic entanglement identified as the basic characteristic of life (see this. In the middle of oleic anhydride there =O-O-=O structure and its splitting in hydration liberates energy. This suggests that this structure also now carries the negentropic entanglement and the metabolic energy. The splitting process of oleic anhydrite occurring at the hotspot would be analogous to ATP→ ADP process involving splitting of PO4 molecule from ATP.

  2. Oleic acid is a lipid containing at its second end the characteristic (C=O)-OH group assumed to serve as a terminal for the magnetic flux tubes in the model of DNA-cell membrane system as quantum computer. In the presence energy feed one could imagine that the inverse process transforming oleic acid to oleic anhydride takes place and a primitive version of the metabolic cycle involving photosynthesis and cellular breathing can be imagined. Metabolic and quantum information processing would be very intimately related. By DNA as topological quantum computer analogy the magnetic flux tubes connecting oleic anhydride molecules would make be responsible for primitive topological quantum computation if present in the system.

  3. Also when the tar from Urey-Miller experiment replaces oleic anhydrite small amount of oleic anhydride was used to build a film around oil droplet to lower surface tension. This suggests that the oleic anhydride has a deeper purpose and defines the analog of cell membrane and make possible for the magnetic flux tubes from the interior of the droplet to attach to the lipids? This could occur at least in the hot spot and at point opposite to it so that magnetic flux tubes would connect the diametrically opposite points of the droplet. Oleic anhydride would therefore serve a dual purpose serving both asa metabolic resource and a building brick of the protocell membrane: metabolic energy would be accompanied by information. Also in real life lipids -about which fats are a special case- have this double role.

  4. The process occurs for both for hydrogen cyanide and mineral oil and and this raises obvious objections against the model since the energy and information carrying =O-C-O-C=O structures making also possible the flux tube connects are not present in the ideal situation. One must however remember that the situation in real life is far from ideal and the most obvious idea is that the polymers as such are not enough: oxygen is the basic metabolic resource and oxygenation serving as the loading of metabolic batteries might be the crucial element.

    1. The backbone of both lipids and of mineral oil polymers is CH2 sequence. If some fraction of mineral oil polymers contain (C=O)-O -(C=O):s serving as carriers of metabolic energy and information the situation reduces to that for oleic anhydride apart from effects caused by the fact that the density of metabolic energy per volume is expected to be lower, which would explain why the motion is slower.
    2. Also in the case of hydrogen cyanide polymers one can imagine the presence of similar defect structures due to oxygenation. A portion of ...(H-C)-(H-C)-(H-C).... sequence would be replaced with ....(H-C)-(C=O)-O-(C=O)-(H-C)... with three carbons lacking. The nitrogen sequence ...N-N-N-N-N.. would split to ...N-OH and OH-N... so that three nitrogens would be lacking.

Under these assumptions the model explains all three cases using hydration as the basic mechanism of metabolism as well as the conditions required by DNA as topological quantum computer model. Note that the process consumes oxygen just as the ordinary breathing.

What about genetic code and counterpart of DNA?

Consider next the possible realization of the genetic code. The first thing to notice is that even in the case that genetic code is not realized the braiding would make possible topological quantum computation like processes and a realization of memory in terms of braiding patterns. Furthermore, chemical realization of the genetric code is not possible so that dark nucleons remain the only possibility in TGD framework. The challenge is to try imagine whether DNA like structures having flux tube connections with the counterparts of lipids in the cell membrane could exist. The following suggestion is a product of free imagination based on analogies and reflects my amateurish skills in biochemistry.

  1. Aromatic rings are an essential element of both phosphate deoxiribose backbone of DNA and of DNA letters itself. Nitrobenzene molecule obeys chemical formula (C6H-5)-NO2 and contains benzene ring to which NO2 nitro group is attached. The oily character is due to the benzene ring. Benzene rings could serve as a counterpart for the hydrocarbon 5-cycles appearing in phosphate deoxiribose backbone. Note however that in deoxiribose ring one carbon is replaced with O and two hydrogens with OH. Moreover, single benzene molecule would correspond to the counterpart of DNA triplet rather than single nucleoside. One could however argue that only a backbone is in question so that the differences might not matter.

