Sunday, March 25, 2012

Riemann zeta and quantum theory as square root of thermodynamics

Ulla mentioned in the comment section of the earlier posting an intervew of Matthew Watkins. The pages of Matthew Watkins about all imaginable topics related to Riemann zeta are excellent and I can only warmly recommend. I was actually in contact with him for years ago and there might be also TGD inspired proposal for strategy proving Riemann hypothesis at the pages of Matthew Watkins.

The interview was very inspiring reading. MW has very profound vision about what mathematics is and he is able to express it in understandable manner. MW tells also about the recent work of Connes applying p-adics and adeles(!) to the problem. I would guess that these are old ideas and I have myself speculated about the connection with p-adics for long time ago.

MW tells in the interview about the thermodynamical interpretation of zeta function. Zeta reduces to a product ζ(s)= ∏pZp(s) of partition functions Zp(s)=1/[1-p-s] over particles labelled by primes p. This relates very closely also to infinite primes and one can talk about Riemann gas with particle momenta/energies given by log(p). s is in general complex number and for the zeros of the zeta one has s=1/2+iy. The imaginary part y is non-rational number. At s=1 zeta diverges and for Re(s)≤1 the definition of zeta as product fails. Physicist would interpret this as a phase transition taking place at the critical line s=1 so that one cannot anymore talk about Riemann gas. Should one talk about Riemann liquid? Or - anticipating what follows- about quantum liquid? What the vanishing of zeta could mean physically? Certainly the thermodynamical interpretation as sum of something interpretable as thermodynamical probabilities apart from normalization fails.

The basic problem with this interpretation is that it is only formal since the temperature parameter is complex. How could one overcome this problem?

A possible answer emerged as I read the interview.

  1. One could interpret zeta function in the framework of TGD - or rather in zero energy ontology (ZEO) - in terms of square root of thermodynamics! This would make possible the complex analog of temperature. Thermodynamical probabilities would be replaced with probability amplitudes.

  2. Thermodynamical probabilities would be replaced with complex probability amplitudes, and Riemann zeta would be the analog of vacuum functional of TGD which is product of exponent of Kähler function - Kähler action for Euclidian regions of space-time surface - and exponent of imaginary Kähler action coming from Minkowskian regions of space-time surface and defining Morse function.

    In QFT picture taking into account only the Minkowskian regions of space-time would have only the exponent of this Morse function: the problem is that path integral does not exist mathematically. In thermodynamics picture taking into account only the Euclidian regions of space-time one would only the exponent of Kähler function and would lose interference effects fundamental for QFT type systems.

    In quantum TGD both Kähler and Morse are present. With rather general assumptions the imaginary part and real part of exponent of vacuum functional are proportional to each other and to sum over the values of Chern-Simons action for 3-D wormhole throats and for space-like 3-surfaces at the ends of CD. This is non-trivial.

  3. Zeros of zeta would in this case correspond to a situation in which the integral of the vacuum functional over the "world of classical worlds" (WCW) vanishes. The pole of ζ at s=1 would correspond to divergence fo the integral for the modulus squared of Kähler function.

What the vanishing of the zeta could mean if one accepts the interpretation quantum theory as a square root of thermodynamics?

  1. What could the infinite value of zeta at s=1 mean? The The interpretation in terms of square root of thermodynamics implied following. In zero energy ontology zeta function function decomposition to ∏p Zp(s) corresponds to a product of single particle partition functions for which one can assigns probabilities p-s/Zp(s) to single particle states. This does not make sense physically for complex values of s.

  2. In ZEO one can however assume that the complex number p-sn define the entanglement coefficients for positive and negative energy states with energies nlog(p) and -nlog(p): n bosons with energy log(p) just as for black body radiation. The sum over amplitudes over over all combinations of these states with some bosons labelled by primes p gives Riemann zeta which vanishes at critical line if RH holds.

  3. One can also look for the values of thermodynamical probabilities given by |p-ns|2= p-n at critical line. The sum over these gives for given p the factor p/(p-1) and the product of all these factors gives ζ (1)=∞. Thermodynamical partition function diverges. The physical interpretation is in terms of Bose-Einstein condensation.

  4. What the vanishing of the trace for the matrix coding for zeros of zeta defined by the amplitudes is physically analogous to the statement ∫ Ψ dV=0 and is indeed true for many systems such as hydrogen atom. But what this means? Does it say that the zero energy state is orthogonal to vacuum state defined by unit matrix between positive and negative energy states? In any case, zeros and the pole of zeta would be aspects of one and same thing in this interpretation. This is an something genuinely new and an encouraging sign. Note that in TGD based proposal for a strategy for proving Riemann hypothesis, similar condition states that coherent state is orthogonal to a "false" tachyonic vacuum.

