Monday, April 15, 2013

Riemann hypothesis and quasilattices

Freeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1-dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture.

Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below.

Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2-D case (Penrose tilings) there is n-fold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1-D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1-D QCs is at least as rich a structure as PV numbers and probably much richer.

Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1-D quasi-crystals.

  1. If Riemann Hypothesis is true, the spectrum for the Fourier transform of the distribution of zeros of Riemann zeta is discrete. The calculations of Andrew Odlycko indeed demonstrate this numerically, which is of course not a proof. From Dyson's explanation I understand that it consists of sums of integer multiples nlog(p) of logarithms of primes meaning that the non-vanishing Fourier components are apart from overall delta function (number of zeros) proportional to

    F(n)= ∑sk n-iskD(isk) , sk=1/2+iyk ,

    where sk are zeros of Zeta. ζD could be called the dual of zeta with summation over integers replaced with summation over zeros. For other "energies" than E=log(n) the Fourier transform would vanish. One can say that the zeros of Riemann Zeta and primes (or p-adic "energy" spectrum) are dual. Dyson conjectures that each generalized zeta function (or rather, L-function) corresponds to one particular 1-D QC and that Riemann zeta corresponds to one very special 1-D QC.

There are also intriguing connections with TGD, which inspire quaternionic generalization of Riemann Zeta and Riemann hypothesis.
  1. What is interesting that the same "energy" spectrum (logarithms of positive integers) appears in an arithmetic quantum field theory assignable to what I call infinite primes. An infinite hierarchy of second quantizations of ordinary arithmetic QFT is involved. A the lowest level the Fourier transform of the spectrum of the arithmetic QFT would consist of zeros of zeta rotated by π/2! The algebraic extensions of rationals and the algebraic integers associated with them define an infinite series of infinite primes and also generalized zeta functions obtained by the generalization of the sum formula. This would suggest a very deep connection with zeta functions, quantum physics, and quasicrystals. These zeta functions could correspond to 1-D QCs.

  2. The definition of p-adic manifold (in TGD framework) forces a discretisation of M4× CP2 having interpretation in terms of finite measurement resolution. This discretization induces also dicretization of space-time surfaces by induction of manifold structure. The discretisation of M4 (or E3) is achieved by crystal lattices, by QCs, and perhaps also by more general discrete structures. Could lattices and QCs be forced by the condition that the lattice like structures defines a discrete distributions with discrete spectrum? But why this?

  3. There is also another problem. Integration is a problematic notion in p-adic context and it has turned out that discretization is unavoidable and also natural in finite measurement resolution. The inverse of the Fourier transform however involves integration unless the spectrum of the Fourier transform is discrete so that in both E3 and corresponding momentum space integration reduces to a summation. This would be achieved if discretisation is by lattice or QC so that one would obtain the desired constraint on discretizations. Thus Riemann hypothesis has excellent mathematical motivations to be true in TGD Universe!

  4. What could be the counterpart of Riemann Zeta in the quaternionic case? Quaternionic analog of Zeta suggests itself: formally one can define quaternionic zeta using the same formula as for Riemann zeta.

    1. Rieman zeta characterizes ordinary integers and s is in this case complex number, extension of reals by adding a imaginary unit. A naive generalization would be that quaternionic zeta characterizes Gaussian integers so that s in the sum ζ(s)=∑ n-s should be replaced with quaternion and n by Gaussian integer. In octonionic zeta s should be replaced with octonion and n with a quaternionic integer. The sum is well-defined despite the non-commutativity of quaternions (non-associativity of octonions) if the powers n-s are well-defined. Also the analytic continuation to entire quaternion/octonion plane should make sense and could be performed in a step wise manner by starting from real axis for s, extended to complex plane and then to quaternionic plane.

    2. Could the zeros sk of quaternionic zeta ζH(s) reside at the 3-D hyper-plane Re(q)=1/2, where Re(q) corresponds to E4 time coordinate (one must also be able to continue to M4)? Could the duals of zeros in turn correspond to logarithms ilog(n), n Gaussian integer. The Fourier transform of the 3-D distribution defined by the zeros would in turn be proportional to the dual of ζD,H(isk) of ζH. Same applies to the octonionic zeta.

