Thursday, April 18, 2019

Secret Link Uncovered Between Pure Math and Physics

I learned about a possible existence of a very interesting link between pure mathematics and physics (see this). The article told about ideas of number theorist Minhyong Kim working at the University of Oxford. As I read the popular article, I realized it is somethin g very familiar to me but from totally different view point.

Number theoretician encounters the problem of finding rational points of an algebraic curve defined as real or complex variant in which case the curve is 2-D surface and 1-D in complex sense. The curve is defined as root of polynomials polynomials or several of them. The polynomial have typically rational coefficients but also coefficients in extension of rationals are possible.

For instance, Fermat's theorem is about whether xn+yn=1, n=1,2,3,... has rational solutions for n≥1. For n=1, and n= 2 it has and these solutions can be found. It is now known that for n>2 no solutions do exist. Quite generally, it is known that the number is finite rather than infinite in the generic case.

A more general problem is that of finding points in some algebraic extension of rationals. Also the coefficients of polynomials can be numbers in the extension of rationals.

Connection with TGD and physics of cognition

The problem is extremely difficult even for mathematicians - to say nothing about humble physicist like me with hopelessly limited mathematical skills. It is however just this problem which I encounter in TGD inspired vision about adelic physics. Recall that in TGD space-times are 4-surfaces in H=M4×CP2, preferred extremals of the variational principle defining the theory.

  1. In this approach p-adic physics for various primes p provide the correlates for cognition: there are several motivations for this vision. Ordinary physics describing sensory experience and the new p-adic physics describing cognition for various primes p are fused to what I called adelic physics. The adelic physics is characterized by extension of rationals inducing extensions of various p-adic number fields. The dimension n of extension characterizes kind of intelligence quotient and evolutionary level since algebraic complexity is the larger, the larger the value of n is. The connection with quantum physics comes from the conjecture that n is essentially effective Planck constant h_eff/h_0=n characterizing a hierarchy of dark matters. The larger the value of n the longer the scale of quantum coherence and the higher the evolutionary level, the more refined the cognition.

  2. An essential notion is that of cognitive representation. It has several realizations. One of them is the representation as a set of points common to reals and extensions of various p-adic number fields induced by the extension of rationals. These space-time points have points in the extension of rationals considered defining the adele. The coordinates are the imbedding space coordinates of a point of the space-time surface.

  3. The gigantic challenge is to find these points common to real number field and extensions of various p-adic number fields appearing in the adele.

  4. If this were not enough, one must solve an even tougher problem. In TGD the notion of "world of classical worlds" (WCW) is also a central notion. It consists of space-time surfaces in imbedding space H =M4× CP2, which are so called preferred extremals of the action principle of theory. Quantum physics would reduce to geometrization of WCW and construction of classical spinor fields in WCW and representing basically many-fermion states: only the quantum jump would be genuinely quantal in quantum theory.

    There are good reasons to expect that space-time surfaces are minimal surfaces with 2-D singularities, which are string world sheets - also minimal surfaces. This gives nice geometrization of gauge theories since minimal surfaces equations are counterparts for massless field equations.

    One must find the algebraic points, the cognitive representation, for all these preferred extremals representing points of WCW (one must have preferred coordinates for H - the symmetries of imbedding space crucial for TGD and making it unique, provide the preferred coordinates)!

  5. What is so beautiful is that in given cognitive resolution defined by the extension of rationals inducing the discretization of space-time surface, the cognitive representation defines the coordinates of the space-time surfaces as a point of WCW. This huge infinite-dimensional space WCW discretizes and the situation can be handled using finite mathematics.

Connection with Kim's work

So: what is then the connection with the work and ideas of Kim. Also he is interested in the above problem of finding rational points of given surface. There has been a lot of progress in understanding the problem: here I an only refer to the popular article.

  1. One step of progress has been the realization that if one uses the fact that the solutions are common to both reals and various p-adic number fields helps a lot. The reason is that for rational points the rationality implies that the solution of equation representable as infinite power series of p contains only finite number powers of p. If one manages to prove the this happens for even single prime, a rational solution has been found.

    The use of reals and all p-adic numbers fields is nothing but adelic physics. Real surfaces and all its p-adic variants form pages of a book like structure with infinite number of pages. The rational points or points in extension of rationals are the cognitive representation and are points common to all pages in the back of the book.

    This generalizes also to algebraic extensions of rationals. Solving the number theoretic problem is in TGD framework nothing but finding the points of the cognitive representation. The surprise for me was that this viewpoint helps in the problem rather than making it more complex. There are however problematic situations in some cases the hypothesis about fintie set of algebraic points need not make sense. A good example is Fermat for x+y=1. All rational points and also algebraic points are solutions. For x2+ y2=1 the set of Pythagorean triangles characterizing the solutions is infinite. How to cope with these situations in which one has accidental symmetries as one might say.

  2. Kim however argues that one can make even further progress by considering the situation from even wider perspective by making the problem even bigger. Introduce what popular article calls the space of spaces. The space of string world sheets is what string models suggests. The "world of classical worlds", WCW is what TGD suggests. One can get a wider perspective of the problem of finding algebraic points of a surface by considering the problem in the space of surfaces and at this level it might be possible to gain much more understanding. The notion of WCW would not mean horrible complication of a horribly complex problem but possible manner to understand the problem!

  3. A further TGD based simplification would be M8-H (H=M4×CP2) duality in which space-time surfaces at the level of M8 are algebraic surfaces which are mapped to surfaces in H identified as preferred extremals of action principle by the M8-H duality. Algebraic surfaces satisfying algebraic equations are very simple as compared to preferred extremals satisfying partial differential equations but "preferred" is what makes possible the duality. This huge simplification of the solution space of field equations guarantees holography necessitated by general coordinate invariance implying that space-time surfaces are analogous to Bohr orbits. It would also guarantee the huge symmetries of WCW making it possible to have Kähler geometry.

    This suggests in TGD framework that one finds the cognitive representation at the level of M8 using methods of algebraic geometry and maps the points to H by using the M8-H duality. TGD and octonionic variant of algebraic geometry would meet each other.

    It must be made clear that now solutions are not points but 4-D surfaces and this probably means also that points in extension of rationals are replaced with surfaces with imbedding space coordinates defining function in extensions of rational functions rather than rationals. This would bring in algebraic functions. This might provide also a simplification by providing a more general perspective. Also octonionic analyticity is extremely powerful constraint that might help.

