surprising connections between number theory and physics The idea of this website is to document all known research which in some way links number theory and physics. Although there have been a few conferences and subsequently published proceedings on this topic, these have only been able to touch on a small part of the overall body of work which has gone on. The contents of the site should be of interest to both number theorists and physicists. In recent times we have seen, somewhat unexpectedly, number theory being applied by physicists to solve physical problems and, perhaps even more unexpectedly, techniques developed by physicists applied to problems in number theory. Material relevant to all such developments is archived in the sections linked from the upper part of the front page. There is other "secondary" material, organised into categories linked from the lower part of the front page. "Probability and statistics" are not in themselves part of physics, but have been developed in reference to "events", physical phenomena, and measurements which in some way vary or fluctuate. The fact that they should be applicable to something so profoundly un-physical and unchanging as the theory surrounding the behaviour of the prime numbers is something widely acknowledged as "curious", "remarkable", "surprising", etc. To treat the occurrence of a prime number as a kind of "random event" is to apply to the pure, eternal world of number a type of thinking inspired by the ever changing physical world. Hardy and Littlewood commented in [HL] that "Probability is not a notion of pure mathematics, but of physics or philosophy." Still, as T. Gowers has observed in [Go] "Although the prime numbers are rigidly determined, they somehow feel like experimental data." The section concerning "fractality" is perhaps less directly physics-related, but a significant part of the content is the work of physicists. The archive has naturally expanded in some directions which lie outside its originally envisaged scope. Some of the sections (Bernoulli numbers, golden mean,...) are not particularly physics-related, but contain unorthodox, innovative, speculative, or just curious approaches to number theoretic issues. As well as links and archived electronic articles relating to in-depth material, I have included some introductory number theory resources (follow the tutorial link from the main page). These are for the benefit of students, interested amateurs who wish to educate themselves in these matters, and research physicists whose work has begun to reveal some unexpected connection with number theory and who therefore need to quickly learn the basics. My own attraction to the subject matter stems from a vague, but deeply-rooted feeling that these newly emerging connections are beginning to reveal something quite extraordinary and unexpected about the very nature of the number system, something which has been hitherto inaccessible. I am overwhelmed with a sense of mystery when I browse the contents of this archive. It's not just me – the prime numbers unto themselves have inspired some remarkable quotations from those who have studied them (collected here). The unexpected connections with physics, I feel, significantly compound this widely acknowledged mystery. Although I am fully aware that my understanding is limited and possibly misguided to some extent, my intuition and enthusiasm compel me to dedicate a significant amount of time to this project, despite the lack of any financial support. It is my hope that this site will catalyse and accelerate the cross-fertilisation of ideas between number theory and physics, as well as inspiring others to use the Web in a similar spirit. The future of mathematics could be essentially a single, enormous, web of hypertext documents, potentially accessible to the whole of humanity. Putting aside organisational difficulties, this project happily looks forward to its possible destiny as a small piece of this great web of knowledge. Contributions are welcomed. I am always happy to see this project expand and
would like to involve as many other people as possible.
Here's an excellent survey article: D. Schumayer and D.A.W. Hutchinson, "Physics of the Riemann Hypothesis" (preprint 01/2011)
Below is a fairly brief overview of some of the more significant content of the archive.
Bernard Julia of the Laboratoire de Physique Théorique de l' Ecole Normale Supérieure in Paris has reinterpreted the (pure mathematical) Riemann zeta function as a (thermodynamic) partition function by defining an abstract numerical "gas" using the prime numbers ([J]). The free Riemann gas is a surprisingly natural concept, and its partition function is identical to the zeta function. In statistical mechanics, the partition function is the fundamental mathematical object of study; in the analytic theory of the distribution of primes, the zeta function is the fundamental object. Hence this unorthodox interpretation of the zeta function as a partition function points to a possible link of fundamental significance between the distribution of primes and this branch of physics. Julia has further linked the pole of the zeta function at s = 1 with the physical phenomenon known as a Hagedorn catastrophe which occurs when a system reaches a critical Hagedorn temperature ([H1-3]). Physical phenomena such as Bose condensation and ladders of fermion models are also indirectly linked to the zeta function by Julia in this paper. Donald Spector has informed me that he independently and simultaneously recognised the connection with partition functions, the Hagedorn temperature, etc. ([Sp]). His own papers draw on the theory of supersymmetry and suggest several intruiging parallels between phenomena in physics and multiplicative number theory. Bost and Connes groundbreaking paper [BC] was partly inspired by the work of Julia, and also involves a model in which the Riemann zeta function is interpreted as a partition function. In this model, the pole of zeta at 1 is understood in terms of a spontaneous breaking of symmetry. D. Fivel has independently interpreted the Riemann zeta function as a partition function in an entirely different context – that of quantum entanglement ([F]). Andreas Knauf and others have been studying spin chains (essentially one-dimensional lattice models) and in many cases the relevant partition functions involve the Riemann zeta function and its derivative in various combinations. In a 1999 lecture in Budapest ([W1]), Marek Wolf, a Wroclaw University physicist, presented a variant on Julia's idea, defining another kind of "prime gas", this time where the partition function resembles that of a quantum harmonic oscillator. Wolf's gas differs from Julia's in that the gaps between the primes are taken to be the fundamental quantities, rather than primes themselves. A summary of Wolf's lecture can be found here and provides a good review of much work done in this area. All of this work (and more) is documented in the section of the number theory and physics archive dedicated to statistical mechanics. Whereas everything mentioned thus far relates to multiplicative number theory, there is an
additional body of work going back to the 1940's which uses the methods of statistical
mechanics to investigate the problems of additive number theory, in particular the
partitioning of integers.