  2. One would naively expect that both nitrogen and phosphorus have same valence equal to three. In PO4 phosporus has 5 valence bonds as a rule and the interpretation is that phosphorus tends to donate its valence electrons to get empty shell. This kind of states are known as oxidation states and are possible also for nitrogen: hydroxylamine NO2H is one exmple of this kind of state. In fact, from the from structural formula of nitrobenzene one finds that nitrogen gives one electron to second oxygen so that also this state can be regarded as an oxidation state. This inspires the idea that nitrogen takes the role of phosphorus at least partially.

  3. If one does not allow oxidation states, the simplest manner to construct the analog of phosphate deoxiribose backbone is as structure ...X-X-X..., with X= R-O- (R1-N)-O, where R denotes oleic anhydride and R1 is for benzene residue. The bridges connecting benzene rings would be reflection symmetric. The breaking of reflection symmetry is however essential since it determines the reading direction of DNA.

  4. If one accepts oxidation states, the simplest option is that in benzene-NO2 complex NO2 is replaced with (N=0)-O and the counterpart of phosphate deoxiribose backbone would have the structure ...X-X-X-..., X=R- (R1-N=0)-O with R denoting oleic anhydride and R1 benzene. Oleic anhydride has valence bond to N so that N has 5 valence bonds as phosphorus in phosphate. Also the crucial =O is present. The units connecting subsequent benzene rings are not reflection symmetric anymore as indeed required. There is however no charged oxygen as in the case of ordinary DNA. Note that the analogs for AMP, ADP, ATP make sense since one can single replace P by N phosphate PO4.

  5. An interesting question is whether the nitrogen based metabolism could be realized as a primordial metabolism. Nitroglycerin is analogous to tri-phosphate although the nitrates are not arrranged linearly as in ATP and is used as both heart medicine and as an active incredient of explosives. The latter use conforms with the idea about the presence of high energy nitrate bond in NO4.
  6. The two mirror image branches of oleic anhydride molecule consist of 15 carbon atoms and the structure is rather long as compared to the basic unit of phosphat edeoxiribose backbone so that the distance between subsequent benzene units would be rather long- of order 10 Angstroms. On the other hand, 10 DNA codons correspond to 10 nm length in a good accuracy so that one codon would take 1 nm length also in this case. If double strand is formed, twisting is possible so that the scales could be the same. The size scale of the dark nucleon representing single DNA codon should correspond to the size scale of single oleic anhydride molecule and the required value of Planck constant would be of order 106 as the ratio of this scale and nucleon size of order 10-15 meters.

  7. The counterparts of DNA nucleotides forming a linear structure should join to the benzene rings. Dark nucleon sequences remain the only possibility if one wants a realization of genetic code. Each dark codon represented by dark nucleon would be connected by three flux tubes with quark and antiquark at their ends to single unit of the proposed structure. There would be three =O:s per single benzene ring. Since single benzene ring corresponds to single DNA codon three =O:s are indeed expected. Therefore =O:s could indeed correspond to terminals for flux tubes coming from single dark nucleon representing single DNA codon.

  8. The division of oil droplet would be the analog of cell replication and would involve at the deeper level the replication of dark nucleon sequences. This requires the analog of DNA double strand and the analogs of DNA codons would be dark nucleons. Genetic codons could be realized in terms of flux tubes connecting dark nucleon sequences to the oleic acids or oleic anhydrides at the surface of the droplet. It remains to be seen whether the division can be achieve in real world.

To sum up, this model is rather direct application of TGD based vision about life and the killer test is whether the mineral oil oil molecules and hydrogen cyanide molecules are not ideal but actually contain the (C=O)-O-(C= O) pieces carrying energy and information and serve as terminals for the magnetic flux tubes.

For background see the chapter Evolution in Many-Sheeted Space-time of "Genes and Memes". See also the article Oil droplets as a primitive life form?