Can one make Kim's idea about the role of symmetries more concrete in TGD framework?

The crux of the Kim's idea is that somehow symmetries of space of spaces could come in rescue in the attempts to understand the rational points of surface. The notion of WCW suggest in TGD framework rather concrete realization of this idea that I have discussed from the point of view of construction of quantum states.

  1. A little bit more of zero energy ontology (ZEO) is needed to follow the argument. In ZEO causal diamonds (CDs) defined as intersections of future and past directed light-cones with points replaced with CP2 and forming a scale hierarchy are central. Space-time surfaces are preferred extremals with ends at the opposite boundaries of CD indeed looking like diamond. Symplectic group for the boundaries of causal diamond (CD) is the group of isometries of WCW. Its Lie algebra has structure of Kac-Moody algebra with respect to the light-like radial coordinate of the light-like boundary of CD, which is piece of light-cone boundary. This infinite-D group plays central role in quantum TGD: it acts as WCW isometries and zero energy states are invariant under its action at opposite boundaries.

  2. As one replaces space-time surface with a cognitive representation associated with an extension of rationals, WCW isometries are replaced with their infinite discrete subgroup acting in the number field define by the extension of rationals defining the adele. These discrete isometries do not leave the cognitive representation invariant but replace with it new one having the same number of points and one obtains entire orbit of cognitive representations. This is what the emergence of symmetries in wider conceptual framework would mean.

  3. One can in fact construct invariants of the symplectic group. Symplectic transformations leave invariance the Kaehler magnetic fluxes associated with geodesic polygons with edges identified as geodesic lines of H. The simplest polygons are geodesic triangles. The symplectic fluxes associated with the geodesic triangles define symplectic invariants characterizing the cognitive representation. For the twistor lift one must allow also M4 to have analog of Kähler form and it would be responsible for CP violation and matter antimatter asymmetry. Also this defines symplectic invariants so that one obtains them for both M4 and CP2 projections and can characterize the cognitive representations in terms of these invariants.

    More complex cognitive representations in an extension containing the given extension are obtained by adding points with coordinates in the larger extension and this gives rise to new geodesic triangles and new invariants.

  4. Also in this framework one can have accidental symmetries. For instance, M4 with CP2 coordinates taken to be constant is a minimal surface, and all rational and algebraic points for given extension belong to the cognitive representation so that they ar infinite. Could this has something to do with the fact that we understand M4 so well and have even identified space-time with Minkowski space! Linear structure would be cognitively easy for the same reason and this could explain why we must linearize.

    CP2 type extremals with light-like M4 geodesic as M4 projection is second example of accidental symmetries. The number of rational or algebraic points with rational M4 coordinates at light-like curve is infinite - the situation is very similar to x+y=1 for Fermat. Simplest cosmic strings are geodesic sub-manifolds, that is products of plane M2 ⊂ M4 and CP2 geodesic sphere. Also they have exceptional symmetries.

    What is interesting from the point of view of proposed model of cognition is that these cognitively easy objects play a central role in TGD: their deformations represent more complex dynamical situations. For instance, replacing planar string with string world sheet replaces cognitive representation with a discrete or perhaps even finite one in M4 degrees of freedom.

See the article Secret Link Uncovered Between Pure Math and Physics or the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry?.

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

Articles and other material related to TGD.

Sunday, April 14, 2019

About the physical interpretation of ramified primes in TGD framework

Adelic physics corresponds to a hierarchy of extensions of rationals inducing extensions of p-adic number fields and the proposal is that ramified primes of extension correspond to preferred p-adic primes.

  1. Adelic physics suggests that prime p and quite generally, all preferred p-adic primes, could correspond to ramified primes for the extension of rationals defining the adele. Ramified prime divides discriminant D(P) of the irreducible polynomial P (monic polynomial with rational coefficients) defining the extension (see this).

    Discriminant D(P) of polynomial whose, roots give rise to extension of rationals, is essentially the resultant Res(P,P') for P and its derivative P' defined as the determinant of so called Sylvester polynomial (see this). D(P) is proportional to the product of differences ri-rj, i≠ j the roots of p and vanishes if there are two identical roots.

    Remark: For second order polynomials P(x)=x2+bx+c one has D= b2-4c.

  2. Ramified primes divide D. Since the matrix defining Res(P,P') is a polynomial of coefficients of p of order 2n-1, the size of ramified primes is bounded and their number is finite. The larger coefficients P(x) has, the larger the value of ramified prime can be. Small discriminant means small ramified primes so that polynomials having nearly degenerate roots have also small ramifying primes. Galois ramification is of special interest: for them all primes of extension in the decomposition of p appear as same power. For instance, the polynomial P(x)=x2+p has discriminant D=-4p so that primes 2 and p are ramified primes.

  3. What does ramification mean algebraically? The ring O(K)/(p) of integers of the extension K modulo p=πiei can be written as product ∏i Oiei (see this). If p is ramified, one has ei>1 for at least one i. Therefore there is at least one nilpotent element in O(K)/(p).

Could one interpret nilpotency quantum physically?
  1. For Galois extensions one has ei=e>1 for ramified primes. e divides the dimension of extension. For the quadratic extensions ramified primes have e=2. Quadratic extensions are fundamental extensions - kind of conserved genes -, whose further extensions give rise to physically relevant extensions.

    On the other hand, fermionic oscillator operators and Grassmann number used to describe fermions "classically" are nilpotent. Could they correspond to nilpotent elements of order ei=e=2 in O(K)/(p)? Fermions are building bricks of all elementary particles in TGD. Could this number theoretic analogy for the fermionic statistics have a deeper meaning?

  2. What about ramified primes with ei=e>2? Could they correspond to para-statistics (see this) or braid statistics (see this)?

    Both parabosonic and parafermionic fields of order n have the representation Ψ=∑i=1n Ψi. For parafermion field one has {Ψi(x),Ψi(y)}=0 and [Ψi(x),Ψj(y)]=0, i≠ j, when x and y have space-like separation. For parabosons the roles of commutator and anti-commutator are changed.