    3. The assumption that n is ordinary integer in ζH would trivialize the situation. One obtains the distribution of zeros of ordinary Riemann zeta at each line s= 1/2+ yI, I any quaternionic unit and the loci of zeros would correspond to entire 2-spheres. The Fourier spectrum would not be discrete since only the magnitudes of the magnitudes of the quaternionic imaginary parts of "momenta" would be imaginary parts of zeros of Riemann zeta but the direction of momentum would be free. One would not avoid integration in the definition of inverse Fourier transform although the integrand would be constant in angular degrees of freedom.


At 5:56 AM, Blogger ThePeSla said...

Hi Matti,

It has been awhile... this is a good post... if Dyson is considering such things we are in good company. Lubos acknowledges p-adics but dismisses these as new properties and wider space in his qm does not need any revision and is under attack by negative minds in his recent post.

Now if you have managed to read my posts you will see I have covered pretty far into this sort of quasi crystal idea (again my quasics as a term has to be distinguished in places of the theory carefully) into wider spaces... after all the idea was part of Coxeters great book Regular Polytopes. In my discussion of fractal paths (superfractals) I relate to the finite (quasifinite) transforms by the usual Frouier discrete functions and draw a picture of how this relates to the meaning of the zeta of Riemann. Both the 3+1 and M4 formalism applies.

Now Lubos sees a war against the dark matter particle discovery so he says... this morning I add as a hypotheses that at the foundations such a dark matter particle would be the same as what we mean by a monopole in the quasic context... the questions of charge as fundamental, of matter, neutral and antimatter, time flow, even of what some actually mean by mini-black holes and so on that begs for a unified theory.

Enjoy and be aware of these ever more concrete new physics- I am glad to see you are holding the line as time mysteriously moves forward so to speak.


At 10:57 AM, Anonymous Anonymous said...

I just shredded some of my old papers on zeta... it's too tedious of a problem. Enjoy, relax, take it easy!--Stephen

At 12:52 PM, Blogger Stephen said...

Wow, a truly excellent essay by Freeman Dyson... stirring, really, and comforting somehow...that I'm not the only one who has thought these things (the need to focus on sanity and be a force for peace and reconciliation)

At 8:34 PM, Anonymous said...

Dyson is free soul. One of the very few birds rather than frog;-).

At 9:56 AM, Blogger Stephen said...

Saw this on slashdot.

From the abstract: "The recent Planck satellite combined with earlier results eliminate a wide spectrum of more complex inflationary models and favor models with a single scalar field, as reported in the analysis of the collaboration. More important, though, is that all the simplest inflaton models are disfavored by the data while the surviving models -- namely, those with plateau-like potentials -- are problematic. We discuss how the restriction to plateau-like models leads to three independent problems: it exacerbates both the initial conditions problem and the multiverse-unpredictability problem and it creates a new difficulty which we call the inflationary "unlikeliness problem." Finally, we comment on problems reconciling inflation with a standard model Higgs, as suggested by recent LHC results. In sum, we find that recent experimental data disfavors all the best-motivated inflationary scenarios and introduces new, serious difficulties that cut to the core of the inflationary paradigm. Forthcoming searches for B-modes, non-Gaussianity and new particles should be decisive. "

*queue dramatic music* dun dun dun



At 8:27 PM, Anonymous said...

Hi Stephen,

maybe I have written something about the problems of inflation scenario. At least I wrote about TGD inspired approach replacing inflation: .

I have always wondered how anyone in his or her right mind can take so horribly ugly theory as inflation seriously. The plateu makes me vomit;-). But fashions are fashions and defy rational mind.

The basic fact is that 3-space has vanishing curvature scalar in good approximation. This we must explain.

a) In GRT based approach this is obtained from exponential expansion.

b) In TGD framework quantum criticality assignable to a phase transition. In the phase transition for some space-time sheets an increase of Planck constant takes place.