Around 1915, about 56 years after Riemann published his famous hypothesis, Hilbert and Pólya independently suggested that it could be proved if the nontrivial (complex) zeros of zeta could be shown to correspond directly to the spectrum of eigenvalues of some hermitian operator on a Hilbert space. Evidence for the validity of this "spectral interpretation" of the Riemann zeta function has since come from two sources:
The Hilbert–Pólya conjecture, as it has become known, still seems to be the most promising approach to proving the Riemann hypothesis. In [J] Julia acknowledges the celebrated work of Sir Michael Berry and Jon Keating ([BK]) of Bristol University, who have also been looking at the Riemann zeta function from a dynamical viewpoint. Theirs is certainly the most widely known of all the research discussed in the present survey. Berry's background involves quantum chaology, a branch of physics which seeks to identify signatures of chaos in the spectra of physical systems on the border of the quantum and classical worlds. Random matrix theory and the Gaussian Unitary Ensemble in particular turn out to play a significant role in this. Having investigated the various connections between quantum chaos, random matrices and the nontrivial zeta zeros, Berry has conjectured that the nontrivial zeta zeros correspond to the spectrum of eigenvalues (energy levels) of a Hamiltonian governing a quantum mechanical system whose underlying classical mechanics are chaotic and time-irreversible. Remarkably, if such a dynamical system could be identified, that is, one whose spectrum corresponds exactly to the set of nontrivial zeta zeros, then the Hilbert–Pólya conjecture, and hence the Riemann hypothesis would be proven. An excellent, reasonably non-technical article documenting this quantum chaological approach to the zeta function, "A Prime Case of Chaos" by Barry Cipra, can be found on the American Mathematical Society website (in PDF format). More recently, a simpler popular exposition of these matters appeared in New Scientist (11/11/00) ([K]). The proof of the Riemann hypothesis has been called "the single most desirable achievement for a mathematician" ([G]), even compared to the holy grail. Astonishingly, the problem has effectively been reduced to a quest for a dynamical system which is in some sense "implied" by the distribution of prime numbers (via the intimately connected zeros of the zeta function). Berry refers to this hypothetical system as the Riemann dynamics. Needless to say, it has not yet been found, although many of its properties are known, if indeed it exists. In [K], Berry is quoted as says that if the dynamical system can be identified, then he is ". . . absolutely sure that . . . someone will find a clever way to make it in the lab. Then you'll get the Riemann zeros out just by observing its spectrum." ""Finding this system could be the discovery of the century," [Berry] says. It would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos. Until [it is found] we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It is a tantalising thought." (from [B]) Several interesting attempts have been made to produce the required Hamiltonian. J.V. Armitage has published notes [A] involving diffusion processes, Brownian motion and the Fokker–Planck equation. Bhaduri, Khare, et. al. have linked the problem to the scattering of partial waves in the analysis of resonances, e.g. in pion-nucleon scattering ([BhK]). The related work of Christopher Deninger ([D1-2]) studies dynamical systems (flows) on foliated manifolds. Much like the Selberg–Weil coincidence mentioned above, he has identified a similarity between
Helsinki physicist M. Pitkänen has suggested an approach involving superconformal invariance ([P]), and in a related work, C. Castro has argued that supersymmetric QM should instead be used, in combination with Brownian motion and p-adic fractal strings, to reduce the Riemann hypothesis to an inverse scattering problem ([C]). More references and commentary on the "spectral interpretation" of the Riemann zeta function can be found in the relevant
section of the number theory and physics archive.