    The states containing N identical parafermions are described by a representation of symmetric group SN with N rows with at most e columns (anti-symmetrization). For states containing N identical parabosons one has N columns and at most e rows. For parafermions the wave function is symmetric in horizontal direction but the modes are different so that Bose-Einstein condensation is not possible.

    For parafermion of order n operator ∑i=1n Ψi one has (∑i=1n Ψi)n= ∏ Ψ1 Ψ2...Ψn and higher powers vanish so that one would have e-nilpotency. Therefore the interpretation for the nilpotent elements of order e in O(K)/(p)$ in terms of parafermion of order n=e-1 might make sense.

    It seems impossible to build a nilpotent operator from parabosonic field Ψ= ∑iΨi: the reason is that the powers Ψin are non-vanishing for arbitrarily high values of n.

  3. Braid statistics differs from para-statistics and is assigned with quantum groups. It would naturally correspond to quantum phase exp(iπ/p) assignable to the exchange of particles by braid operation regarded as a homotopy permuting braid strands. Could ramified prime p would correspond to braid statistics and the index ei=e characterizing it to para-statistics of order e-1? This possibility cannot be excluded since this exotic physics would be associated in TGD framework to dark matter assigned to algebraic extensions of rationals whose dimension n equals to heff/h0.

Why the primes, which do not split maximally in given extension - the ramified ones - would be physically special?
  1. Do ramified primes possess exceptional evolutionary fitness and are ramified primes present for lower-dimensional extensions present also for higher-dimensional extensions? If higher extensions are formed as extensions of already existing extensions, this is the case. Hierarchy of polynomials of polynomials would to this kind of hierarchy with ramified primes of starting point polynomials analogous to conserved genes.

  2. Quadratic extensions are the simplest ones and could serve as starting point extensions. Polynomials of form x2-c are the simplest among them. Discriminant is now D= -4c.

  3. Why c= Mn=2n-1 allowing p=2 and Mersenne prime p=Mn as ramified primes would be favored? Extension of rationals defined by x=2n is non-trivial for odd n and is equivalent with extension containing 21/2. c=Mn=2n-1 as a small deformation of c=2n gives an extension having both 2 as Mn as ramified primes.

    For c=Mn the number of ramified primes is smallest possible and equal to 2: why minimal number of ramified primes would give rise to a fittest extension? Why smallest number of fermionic p-adic mass scales assignable to the ramified primes would be the fittest option?

    The p-adic length scale corresponding ro Mn would be maximal and mass scale minimal. Could one think that other quadratic extension are unstable against transforming to Mersenne extensions with smallest p-adic mass scale?

See the appendix of the article Shnoll effect decade later or of the chapter A possible explanation of Shnoll effect of "Hyper-finite Factors and Dark Matter Hierarchy: part II" .

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

Articles and other material related to TGD.

Thursday, April 11, 2019

Shnoll effect decade later

As I wrote the first version of this chapter about Shnoll effect for about decade ago I did not yet have the recent vision about adelic physicsas a unification of real physics and various p-adics and real number based physics to describe the correlates of both sensory experience and cognition.

The recent view is that the hierarchy of extensions of rational numbers induces a hierarchy of extensions of p-adic number fields in turn defining adele. This hierarchy gives rise to dark matter hierarchy labelled by a hierarchy of Planck constants and also evolutionary hierarchy. The hierarchy of Planck constants heff=n× h0 is and essential element of quantum TGD and adelic physics suggests the identification of n as the dimension of extension of rationals. n could be seen as a kind of IQ for the system.

What is also new is the proposal that preferred p-adic primes labelling physical systems could correspond to so called ramified primes, call them p, of extension of rationals for which the expression of the rational p-adic prime as product of primes of extension contains less factors than that the dimension n of extension so that some primes of extension appear as higher powers. This is analog for criticality as the appearance of multiple roots of a polynomial so that the derivative vanishes at the root besides the polynomial itself.

Before continuing it is good to make some confessions. Already in the earlier approach I considered two options for explaining the Shnoll effect: either in terms of p-adic fractality or in terms of quantum phase q of both. I however too hastily concluded that the p-adic option fails and choose the quantum phase option.

In the following both options are seen as parts of the story relying on a principle: the approximate scaling invariance of probability distribution P(n) for fluctuations under scalings by powers of p-adic prime p implying that P(n) is approximately identical for the divisions for which the interval Δ defining division differs by a power of p.

Second new idea is the lift of P(n) to wave function Ψ(n) in the space of counts. For quantum phase qm, m=p, Ψm would have quantum factor proportional to a wave function in finite field Fp, and the notion of counting modulo p suggests that the wave function corresponds to particle in box - standing wave - giving rise to P(n) representing diffraction pattern.

Basic facts about Shnoll effect

Usually one is not interested in detailed patterns of the fluctuations of physical variables, and assumes that possible deviations from the predicted spectrum are due to the random character of the phenomena studied. Shnoll and his collaborators have however studied during last four decades the patterns associated with random fluctuations and have discovered a strange effect described in detail in the articles of Shnoll (references can be found in
A Possible Explanation of Shnoll Effect).

  1. Some examples studied by Shnoll and collaborators are fluctuations of chemical and nuclear decay rates, of particle velocity in external electric field, of discharge time delay in a neon lamp RC oscillator, of relaxation time of water protons using the spin echo technique, of amplitude of concentration fluctuations in the Belousov-Zhabotinsky reaction. Shnoll effect appears also in financial time series, which gives additional support for its universality. Often the measurement reduces to a measurement of a number of events in a given time interval τ. More generally, it is plausible that in all measurement situations one divides the value range of the studied observable observable to intervals of fixed length and counts the number of events in each interval to get a histogram representing the distribution N(n), where n is the number of events in a given interval and N(n) is the number of intervals with n events. These histograms allow to estimate the probability distribution P(n), which can be compared with theoretical predictions for the spectrum of fluctuations of n. Typical theoretical expectations for the fluctuation spectrum are characterized by Gaussian and Poisson distributions.

  2. Contrary to the expectations, the histograms describing the distribution of N(n) has a distribution having several maxima and minima (see the figures in the article of Shnoll and collaborators (see this). Typically -say for Poisson distribution - one expects single peak. As the duration of the measurement period increases, this structure becomes gets more pronounced: standard intuition would suggest just the opposite to take place. The peaks also tend to be located periodically. According to Shnoll the smoothed out distribution is consistent with the expected distribution in the case that it can be predicted reliably.