Criticality of the phase transition implies a vanishing of curvature scalar as something dimensional and means critical gravitational mass density for vacuum extremal of Kahler action. The interpretation is that gravitational mass represents topologically condensed matter.

This does not imply exponential expansion, only accelerated one. The Robertson-Walker metric is unique apart from the its finite duration serving as a parameter from imbeddability so that the killer problems of inflationary scenario are avoided.

At 8:44 AM, Blogger Stephen said...

Nice paper. Interesting ideas. The debate makes a lot more sense to me now. Take it easy! --Stephen

At 9:27 AM, Blogger Stephen said...

Well that's interesting.. I guess!

At 9:10 PM, Anonymous said...

M_89 hadron physics in action?

At 11:25 PM, Blogger Hamed said...

Dear Matti,

If symplectic transformatons of deltaM4 time to CP2 acts as isometries, the value of kahler form at partonic 2-surfaces are zero modes.
Symplectic transformations leaving kahler form invariant and isometries leaving distances between points invariant. How a symplectic transform can acts as isometric transformation? In really, relation between them is not clear for me.

At 8:16 AM, Blogger Stephen said...


At 12:26 PM, Blogger Stephen said...

Old but might have not been familiar with it... Higgs bundles

"Higgs bundles have a rich structure and play a role in many different areas including gauge theory, Kähler and hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, the Deligne–Simpson problem on products of matrices, and (most recently) mirror symmetry and Langlands duality.
In this essay we will touch lightly on a selection of these topics."

At 9:26 PM, Anonymous Anonymous said...

"Higgs bundle" is a slightly misleading term, not quite what "Higgs" would suggest. In 2-D case "Higgs" is 1-form. I recall that Teichmueller parameters characterizing the conformal equivalence class od 2-surface (partonic 2-surface in TGD) are integrals of one forms of cycles of 2-D Riemann surface. Elementary particle vacuum functionals are functionals in the space of conformal equivalence classes.


At 2:21 AM, Anonymous said...

Dear Hamed,

You say:
"If symplectic transformatons of deltaM4 time to CP2 acts as isometries, the value of kahler form at partonic 2-surfaces are zero modes. Symplectic transformations leaving kahler form invariant and isometries leaving distances between points invariant. How a symplectic transform can acts as isometric transformation? In really, relation between them is not clear for me. "

The point is that symplectic transformations of delta M^4xCP_2 act as isometries of "world of classical worlds", WCW! Not imbedding space! This is of course a very natural conjecture about isometries of WCW geometry. The motivation is that these symplectic transformations are approximate symmetries of Kahler action for surfaces representable as maps from M^4 to CP_2 broken only gravitationally: that is due to the deviation of the induced metric from flat Minkowski metric. For vacuum externals they are exact symmetries.

A note about "symplectic". Strictly speaking, in delta M^4 the "symplectic transformation" should be replaced with "contact transformation". Delta M^4 is effectively 2-D metrically and thus allows natural symplectic structure J^2=-g by its metric 2-D property. Symplectic transformations are those of S^2 made local with respect to radial light-like direction (delta M^4= S^2xR_+). Lorentz transformations define different composition of this kind and one can speak about moduli space for symplectic structures.

Hamiltonians for the transformations can be expressed as products of delta M^4 Hamiltonians and CP_2 Hamiltonians and can be assumed to possess well-defined colour and spin quantum numbers and correspond to irreducible representations of SO(3)xSU(3)/Z^3. They can be thought of being made local gauge group by localising with respect to the light-like radial coordinate. This gives rise to the counterpart of conformal invariance.

At 11:50 AM, Blogger Ulla said...
A difference between properties of matter and antimatter, named CP violation by particle physicists, was observed for the first time in the decays of Strange Beauty particles, the B0s mesons composed of a beauty antiquark b bound with a strange quark s.