  5. RH would state in this framework that all zeros of ζ correspond to zero energy states for which the thermodynamical partition function diverges. Another manner to say this is that the system is critical. (Maximal) Quantum Criticality is indeed the key postulate about TGD Universe and fixes the Kähler coupling strength characterizing the theory uniquely (plus possible other free parameters). Quantum Criticality guarantees that the Universe is maximally complex. Physics as generalized number theory would suggest that also number theory is quantum critical! When the sum over numbers proportional to propabilities diverges, the probabilities are considerably different from zero for infinite number of states. At criticality the presence of fluctuations in all scales implying fractality indeed implies this. A more precise interpretation is in terms of Bose-Eisntein condensation.

  6. The postulate that all zero energy states for Riemann system are zeros of zeta and critical in the sense being non-normalizable combined with the fact that s=1 is the only pole of zeta implies that the all zeros correspond to Re(s)=1/2 so that RH follows from purely physical assumptions. The behavior at s=1 would be an essential element of the argument. Note that in ZEO coherent state property is in accordance with energy conservation. In the case of coherent states of Cooper pairs same applies to fermion number conservation.

    With this interpretation the condition would state orthogonality with respect to the coherent zero energy state characterized by s=0, which has finite norm and does not represent Bose-Einstein condensation. This would give a connection with the proposal for the strategy for proving Riemann Hypothesis by replacing eigenstates of energy with coherent states and the two approaches could be unified. Note that in this approach conformal invariance for the spectrum of zeros of zeta is the axiom yielding RH and could be seen as counterpart for the fundamental role of conformal invariance in modern physics and indeed very natural in the vision about physics as generalized number theory.


At 8:06 PM, Blogger ThePeSla said...

I looked up the 1973 Kahler, Morse and wonder if it is indeed a subset that distinguishing manifolds is an incorporation or subset of TGD - the issue is not clearly shown to me or I cannot see it explicitly.
I do understand, in a vague way such a simple topology in rather abstract descriptive but vague terms is in that sense incorporated in quasics or higher systems...susy,p-adics, surreal and so on.

Now to start with you could mean the fourth root, or some partition root and almost certainly mean the Mersenne root for example.

This is compatible with the idea exploding or expanding a central point in an abstract space.

Now, can we prove or disprove, well we can understand the Riemann hypothesis in a better context and one that of course relates to such things as Poincare conjectures and so on but relative only to what we may consider a local space. One has to include more than complex analysis here for that is already a subset of the topology if the world is reasonable at all.

We in effect consider composite grounds also at the fundamental level in the height of generalization of which you speak.

As I have shown lately, at the loss of general readership it seems, these are issues of a more general idea beyond the distinctions of the subjective and objective- of consciousness and the physical world.

Zero and Infinity within themselves as poles each can be distinguished in fluid ideas so too the relations as a subtle difference between them. Can a zero point be relative and yet be a constant value to some manifold?

Our view of space has is so intimate with our physics of our brains and its evolving we have different approaches and each can claim a unity of sorts- but the infinite to one can be the nothing to another. The science magazines have breakthrough technologies lately where with simulation rather than issues of chance or limitations of mechanism a abstract idea like our intuitions on scales of Planck constants apply but not to the standard irreducible concept of a minimum distance or duration- to arrive at such an area or volume is to assert some form of the division by twos at first as the Riemann hypothesis.

Apparently, what is lacking in the theories is the scope and span of imagination not the endless chasing of theories that only utilize partial concepts of our brain and seemingly mind structures.

Crossing over to a deeper and higher generalization is not an endless task of the languages of mathematics- it is a common language written in the universal one of organic chemistry in the flesh. Even beyond string theory without knowing how we know why we need to make more rigorous the third thermodynamic law.

I started with reinventing the wheel of philosophy and eventually grew past the forums and so on- now I feel I have grown past the blogging. If you desire we can continue on new domains as our work hopefully progresses further.


At 10:32 PM, Anonymous said...

To Pesla:

Just a technical note: technical details are important;-).

That Kahler and Morse have natural place in TGD was as such a totally trivial observation. For the Minkowskian signature of induced metric Kahler action is naturally imaginary since sqrt(det(g)) is square root of a negative number. I refused to take this as fact and replaced det(g) with its absolute value! Rather stupid and shows how dangerous "knowing" is. I missed second beautiful half of the theory.