  3. There are also other strange features involved with the effect. The anomalous distribution for the number n of events per fixed time interval (or more general value interval of measured observable) seems to be universal as the experiments carried out with biological, chemical, and nuclear physics systems demonstrate. The distribution seems also to be same at laboratories located far away from each other. The comparison of consecutive histograms shows that the histogram shape is likely to be similar to the shape of its nearest temporal neighbors. The shapes of histograms tend to recur with periods of 24 hours, 27 days, or 365 days. The regular time variation of consecutive histograms, the similarity of histograms for simultaneous independent processes of different nature and occurring in different geographical positions, and the above mentioned periods, suggest a common reason for the phenomenon possibility related to gravitational interactions in Sun-Earth and Earth-Moon system.

In the case that the observable is number n of events per given time interval, theoretical considerations predict a distribution characterized by some parameters. For instance, for Poisson distribution the probabilities P(n) are given by the expression

P(n|λ)= exp(-λ) λn/n! .

The mean value of n is λ>0 and also variance equals to λ. The replacement of distribution with a many-peaked one means that the probabilities P(n| λ) are modified so that several maxima and minima result. This can occur of course by the randomness of the events but for large enough samples the effect should disappear.

The universality and position independence of the patterns suggest that the modification changes slowly as a function of geographic position and time. The interpretation of the periodicities as periods assignable to gravitational interactions in Sun-Earth system is highly suggestive. It is however very difficult to imagine any concrete physical models for the effect since distributions look the same even for processes of different nature. It would seem that the very notion of probability somehow differs from the ordinary probability based on real numbers and that this deformation of the notion of probability concept somehow relates to gravitation.

Quantum group inspired model for Shnoll effect

Usually quantum groups are assigned with exotic phenomena in Planck length scale. In TGD they are assignable to a finite measurement resolution. TGD inspired quantum measurement theory describes finite measurement resolution finite measurement resolution in terms of inclusions of hyper-finite factors of type II1 (HFFs) and quantum groups related closely to the inclusions and appear also in the models of topological quantum computation quantum computation based on topological quantum field theories.

Consider first the original version of the proposed model. If I would rewrite it now correcting also the small errors, the summary would be as follows. This slightly revised model can be included as such to the new model.

  1. The possibility that direct p-adic variants of real distribution functions such as Poisson distribution might allow to understand the findings was discussed also in the original version. The erratic conclusion was that this cannot the case. In fact, for λ=1/pk the sum of probabilities P(n) without normalization factor is finite, and the appriximate scaling symmetry P(n)≈ P(prn) emerges for k=1. p-Adicity predicts approximate p-periodicity corresponding to the periodic variation of nR with the lowest pinary digit of n.

  2. It was argued that one should replace the integer n! in P(n) with quantum integer (n!)qm, q=exp(iπ/m), identified as the product of quantum integers rqm=(qr-q-r)/(q-q-1), r<n.

    This however leads to problems since rqm can be negative. The problem can be circumvented by interpreting n! as p-adic number and expanding it in powers of p with pinary digits xk<p. For m=p the replacement of xk with quantum integer yields positive pinary digits.

    The resulting quantum variant of p-adic integer can be mapped to its real counterpart by a generalization of canonical identification x= ∑ xnpn→ ∑ xnp-n. Whatever the detailed definition, quantum integers are non-zero and positive. The quantum replacement r→ rqm of the integers appearing in rational parameters in P(n|λi) might therefore make sense. It however does not make sense in the exponents like λn and λ=pk, k>1,2,.., is forced by convergence condition.

  3. I proposed also another modification of quantum integers xqm, x<p=m appearing in as pinary digits by decomposing x into a product of primes s<p and replacing s with quantum primes sqp so that also the notion of quantum prime would make sense: one might talk about quantum arithmetics. This is possible but is not necessary.

Adelic model for Shnoll effect

At the first re-reading the original model looked rather tricky, and this led to a revised model feeding in the adelic wisdom. One implication hierarchy of Planck constants heff/h0=n with n identified as the dimension of Galois extension.

One also ends up to the proposal that preferred p-adic primes p correspond to so called ramified primes of the extension of rationals inducing the extensions of p-adic number fields defining the adele. This kind of prime would naturally define a small-p p-adicity associated with Shnoll effect, which would thus serve as a direct signature of adelic physics.

  1. The first observation in conflict with the original belief is that one can actually define purely p-adic variant of the Poisson distribution P(n| λ) by replacing 1/n! with its image (n!)R under canonical identification. For instance, for Poisson distribution one must have λ= p-k, k=1,2,.. for both real and p-adic distributions to nake sense. The sum of the probabilities P(n) is finite. Poisson distribution with trivial quantum part is determined uniquely.

  2. One can also consider quantization P(n)=|Ψ(n)|2, suggested by the vision about quantum TGD
    as complex square root of thermodynamics and hierarchy of Planck constants making possible macroscopic quantum coherence in arbitrarily long scales. The complexity of Ψ(n) could genuine quantum interpretation. Quantum factor of Ψ(n) allows interpretation as a wave function in finite field Fp representing the space of counts modulo p. The existence of quantum p-adics requires m=p. Scaling by p is not a symmetry but multiplication by 0<k<p and shift by 0≤ k<p act as symmetries analogous to rotations and translations acting on waves functions in Euclidian 3-space.

  3. The objections against Shnoll effect lead to an additional condition - or should one say principle - stating that the P(n) is approximately invariant under scalings n→ pkn. This could be seen as a manifestation of p-adic fractality in turn reflecting quantum criticality of TGD Universe.

  4. Taking n as the observable simplifies p-adicization crucially since the highly non-unique p-adicization of classical observables is avoided. One could speak of quantum measurement in the space of counts n defining universal observables. In quantum measurements the results are typically expressed as numbers of counts in given bin so that this kind of p-adicization is physically natural. The division of measurement interval would define an ensemble and n would be measured in each interval. State function reduction would localize Ψ(n) to n in each interval.