At 12:19 AM, Blogger ThePeSla said...


such a first time, in the spirit of it if not it was the first in fact after parity surprise preferred directions, was the K meson asymmetry to so ground that idea as a time direction... it evades the mechanism that is the analogy and thus causality where it seems to be- and not just time as illusion only, that mechanism is deep into at least topology and arithmetic. The flat Minkowski metric as such duality is flat only in the sense it is a brane, and integers like Marni's 2/9 do exist in the world only if it expresses directionally in its transendental mirrors... what I called "intertial pi"

At 12:27 AM, Blogger ThePeSla said...

Matti, if someone watched the development of our posts they might conclude we are reading each other but not mentioned the fact as to what is priority of discovery or originality- or we are all sharing some mysterious common knowledge as is the case on the Jungian frontier.

But could we not save some time instead of reinventing the old work if we in fact do read something from somewhere- it is not just we who speak in a less formal and standard setting- the icosahedron, the feynman diagrams I mean it would look better to some of us if you posted on these things before they appear elsewhere- still, if you are reading this from those in standard physics or academia- it is not us who are behind the times- in fact it could be possible that when the few of us get an idea it is then there for all who in what seems isolation are doing physics.

Have you thought of such things, in moments of joy or self questioning- I find the awakening to new ideas far beyond the simple but hard to reach sense of awake so late and little time left if we watch the clock is discouraging- the generation after us could have been the first ones with indefinite lifespans but now they life less abundantly than we did.

At 11:03 AM, Blogger Ulla said...


TGD IS a new idea by itself. Even You have difficulties realizing it, and I know of NOBODY working as hard as Matti, without even a decent salary. To do what you ask for he would need a stab of hundreds of TGD scientists.

At 2:49 PM, Blogger zahid ahmed said...

This comment has been removed by a blog administrator.

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At 11:27 PM, Anonymous tai said...

I solve Riemann Hypothesis.

Please see it.

At 10:56 AM, Anonymous Anonymous said...

Tai, no offense meant but that paper is nonsense. It is formatted so badly and the English is so poor none of it makes sense

At 8:09 PM, Anonymous tai said...

Where is the wrong point?

Please tell me concletely.

At 10:16 AM, Anonymous Anonymous said...

tai, the very first formula is utterly wrong, unless [m/n] means something other than (m/n).

hmm := proc (m) options operator, arrow; sum(mobius(n)*(m/n), n = 1 .. m) end proc;

[1., 1., .5000000000, .6666666667, -.1666666667, .8000000000, -0.6666666667e-1, -0.7619047619e-1, -0.8571428571e-1, .9047619048, -0.4761904762e-2, -0.5194805195e-2, -1.005627706, -0.8298368298e-1, .9110889111, .9718281718, 0.3256743257e-1, 0.3448316389e-1, -.9636011048, -1.014316952]

if your claim was true, all of those would equal to 1

unless you intend [m/n] to mean something else other than simple addition


At 10:17 AM, Anonymous Anonymous said...

er, division, not addition

At 2:56 AM, Anonymous tai said...

So easy point,you say.

[m/n] is m/n 's Gauss sign.

[3/2] =[1.5]=1

I think what you say seriously

Your level is nonsence

At 3:07 AM, Anonymous tai said...

more,Mr. Terence Tao prove

my theorem2,originally

So , no more doubt in proof.

At 8:48 AM, Anonymous Anonymous said...

So gausss sign is just the floor function which truncates the fractional part to the next lowest integer. A good mathematician always clearly explains the symbols he or she is using

At 5:15 PM, Anonymous tai said...

So,what you should say is to Withdrow "crow" or "nonsence" you say.

Your rudeness attitude is very

At 5:31 PM, Anonymous tai said...

more,Many mathematician,at least 3 person,recognise what I write.

also Gaussian sign,

No more talk to you.

Good bye

At 7:20 PM, Anonymous Anonymous said...

My apologies, my style was rude and I offer my humble apology. I was only intending the comment to encourage your work, because I am sure that you did not intend for it to be difficult to interpret. I will check the calculations and post my results, you can disregard if you wish . peace


At 10:14 PM, Anonymous tai said...

Thank you for your apology.


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