When one allows sqrt(deg(g)) to be imaginary, Morse finds place in the theory as also the interference effects assignable directly to vacuum functional and central in QFT.

At 5:54 PM, Anonymous ◘Fractality◘ said...


Electric Emitting Trees:

Soul Associated with Eyes:


At 4:08 AM, Anonymous Orwin said...

Matti, now I see why Dante and the squid caught your attention!

Extended Ginzberg-Landau expansion in small deviations from critical temperature:

Lattice spin in ultracold atoms:

Yukawa cluster crystals and colloids:

At 11:02 AM, Blogger Ulla said...

As usual a bit off topic
Denis Noble, A theory of biological relativity: no privileged level of causation.
Must higher level biological processes always be derivable from lower level data and mechanisms, as assumed by the idea that an organism is completely defined by its genome? Or are higher level properties necessarily also causes of lower level behaviour, involving actions and interactions both ways? This article uses modelling of the heart, and its experimental basis, to show that downward causation is necessary and that this form of causation can be represented as the influences of initial and boundary conditions on the solutions of the differential equations used to represent the lower level processes. These insights are then generalized. A priori, there is no privileged level of causation. The relations between this form of ‘biological relativity’ and forms of relativity in physics are discussed. Biological relativity can be seen as an extension of the relativity principle by avoiding the assumption that there is a privileged scale at which biological functions are determined.

At 12:09 PM, Anonymous Anonymous said...

I don't know what these calculations imply but there is a method of "continuing" zeta at integer values by the method of certain hyper-geometric identities which give a value at s=1 other than infinity, it gives Zeta(1)=Ei(1)-gamma where is the Exponential Integral and gamma is Euler's constant. See my paper on that viXra because I'm not "connected enough" to get my work on arxiv apparently although I haven't tried since I don’t know how to proceed there. . The same method gives another funky value for Zeta(0) and it is not without precedent that "new" values for such functions are discovered.. a similiar thing was done for the Bernoulli numbers, See . If I was really smart I would see if I could find a way to interpolate these "hypergeometric continuations " and see what it gives for the line Re(s)=1/2 ...


At 9:31 PM, Anonymous said...

To Ulla:

This view about causation levels gives new formal new view to the notions and ideas like magnetic body, DNA as topological quantum computer, the role of dark matter in biology, hierarchy of Planck constants, etc...
Reductionism could be seen also as hypothesis that basic level of causation reduce to that for quarks.

To Stephen: your article contains impressive collection of basic facts about various number theoretical special functions. I confess my pure knowledge in these matters.

I understood that you consider representations of zeta(n) in terms of poly-logarithms Li_n(t) at t=1. Also the continuation was between integer arguments of zeta. What about zeta (s) at general complex values? Does the non-standard continuation work for these values? Can one make n complex in Li_n(t)?

At 5:59 AM, Blogger Ulla said...

Well, Nottale belongs to the group too.

At 7:32 AM, Anonymous said...

Amusing. I ended up with the hierarchy of hbars from Nottale's scale relativity applied to astrophysics and applied it then to biology. Now Nottale himself is applying scale relativity to biology.

Could one dare to hope that this is beginning of something?;-) Finally!

The notion of scale hierarchy (or hierarchies: hierarchy of Planck constants and p-adic length scales) provides a highly plausible solution to key problems in both particle physics and biology. In particle physics the scale hierarchy is directly visible but GUT and string unifiers refuse to see it, and try to understand everything in terms of Planck scale and proton Compton length. Also in biology there are obvious fundamental scales. Where do these scales come? This is the question.

Particle physics has been seen as the foremost frontier of science and on experimental side one cannot but admire the achievements. The problem is that theoretical particle physicists in their reductionistic delusion continue to apply old recipes, and know nothing about the fantastic experimental progress in biology and neuroscience.

I think it was Glashow who warned around 1984 about how dangerous it is to make string theory the only game in the town and now we see that he was right.

At 12:44 PM, Blogger Ulla said...

Nottale has done this many years ago. I wrote him last year and wanted to discuss your approach, but he only told he knew of it. Sic! Everyone similar. His first two papers on the subject are from 2008, but not free. This is a short intro

a long list of references if nothing more. The approach is interesting. Not the usual artificial life approach.

At 12:51 PM, Blogger Ulla said...

Look at this! Noble talks sense?