This picture leads to an alternative and simpler view about Shnoll effect. The scaling invariance is an essential additional condition now.
  1. The factorials n! appearing in P(n)=(dnf/dxn)/n! identified as coefficients of Taylor series of its generating function developed in pinary expansion for p=m. In this expansion one must invert powers of p in (n!)R and could also replace the coefficients of powers of p with quantum integers or replace even primes in their prime composition with quantum primes. For given norm (n!)R has period p approximately.

  2. The n:th derivative X(n)= dnf/dxn appearing as coefficient of 1/n! is replaced with X(n)R/X(n)p giving approximate periodicity and scaling invariance n→ pn.

  3. Quantum phase is associated with the ansatz stating P(n)= | Ψ(n)|2. In the "diffractive" situation quantum counterpart corresponds to | (kn)qm|2, 0<k<p-1. This gives rise to periodicity with period m=p.

The universal modifications of the probability distributions P(n|λi) considered predict patterns analogous to the ones observed by Shnoll. The p-adic prime p=m characterizes the deformation of the probability distribution and implies approximate p-periodicity, which could explain the periodically occurring peaks of the histograms for N(n) as function of n.

One can imagine several explanations for the dependence of the time series distribution P(n) on the direction of the momentum of alpha particle and on the dependence of P(n) on time.

  1. The change of ramified prime p induced by the change of the extension of rationals would affect the periods. An interesting question is whether the effects understood in terms of the effect of the measurement apparatus on many-sheeted space-time many-sheeted space-time topology and geometry on p. Can one speak about measurement of p and of extension of rationals?

  2. The extension of rationals (and thus p) need not change. The "quantum factor of Ψ in P(n)=| Ψ(n)|2 has part depending on qp. The dependence on qp could change without change in p so that the extension of rationals need not change. One could speak about measurement of an observable related to the quantum factor of Ψ. A more concrete model relies on wave function proportional to (kn)qp ∝ qmkn+qm-kn - analog to a superposition of plane waves with momenta k propagating to opposite directions in the space of counts and producing in P(n) diffraction pattern proportional to (qn)qp2. Change of momentum k by scaling or shift induced by variation of the gravitational parameters or time evolution could be in question.

The p-adic primes p in question are rather small, not much larger than 100 and the periods of P(n) provide a stringent test for the proposal. If p corresponds to ramified prime as adelic physics suggests, it can be indeed small.

To sum up, I cannot avoid the thought that fluctuations regarded usually as a mere nuisance could be actually a treasure trove of new physics. While we have been busily building bigger and bigger particle accelerators, the truth would have been staring directly at our face and even winking eye to us.

See the updated article Shnoll Effect decade later.

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

Articles and other material related to TGD.

Saturday, April 06, 2019

Number theoretical view about unitarity conditions for twistor lift

Twistorialization leads to the proposal that cuts in the scattering amplitudes are replaced with sums over poles, and that also many-particle states have discrete momentum and mass squared spectrum having interpretation in terms of bound states. Gravitation would be the natural physical reason for the discreteness of the mass spectrum and in string models it indeed emerges as "stringy" mass spectrum. The situation is very similar to that in dual resonance models, which were predecessors of super string theories.

Number theoretical discretization based on the hierarchy of extensions of rationals defining extensions of p-adic number fields gives rise to cognitive representatations as discrete sets of space-time surface and discretization of 4-momenta and S-matrix with discrete momentum labels. In number theoretic discretization cuts reduce automatically to sequences of poles. Whether this discretization is an approximation reflecting finite cognitive resolution or whether finite cognitive representation is a property of physical states reflecting itself as a condition that various parameters characterizing them belong to the extension considered, remains an open question.

One can approach the unitarity conditions also number theoretically. In the discretization forced by the extension of rationals the amplitudes are defined between states having a discrete spectrum of 4-momenta. Unitarity condition reduces to a purely algebraic condition involving only sums. In these conditions the Dirac delta functions associated with the mass squared of the resonances are replaced with Kronecker deltas.

  1. For given extension of rationals the unitary conditions are purely algebraic equations

    i(Tmn+T*nm)= ∑r TmrT*nr = TmnT*nn +TmmTmn + ∑r≠ m,n TmrT*nr .

    where Tmn belongs the extension. Complex imaginary unit i corresponds to that appearing in the extension of octonions in M8-H duality (see this).

  2. In the forward direction m=n one obtains

    2Im(Tmm)= Re(Tmm)2 + Im(Tmm)2+ Pm , Pm= ∑r≠ m TmrT*mr .

    Pm represents total probability for non-forward scattering.

  3. One can think of solving Im(Tmm) algebraically from this second order polynomial in the lowest order approximation in which Tmn=0 for m ≠ n. This gives

    2Im(Tmm)= 1+ (1-Pm-Re(Tmm)2)1/2 .

    Reality requires 1-Re(Tmm)2-Pm≥ 0 giving

    Re(Tmm)2+Pm≤ 1 .

    This condition is identically true by unitarity since the probability for scattering cannot be larger than 1.

    Besides this the real root must belong to the original extension of rationals. For instance, if the extension of rationals is trivial, the quantity 1-Pm-Re(Tmm)2 must be a square of rational y giving 1-Pm= y2+Re(Tmm)2. In the case of extension y is replaced with a number in the extension. I am not enough of number theorist to guess how powerful this kind of number theoretical conditions might be. In any case, the general ansatz for S is a unitary matrix in extension of rationals and this kind of matrices form a group so that there is no hope about unique solution.

  4. One could think of iterative solution of the conditions by assuming in the zeroth order approximation Tmn=0 for m≠ n giving Re(Tmm)2 +Im(Tmm)2= 1 reducing to cos2(θ)+sin2(θ)=1. For trivial extension of rationals θ would correspond to Pythagorean triangle.

    For non-diagonal elements of Tmn one would obtain at the next step the conditions

    i(Tmn+T*nm)= TmnT*nn + TmmT*nm .

    This gives a 2 linear equations for Tmn.

  5. These conditions are not enough to give unique solution. Time reversal invariance gives additional conditions and might help in this respect. T invariance is slightly broken but CPT symmetry could replace T symmetry in the general situation.

    Time reversal operator T (to be not confused with Tmn above) is anti-unitary operator and one has S= T(S). In wave mechanics one can show that T-invariant S-matrix and thus also T-matrix is symmetric. The matrices of this kind do not form a group so that the conditions can be very powerful.

    Combined with the above equations symmetry gives

    2Im(Tmn)= TmnT*nn + TmmT*mn .