Another area of interconnection between number theory and physics, although of a very different flavour, involves the theory of p-adic numbers. Both A. Khrennikov and B. Dragovich have published impressive numbers of articles providing p-adic interpretations of physical systems and phenomena. This appears to be a rapidly developing area of research. Also of possible interest here are Castro and Mahecha's preprints [CM] and [C] linking the Riemann hypothesis to p-adic fractal strings and fractal p-branes. This has been partly inspired by Lapidus and van Frankenhuysen's fascinating book [LvF] which relates the zeta zeros to fractal geometry and a theory of complex dimensions. Castro explains: "Supersymmetry, p-adic stochastic dynamics, Brownian motion, Fokker–Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc...are all deeply interconnected in this paper." M. Pitkanen has also produced some intriguing notes suggesting a p-adic physics-inspired interpretation of the nontrivial zeta zeros, leading to his aforementioned proof strategy for the Riemann hypothesis which involves superconformal invariance, and in particular the Virasoro generator ([P]). Interestingly (although without any p-adic content), A. Petermann's preprint [Pe] also involves the Virasoro generator in an attempt to elucidate the deep reasons for the logarithmic distribution of primes. This involves a breaking of symmetry, namely that of scale invariance, and relates to certain aspects of quantum chromodynamics. Fields medalist Alain Connes, who some
commentators have suggested is the most likely candidate for a
proof of the Riemann hypothesis, has also linked the distribution
of primes to a spontaneous symmetry-breaking, albeit of a different
kind, in the article [BC] mentioned above. His extremely deep work
involves noncommutative geometry, and the theory of adeles
(related to p-adic analysis) in an attempt to produce the
necessary Hamiltonian which will satisfy the Hilbert–Pólya conjecture
discussed above.
Indirectly, as a result of studying nonlinear dynamics Marek Wolf discovered two instances of apparent fractality within the distribution of prime numbers ([W2-3]). These discoveries were realised experimentally using powerful computers. Wolf's resulting interest in the distribution of the primes led him to experimentally discover the presence of 1/f noise when the primes are treated as a "signal" in the sense of information theory ([W4]). This is also a self-similar (scale invariant, or fractal) property of the distribution of primes. 1/f noise, also known as flicker noise or pink noise, is a property of the power-frequency spectrum (obtained through Fourier analysis) of the signal. It has been detected in many diverse physical systems including sunspots, quasars, hourglasses, rivers, electronic components, DNA sequences, written language, weather patterns and stock exchange indices. It has been argued that its presence suggests some kind of "cooperative" effect over a wide range of timescales. Bak, Tang, and Wiesenfeld have offered an explanation for the ubiquity of 1/f noise by developing a simple model of self-organised criticality which has 1/f noise as a "temporal fingerprint" and self-similarity as a "spatial fingerprint" ([BTW]). The implication is that the previously mentioned physical systems could all be examples of self-organised critical systems. Wolf, being aware of this work, ended his article "1/f noise in the distribution of prime numbers" ([W4]) with the astonishing question "Are the prime numbers in a self-organized critical state?" I have since been informed that Bak, et. al. made a fundamental error in their calculations, and consequently the results in [BTW] apply to the less significant phenomenon of 1/f^{2} noise. Still, the possibility that the primes might constitute something akin to a self-organised system is one which I personally find to be quite compelling. Michel Planat of
the Laboratoire de Physique et Métrologie des
Oscillateurs du CNRS in France has since brought to light
more connections between 1/f noise and the distribution of primes,
via the Riemann Hypothesis ([Pl]).
Another relatively recent development linking prime numbers and physics is an article by H. Gopalkrishna Gadiyar and R. Padma called "Ramanujan–Fourier series, the Wiener–Khintchine formula and the distribution of prime pairs" [GGP1]. Its abstract explains: "The Wiener–Khintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a Wiener–Khintchine formula. "Experimental" evidence is given for this." The authors concludes with the following observation: "It is a pleasant surprise that the Wiener–Khintchine formula which normally occurs in practical problems of Brownian motion, electrical engineering and other applied areas of technology and statistical physics has a role in the behaviour of prime numbers which are studied by pure mathematicians." This follows the authors' earlier publication
"Renormalisation and the density of prime pairs" [GGP2]
which uses techniques from quantum field theory to suggest a possible
approach to proving a number theoretic conjecture of Hardy and Littlewood.
"Despite the stunning advances linking Riemann's zeta function to
20^{th} century physics, no one is predicting an imminent
proof of the Riemann hypothesis. Odlyzko's numerical experiments and
evidence amassed by physicists have convinced everyone that a spectral
interpretation of the zeta zeros is the way to go, but number
theorists say they are at least one "big idea" away from even the
beginnings of a proof. Mathematicians aren't yet sure what to aim at,
says [Princeton University mathematician Peter] Sarnak."
"...the Riemann Hypothesis will be settled without any fundamental changes in our
mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea
is missing."
"...there have been very few attempts at proving the Riemann hypothesis, because, simply,
no one has ever had any really good idea for how to go about it."