Genes do not act in isolation either from each other or from the environment, and so I replace the metaphor of the selfish gene with metaphors that emphasise the processes involved rather than the molecular biological components. This may seem a simple shift of viewpoint. In fact it is revolutionary. Nothing remains the same. There is no 'book of life', nor are there 'genetic programs'. The consequences for the study of the brain and the nature of the self are profound. They lead naturally to the concept of anātman, no-self, and to a better understanding of the relation between the microscopic and macroscopic views of the world.

At 8:02 PM, Blogger ThePeSla said...

Indeed Matti, after all we humans only hear half of what is said to us and can sometimes infer the rest- but go to my page before the last on and find the links there to Science Daily for what I think important breakthrough ideas from experiments.

I wonder, if the proteins engage in time differences preferring one or the other of the codon forms that code for the same proteins - and that is a difference up to ten fold- could there be a vague hint of dark like matter structure here?

Alas, we are diverging too much, but I do not see really you lost half the beauty of a theory- in some ways the absolute as a view is what p-adics imagine yes? Or is there such a thing as negative distance and probability and so on?

I hope we have not missed half the beauty of our alternative new physics- then again maybe this is the human condition or state of our minds.

Stephen, good luck on that we do need to explain that fundamental constant as desparately as what may be loss in half the beauty of the imaginary part of Riemann's half value :-)

The PeSla

At 8:19 AM, Anonymous said...

To Ulla:

It is easy to agree with the claims of Noble. I am happy that simple common sense seems to be finally winning in biology and we are getting rid of the reductionistic dogma which is paralyzing science from particle physics to biology.

I would see quantum TGD as a concrete realization for systems biology based on generalization of quantum theory.

At 1:14 PM, Blogger Ulla said...

Look at this

I think this is a way to go. Linked to Nottale, Noble et al. on EvoDevoUniverse. They have a conference next year. I feel excited :)

You cannot think of mailing me? I have not yet learned my lesson? No?

At 10:40 AM, Anonymous Anonymous said...

Matti, I feel that it should be possible to accomplish this continuation for general complex values of s but at this time I do not know how to accomplish it. I think a good start would be the paper by SC Woon "Analytic Continuation of Operators -- operators acting complex s-times -- Applications: from Number Theory and Group Theory to Quantum Field and String Theories" . I would like to try this some time, maybe get that cool million dollar prize so I can retire inside a faraday cage and find a way to make a living that doesn't involve programming and being around too many cell phones and wi-fi "access points". If people really understood what all these radio frequencies were doing too them they would get the hell away from it all. P.s., I sent you a book "In search of the riemann zeros"


At 9:03 PM, Anonymous said...


a theorem saying that the analytic continuation of zeta is unique would resolve the question at once. Any idea about the possible existence of this kind of theorem in the case of zeta?

One could imagine that if the continuation is not unique, various continuations organize to a kind of Riemann surface as in the case of z^(1/n). One can also ask how uniquely the structure of poles and zeros fixes the continuation. Around zeros z^n behavior gives winding number n and z^1/n fractional winding number.

RH hypothesis is grave yard of dreams. Do not make any. My hunch - almost proposal - is that RH is not provable without posing conformal invariance of the spectrum of zeros - in some sense - as an axiom. This axiom would be completely analogous to conformal invariance as a physical axiom.

At 2:48 AM, Blogger Santeri Satama said...

Jasson Vindas: The Prime Number Theorem for Beurling’s Generalized Primes. New Cases
(Beurling generalized primes over th 3/2 limit):

At 3:38 AM, Anonymous said...

The generalized prime number theorem relates to the speculation of Matthew Watkings that primes can be generalized to dynamical objects and the "real" primes correspond to their equilibrium values.

One possibility is based on generalization of p-adicity: one can define q-adic norm for any rational number q, not only for q=p. This gives ring but not number field. In equilibrium situation only primes would remain as stabile rings being at the same time number fields.

Amongst primes in turn some primes, such as certain primes near powers of 2, would be especially stable. I have developed this argument in some detail.

At 4:29 AM, Blogger Ulla said...

Cleaning up and consolidating the lines after FTL

At 7:13 AM, Anonymous Anonymous said...

Matti, my hunch if that there is such a theorem then the "hypergeometric continuation" is something other than an "analytic continuation". After all its getting into the concept of counting and factoring itself so of course its going to introduce some strange concept or another. I've adopted a new attitude so I try not to mix my dreams with the mathematics now :)

At 4:33 AM, Anonymous said...

Thanks to Stephen. I received the book of Lapidus about Riemann Hypothesis.


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