    The two conditions for Tmn in principle fix it completely in this order.

    One obtains from the real part of the equation

    2Im(Tmn)= Re[TmnT*nn + TmmT*mn] .

    The vanishing of the imaginary part gives

    Im[TmnT*nn + TmmT*mn]=0 .

    giving a linear relation between the real and imaginary parts of Tmn. No new number theoretical conditions emerge. This relation requires that real and imaginary parts belong to the extension.

  6. At higher orders one must feed the resulting ansatz to the unitarity conditions for the diagonal elements Tnn. One can hope that the lowest order ansatz leads to rather unique solution by iteration of the unitarity conditions. In higher order conditions the higher order corrections appear linearly so that no new number theoretic conditions emerge at higher orders.

    Physical picture suggests that the S-matrices could be obtained by an iterative procedure. Since infinitely long procedure very probably leads out of the extension, one can ask whether the procedure should stop after finite steps. This property would pose an additional conditions to the S-matrix.

  7. Diagonal matrices are solutions to the conditions and for then the diagonal elements are roots of unity in the extension of rationals considered. The automorphisms Sd→ USdU-1 produce new S-matrices and if the unitary matrix U is orthogonal real matrix in algebraic extension satisfying therefore UUT=1, the condition S=ST is satisfied. There are therefore a large number of solutions.

    S-matrices diagonalizable in the extension are not the only solutions. The diagonalization of a unitary matrix S=ST in general gives a diagonal S-matrix, for which the roots of unity in general do not belong to the extension. Also the diagonalizating matrix fails to be in the extension. This non-diagonalizability might have deep physics content and explain why the physically natural state basis is not the one in which S-matrix is diagonal. In the case of density matrix it would guarantee stability of entanglement.

To sum up, number theoretic conditions could give rise to highly unique discrete S-matrices, when CPT symmetry can be formulated purely algebraically and be combined with unitarity. CPT symmetry might not however allow formulation in terms of automorphisms of diagonal unitary matrices analogous to orthogonal transformations.

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

Friday, April 05, 2019

Icosa-tetrahedral and icosa-dodecahedral bioharmonies as candidates for genetic code

Both the icosa-tetrahedral (see this) and icosa-dodecahedral harmony to be discussed below can be considered as candidates for bio-harmony as also the harmony involving fusion of 2 icosahedral harmonies and toric harmony (see this). The basic reason is that the third harmony corresponds to doublets. One cannot exclude the possibility of several equivalent representations of the code.

Icosa-tetrahedral harmony

Icosahedral harmonies can be characterized by a subgroup of icosahedral isometries A5 having 60 elements. If reflections are included the isometry group, oneas A5× Z2 with 120 elements. The group of symmetries is Z6,Z4, or Z2. There are two choices for Z2 and the interpretation has been that Z2 correspond to either reflection or rotation by π. A5 however allows also Z2× Z2 as subgroup. Amino-acids (AAs) correspond to orbits of the symmetry group and DNA codons coding for the AA correspond to triangles (3-chords) at the orbit. In purely icosahedral model on obtains 20+20+20 codons. A fusion with tetrahedral harmony gives 64 codons.

  1. Z6 gives rise to 3 AAs coded by 6 codons each (leu,se,arg) and 2 AAs coded by 2 codons: the choice of the doublet would require additional conditions. One option is ile doublet.

  2. Depending on whether one includes reflection or not, one can have either Z4⊂ A5 (60=4× 15) or Z4=Z2,rot× Z2 ⊂ A5× Z2. I have assumed that Z4=Z2,rot× Z2 but the recent argument suggests the first option. This does not have any implications for the earlier model. Icosahedral Z4 gives rise to 5 AAs coded by 4 codons each (5× 4=20). This leaves 11 AAs and 3 "empty" AA formally coded by stop codons.

  3. Icosahedral Z2 gives rise to 10 dublets. These 4-plets would correspond to (phe, tyr, his, gln, asn, lys, asp, glu, cys, stop-doublet) This leaves (stop,trp) double and (ile,met) doublet with broken Z2 symmetry.

    The fusion with tetrahedral code with 4- codons and 4 AAs should explain these 4 AAs. Tetrahedral isometries form group S3 and reduce to group Z3 for tetrahedral cycle.

    1. One could argue that ile-triplet and and met correspond to 3-element orbits with 1-element orbit. (stop,trp) would be formed by Z2 symmetry breaking from trp doublet and there is no obvious mechanism for this.

    2. If one tetrahedral face is fixed as a face shared with icosahedron, the symmetry group of tetrahedral cycle reduces to Z1. This would give 4 singlets identifiable as (ile,met) and (stop,trp) symmetry broken doubles. Since ile appears also in doublet, tetrahedral 1-orbit and icosahedral 2-orbit must have a common doubled triangle identifiable as the common face of icosahedron and tetrahedron. The doubling of the common triangle replaces ile-doublet with ile-triplet. This option looks rather reasonable.

Dodecadedral harmony

Dodecahedral harmony correspond to the unique Hamilton cycle at dodecahedron. Dodecahedral harmony as 20 notes and and 12 5-chords. If one assumes that the octave divides to 20 notes, this brings in mind "eastern" view about harmony.

The obvious objection against dodecahedral harmony is that dodecahedral faces are pentagons so that dodecahedral chords would be 5- rather than 3-chords so that the correspondence between chords and DNA codons would be lost. The situation changes if 3 notes - 3-chord - determine the 5-chord completely and one can assign a unique 3-chord to each pentagon. This is indeed the case!

  1. 3-edges meet in every dodecahedral vertex (this makes the dodecahedral cycle unique apart from rotations) and each edge pair in the vertex belongs to same pentagon (in the case of icosahedron there are 5 edges per vertex so that this is not true). Therefore each pentagon must contain at least 2 edges of Hamilton's cycle.

    The cycle must visit all vertices of pentagon, and the visit to the vertex means that the cycle shares at least one edge with pentagon. Since all vertices of the pentagon must be visited, there are two options. For option a) given pentagon shares with the cycle disjoint 2-edge with 3 vertices and 1-edge with two vertices. For option b) the pentagon shares with the cycle 4-edge with 5 vertices.