"I still think that some major new idea is needed here"
"Sometimes I think that we essentially have a complete proof of the Riemann Hypothesis
except for a gap . The problem is, the gap occurs right at the beginning, and so it's hard to
fill that gap because you don't see what's on the other side of it."
conclusion
references[HL] G.H Hardy and J.E. Littlewood, "Some problems of 'partitio numerorum': III: on the expression of a number as a sum of primes" Acta Mathematica 44 (1922) 1–70 [Go] T. Gowers, "Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.118 [J] B.L. Julia, "Statistical theory of numbers", from Number Theory and Physics (eds. J.M. Luck, P. Moussa, and M. Waldschmidt ), Springer-Verlag, 1990 [H1] R. Hagedorn, Suppl. Nuovo Cimento 3 (1965) 147. See also Yu. B. Rumer, Journal of Experimental Theoretical Physics 38 (1960) p.1899 [H2] R. Hagedorn, Lecture Notes in Physics, Vol. 221 (Springer, Berlin, 1985), p.53 [H3] R. Hagedorn, "Hadronic matter near the boiling point", Nuovo Cimento 56A, (1968) 1027–1057 [Sp] D. Spector, "Supersymmetry and the Mobius Inversion Function", Communications in Mathematical Physics 127 (1990) p.239 [BC] J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411–457 [F] D. Fivel, "The prime factorization property of entangled quantum states" (preprint hep-th/9409150) [BK] M.V. Berry and J.P. Keating "The Riemann zeros and eigenvalue asymptotics" SIAM Review, 41, No. 2 (1999), 236–266 [K] E. Klarreich, "Prime Time", New Scientist, 11 November 1999 [G] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, 1990), p.308 [B] J. Brown, "Where Two Worlds Meet", New Scientist (16 May 1996) [A] J.V. Armitage, "The Riemann Hypothesis and the Hamiltonian of a quantum mechanical system" (section 5: "A random walk approximation to the Riemann Hypothesis"), from Number Theory and Dynamical Systems, eds. M.M. Dodson and J.A.G. Vickers (LMS Lecture Notes, series 134, Cambridge University Press), 153–172 [BhK] R.K. Bhaduri, A. Khare, S.M. Reimann and E.L. Tomusiak, "The Riemann zeta function and the inverted harmonic oscillator", Annals of Physics 254 no. 1 (1997) [D1] C. Deninger, "Some ideas on dynamical systems and the Riemann zeta function" (preprint from proceedings of the 1997 ESI conference on the Riemann zeta function) [D2] C. Deninger, "Some analogies between number theory and dynamical systems on foliated spaces", Documenta Mathematica, Extra Volume ICM I (1998) 163–186 [D3] C. Deninger, "Motivic L-functions and regularized determinants", Proc. Symp. Pure Math. 55 1 (1994) 707–743 [P] M. Pitkänen, "Riemann hypothesis and super-conformal invariance" (preprint math.GM/0102031) [C] C. Castro (Perelman), "p-Adic stochastic dynamics, supersymmetry and the Riemann conjecture" (preprint physics/0101104) [CM] C. Castro and J. Mahecha, "Comments on the Riemann conjecture and index theory on Cantorian fractal space-time", Chaos, Solitons and Fractals 13 (2002) 1407 [LvF] M.L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Fractal Strings and Zeros of Zeta Functions (Birkhauser, 2000) [Pe] A. Petermann, "The so-called renormalization group method applied to the specific prime number logarithmic decrease", European Physical Journal C 187 (2000) 367–370 [W1] M. Wolf, "Applications of statistical mechanics in prime number theory", preprint. Available on request from the author, and summarised here [W2] M. Wolf, "Multifractality of prime numbers", Physica A 160 (1989), 24–42 [W3] M. Wolf, "Random walk on the prime numbers", Physica A 250 (1998), 335–344 [W4] M. Wolf, "1/f noise in the distribution of prime numbers", Physica A 241 (1997), 493–499 [BTW] P. Bak, C. Tang, and K. Wiesenfeld, "Self-organized criticality", Physical Review A 38 (1988), 364–374 [Pl] M. Planat, "1/f frequency noise in a communication receiver and the Riemann Hypothesis" [GGP1] H. Gopalkrishna Gadiyar and R. Padma, "Ramanujan–Fourier series, the Wiener–Khintchine formula and the distribution of prime pairs", Physica A 269 (1999) 503–510 [GGP2] R. Padma and H.Gopalkrishna Gadiyar, "Renormalisation and the density of prime pairs" (preprint, 06/98) [S] K. Sabbagh, Dr.
Riemann's Zeros: The Search for the $1 Million Solution to the Greatest Problem in Mathematics (Atlantic Books, 2002)
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