  2. The numbers na of pentagons with 4-edges and nb=12-na 2-edge+ 1-edge (making 3 edges) can be deduced easily. Cycle has 20 edges. Pentagon of type a) shares 3 edges with the cycle and the edge is shared by 2 pentagons. This gives 3na/2 edges. Pentagon of type b) shares 4 edges with the cycle. This gives 2nb= 2(12-na) edges. The total number of edges is 3na/2+2nb= 20, which gives na=8 and nb=4. Dodecahedral Hamilton's cycle can be found from web (see this). The structure is as deduced here.

    For case a) the 3-chords correspond naturally to the 3 vertices of the 2-edge shared with the cycle. Therefore it is possible to assign unique 3-chords to the dodecahedral harmony and to obtain connection with codons in this case. One however obtains also 12 2-chords: could they have some genetic counterpart?

    What about 5-chords for pentagons of type b)? Hamiltonian cycle can be oriented and this is induces orientation of the pentagons. One can say that the first vertex in the 4-edge is the vertex at which cycle arrives to the pentagon and identify the 3-chord as the first three vertices. It turns out that for the replacement of quint cycle this is not actually necessary.

Is icosa-dodecahedral harmony consistent with the genetic code?

One must check whether icosa-dodecahedral harmony is consistent with the degeneracies of the genetic code.

  1. A fusion of 2 icosahedral harmonies and 2 copies of dodecahedral harmony would be in question. As in the case of icosahedral harmony already discussed, the two icosahedral harmonies would have symmetry groups Z6 and Z4 and give the codons coding for 3 6-plets and 1 doublet+ 5 4-plets + two copies of dodecahedral harmony.

  2. Can the model predict correctly the numbers of codons coding for AAs? It is known that dodecahedral Hamilton cycle divides dodecahedron to two congruent pieces related by Z2 symmetry (see this). Also the Hamiltonian cycle defining the common boundary has Z2 symmetry. A good guess is that these Z2:s corresponds to reflection symmetry and rotation by π but I am not able to exclude Z4⊂ G0, where G0 consists of 60 orientation preserving isometries. In this case some orbits - presumably all 3 of them - could contain 4 pentagons. This is not consistent with the condition that one has doublets and singlets.

    If the second symmetry corresponds to reflection, it can be excluded by simply assuming that reflections change the orientation of the cycle.

  3. Rotation by π has two fixed points corresponding to opposite poles so that one has 5 2-orbits and 2 1-orbits giving 12 triangles for each copy. Two copies of dodecahedral harmony would give 5+5=10 doublets and 2+2=4 singlets. A possible interpretation would be as (ile,met) and (stop,trp).

Consider now objections against dodecahedral harmony.
  1. Why two copies of dodecahedral code? What distinguishes between them? If imirror symmetry leaves the cycle invariant apart from orientation the copies could be mirror images and consist of same faces. The second option is that they related by a rotation?

  2. The number of dodecahedral AAs is 24 rather than 20. Could the additional 4 AAs as orbits have interpretation as AAs in some sense. Could the "empty" AAs coded by stop codons be counted as AAs exceptional in some sense. In TGD framework one can consider the possibility that although AA is "empty", there is analog of AA as physical signature for the end of protein telling what stopping codon it corresponds. The magnetic body of protein is a good candidate.

    Genetic code has several slightly differing variants. Could the 2 additional exotic AAs Pyl and Sec correspond in some situations to the additional AAs?

  3. Essential for the bio-harmony as a fusion of harmonies is that one can select from each orbit single face as a representative of the AA it codes - kind of gauge choice is in question - and that the orbits corresponding to different AAs can be chosen to be disjoint. Otherwise codons belonging to the orbits of different Hamilton cycles can code for the same AA if the AA can be chosen to be in intersection. If not, the same codon can code for 2 different AAs - this can indeed occur in reality (see this)!

    The condition that orbits of different cycles do not interesect seems quite stringent but has not been proven. But what if it is actually broken? Indeed, in the case of icosahedral harmony with Z1 symmetry tetrahedron and icosahedron could have common a doubled face the breaking of this condition would geometrically explain why ile belongs to both icosahedral and tetrahedral orbit.

    Ile is the problem also in the case if icosa-dodecahedral harmony. Dodecahedral singlet codes for ile as also icosahedral doublet. Could one talk about doubling of ile face so that it corresponds to a pair of triangle and pentagon (in 1-1 correspondence with triangle as chord).

  4. The two copies of the dodecahedral code should correspond to 5 doublets and 2 singlets each. One expects that together they give rise to 10+2 +10+2 =24 faces. Do they? Mirror symmetry and rotation by π act as symmetries of the cycle so that neither can map the two cycles to each other. Dodecahedral (equivalently icosahedral) rotations give rise to new equivalent cycles. The action on pentagons corresponds to the action on vertices of icosahedron so that it easy to understand what happens.

    Each symmetry corresponds to a rotation around some axis and has opposite icosahedral vertices at this axis as fixed points. Hence any two cycles obtained in this manner have 2 common pentagons. This means reduction 24→ 22 unless one interprets the situation in terms doubled faces? Could the disappearing doublet correspond to stop-doublet? What about the remaining stop of the vertebrate code pairing with trp? Why does second singlet correspond to empty AA and not something else such as exotic AA.

  5. There is also further problem. Suppose that an intersection of orbits takes place at single triangle. Suppose that one cannot choose this triangle to be "AA" triangle for both orbits. In this case it is not clear to which AA the codon codes. This kind of phenomenon actually takes place in some cases and is known as homonymy. It is associated with the deviations of the code from the vertebrate code and involves exotic AAs Pyl and Sec. Codons can serve as a stop codon or code for an exotic AA.

Clearly, the notion of bio-harmony involves many unclear aspects but my strong feeling is that there is very beautiful mathematics involved.

See the article Hachimoji DNA from TGD perspective.

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

Articles and other material related to TGD.

Tuesday, April 02, 2019

Quantum scarring from TGD point of view

Quantum scarring (QS) and its many-body counterpart MBQS are very interesting challenges for TGD. Eigenstate thermalization hypothesis (ETH) states the time development of energy eigenstate to a superposition of large number of eigenstates with the same energy gives rise to thermalization. In MBQS the thermalization is however very slow for most states and there are states for which it it does not occur at all and the system returns to the original state periodically. Integrable systems for which the energies of the sates are rational multiples of a finite number of "fundamentals" have this property. MBQS in the case considered occurs for an array containing ground state atoms and their Rydberg counterparts.

In TGD framework one can consider the possibility that instead of Rydberg atoms one has pseudo Rydberg atoms having non-standard value heff=nh0 of Planck constant such that heff=mh is true (also fractional effective principal quantum number is possible and could serve as a test for the proposal). In this framework the exchange force between valence electrons would be scaled by factor (heff/h)2 and promote localization in turn forcing the periodic orbits. Even if this effect is not involved in the case considered, it could make possible to have dark variant of MBQS at higher temperatures.

See the article Quantum scarring scarring from TGD point of view or the chapter Criticality and dark matter of "Hyper-finite Factors and Dark Matter Hierarchy".

Monday, March 25, 2019

Why the interstellar gas is ionized?

I became aware about new-to-me cosmological anomaly. FB really tests by tolerance threshold but it is also extremely useful.

The news is that the sparsely distributed hot gas in the space between galaxies is ionized. This is difficult to understand: as universe cooled below the temperature at which hydrogen atoms became stable, it should neutralized in standard cosmology.

In biosystems there is similar problem. Why biologically important ions are indeed ions at physiological temperatures? Even the understanding of electrolytes is plagued by a similar problem. It sounds like sacrelege to even mention to a fashionable deeply-reductionistic popular physicist talking fluently about Planck scale physics, multiverses, and landscape about the scandalous possibility that electrolytes might involve new physics! The so called cold fusion is however now more or less an empirical fact (see this) and takes place in electrolytes - also living matter is an electrolyte.

TGD explanation is based on the hierarchy of Planck constants heff=n×h0 predicted by adelic physics as kind of IQ of the system.

  1. The energy of radiation with very low frequencies - such as EEG frequencies - can be in the range of ionisation energies of atoms by E=heff×f - typically in UV range. Hence interaction between long and short length scales characterized by different values of heff becomes possible and in TGD magnetic body (MB) in long scales would indeed control bio-matter at short scales in this manner. Cyclotron radiation from magnetic flux tubes of MB carrying dark ions would be used as control tool and Josephson radiation from cell membrane would be utilized to transfer sensory input to MB.

  2. TGD variant of Nottale's hypothesis predicts really large values of heff. One would have heff= hgr= GMm/v0 at the magnetic flux tubes connecting masses M and m and carrying gravitons (v0 <c is a parameter with dimensions of velocity). What is important that at gravitational flux tubes cyclotron frequencies would not depend on m being thus universal. For instance, biophotons with energies in UV and visible range would result from dark photons with large heff= hgr for frequencies even in EEG range and below.

The ordinary photons resulting from dark photons would ionize biologically important atoms and molecules. In the interstellar space the situation would be the same: dark photons transforming to ordinary higher energy photons would ionize the intersellar gas.

This relates closely to another cosmological mystery.

  1. Standard model based cosmology cannot explain the origin of magnetic fields appering in all scales. Magnetic fields require in Maxwell's theory current and in cosmology thermal equilibrium does not allow any currents in long length scales. In TGD however magnetic flux tubes carrying monopole fluxes are possible by the topology of CP2. They would have closed 2-surface as cross section rather than disk. They are stable and do not require current to generate the magnetic field. These flux tubes would be carriers of dark matter generating the dark cyclotron radiation ionizing interstellar gas in the scale of wavelength, which would be astrophysical.

  2. There are also another kind of magnetic flux tubes for which cross section is sphere but the flux vanishes since the sphere is contractible. hese flux tubes are not stable against splitting. There would be no magnetic field in the scale of flux tube. Magnetic field is however non-vanishing and ions in it generate dark cyclotron radiation. These flux tubes would naturally carry gravitons and photons. These flux tubes could could mediate gravitational and electromagnetic interactions: gravitons and photons (also dark) would propagate along them.

  3. This picture leads to a model for the formation of galaxies as tangles of long monopole flux carrying cosmic strings looking like dipole field in the region of galaxy (for TGD based model of quasars see this): the energy of these tangle would transform to ordinary matter as the cosmic strings would gradually thicken - this corresponds to cosmic expansion. The process would be the analog of inflation in TGD. Also stars and even planets could be formed in this manner, and thickeded cosmic strings would be carriers of dark matter in TGD sense. The model explains the flat galactic rotation curves trivially.

  4. Dark ions responsible for the intergalactic ionization could reside at these monopole flux tubes or at the
    flux tubes which vanishing magnetic flux carrying mediating gravitational interactions. Which option is correct? Or can one consider both options?

    I learned some time ago that T= 160 minute period appears in astrophysics in many scales from stars to quasars (see this. Its origin is not known. The observation is hat dark cyclotron photons created by Fe2+ ions in interstellar magnetic field about .2 nT have period of 160 minutes.

    1. In TGD inspired biology the endogenous magnetic field is about .2 Gauss and now the time scale is t=.1 seconds which corresponds to alpha rhythm, the fundametal biorhyth. 160 minutes would correspond to cosmic alpha rhythm! Also cyclotron photons with this frequency could induce ionization of interstellar scales. This would require hgr which is by a factor T/t= 105 higher. The mass M would be now 105 larger than for ordinary alpha frequency for which M is naturally proportional to the mass of Earth: M= kE ME. Solar mass is 3.33×:105 times mE. Could the dark matter in question be associated with the flux tubes connecting Sun to smaller masses m mediating gravitational interaction? The ratio of Planck constants would be

      hgr,S/hgr,E= (kS/kE)× (v0,E/v0,S)× ( MS/ME).

      This would demand

      (kS/kE)× (v0,E/v0,S)=1/3.33≈ 3 .

    2. Note that the 160 minute period was discovered in the dynamics of Sun: no mechanism is not know for an oscillation coherent in so long length scale. Could this mean that the MB of Sun controls dynamics of Sun just as the MB of Earth controls the dynamics of biosphere? Is Sun a conscious, intelligent, entity?

See the article Could 160 minute oscillation affecting Galaxies and the Solar System correspond to cosmic "alpha rhythm"? or the chapter About the Nottale's formula for hgr and the possibility that Planck length lP and CP2 length R are identical.

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

Articles and other material related to TGD.