p-adic and adelic physics - general
A. Connes, "Formule de trace en geometrie non commutative et hypothese
de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996)
(Abstract) "We reduce the Riemann hypothesis for L-functions on a
global field k to the validity (not rigorously justified) of a trace
formula for the action of the idele class group on the noncommutative
space quotient of the adeles of k by the multiplicative group of
Berry and Keating refer to this article in their "H = xp
and the Riemann zeros", and explain that Connes has devised a Hermitian
operator whose eigenvalues are the Riemann zeros on the critical line.
This is almost the operator Berry seeks
in order to prove the Riemann Hypothesis, but unfortunately the possibility of zeros off the
critical line cannot be ruled out in Connes' approach.
His operator is
the transfer (Perron-Frobenius) operator of a classical transformation.
Such classical operators formally resemble quantum Hamiltonians, but
usually have complicated non-discrete spectra and singular eigenfunctions.
Connes gets a discrete spectrum by making the operator act on an
abstract space where the primes appearing in the Euler product for the
Riemann zeta function are built in; the space is constructed from
collections of p-adic numbers (adeles) and the associated units
(ideles). The proof of the Riemann Hypothesis is thus reduced to
the proof of a certain classical trace formula.
of the Real Prime (OUP, 2001)
[publisher's description:] "In this important and original monograph, useful for both
academic and professional researchers and students of mathematics and physics, the author
descibes his work on the Riemann zeta function and its adelic interpretation. It provides
an original point of view, bringing new, highly useful dictionaries between different
fields of mathematics. It develops an arithmetical approach to the continuum of real
numbers and unifies many areas of mathematics including: Markov Chains, q-series, Elliptic
curves, the Heisenberg group, quantum groups, and special functions (such as the Gamma,
Beta, Zeta, theta, Bessel functions, the Askey-Wilson and the classical orthagonal
polynomials) The text discusses real numbers from a p-adic point of view, first mooted by
Araeklov. It includes original work on coherent theory, with implications for number
theory and uses ideas from probability theory including Markov chains and noncommutative
geometry which unifies the p-adic theory and the real theory by constructing a theory of
quantum orthogonal polynomials."
If you didn't know, the "real prime" is the mysterious thing also referred to as
"the prime at infinity". This is associated with the real valuation (as opposed to any of the
p-adic valuations) on the rationals - the one which completes Q to give R
rather than some Qp. An adele is an object of the form
a11,a13,...).The first term is a real number, corresponds to
'the real prime' or the prime at infinity. The other ap are p-adic
numbers - they correspond to the 'finite' primes p. Adeles are central to Connes'
approach to the Riemann Hypothesis (his all-important trace formula is defined on a non-commutative space
of Adele classes).
R. Pearson, "Number theory and critical exponents",
Phys. Rev. B 22 (1980) 3465-3470
[abstract:] "The consequences of assuming p-adic analyticity for thermodynamic functions are discussed. Rules are
given for determining the denominator of a rational critical exponent from the asymptotic behavior of the coefficients of series
expansions. The example of the Hamiltonian Q-state Potts model is used to demonstrate the ideas of the paper."
M. Planat, "On the cyclotomic quantum algebra of time
perception" (preprint 03/04)
[abstract:] "I develop the idea that time perception is the quantum counterpart to time measurement. Phase-locking
and prime number theory were proposed as the unifying concepts for understanding the optimal synchronization of
clocks and their 1/f frequency noise. Time perception is shown to depend on the thermodynamics of a quantum
algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to
a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers.
The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below,
which are parametrized by the symmetry group for the primitive roots of unity. Squeezing of phase fluctuations close to
the phase transition temperature may play a role in memory encoding and conscious activity."
B. Dragovich and B. Sazdovic "Real, p-adic
and adelic noncommutative scalar solitons" (preprint 06/03)
[abstract:] "We considered real, p-adic and adelic noncommutative scalar solitons and obtained
some new results."
M.V. Altaisky and B.G. Sidharth, "p-Adic
physics below and above Planck scales"
[abstract:] "We present a rewiew and also new possible applications of p-adic
numbers to pre-spacetime physics. It is shown that instead of the extension
Rn to Qpn, which is usually implied in p-adic quantum field theory, it is possible
to build a model based on the Rn to Qp,
where p = n + 2 extension and get rid of loop
divergences. It is also shown that the concept of mass naturally arises in p-adic
models as inverse transition probability with a dimensional constant of proportionality."
A. Abdesselam, A. Chandra and G. Guadagni, "Rigorous quantum field theory functional integrals over the $p$-adics I: Anomalous dimensions" (preprint 01/2013)
[abstract:] "In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional $p$-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group."
N. Makhaldiani, "Adelic universe
and cosmological constant" (preprint, 12/03)
[abstract:] "In the quantum adelic field (string) theory models, vacuum energy - cosmological
constant vanish. The other (alternative?) mechanism is given by supersymmetric theories. Some
observations on prime numbers, zeta-function and fine structure constant are also considered."
S. Kozyrev, "Wavelets
analysis as p-adic harmonic analysis"
[abstract:] "New orthonormal basis of eigenfunctions for the Vladimirov operator of
p-adic fractional derivation is constructed. The map of p-adic numbers
onto real numbers (p-adic change of variables) is considered. This map (for p = 2)
provides an equivalence between the constructed basis of eigenfunctions of the Vladimirov
operator and the wavelet basis in L2(R) generated from the Haar
wavelet. This means that the wavelet analysis can be considered as a p-adic
and M. Marcolli, "Non-commutative geometry,
dynamics, and infinity-adic Arakelov geometry" (to appear in
[abstract:] "In Arakelov theory a completion of an arithmetic
surface is achieved by enlarging the group of divisors by formal
linear combinations of the 'closed fibers at infinity'. Manin
described the dual graph of any such closed fiber in terms of an
infinite tangle of bounded geodesics in a hyperbolic handlebody
endowed with a Schottky uniformization. In this paper we consider
arithmetic surfaces over the ring of integers in a number field, with
fibers of genus g > 2. We use Connes' theory of spectral
triples to relate the hyperbolic geometry of the handlebody to
Deninger's Archimedean cohomology and the cohomology of the cone of
the local monodromy N at arithmetic infinity as introduced by
the first author of this paper."
K. Consani and M. Marcolli, "Triplets spectraux en
geometrie d'Arakelov" Comptes Rendus Acad. Sci. Paris Ser.
I 335 (2002) 779-784
[abstract:] "This note is a brief overview of the results of
We use Connes' theory of spectral triples to provide a connection
between Manin's model of the dual graph of the fiber at infinity of an
Arakelov surface and the cohomology of the mapping cone of the local
B. Dragovich and A. Dragovich, "p-Adic modelling of the genome and the genetic code"
[abstract:] "The present paper is devoted to foundations of p-adic modelling in genomics. Considering nucleotides, codons, DNA and RNA sequences,
amino acids, and proteins as information systems, we have formulated the corresponding p-adic formalisms for their investigations. Each of these systems has its
characteristic prime number used for construction of the related information space. Relevance of this approach is illustrated by some examples. In particular, it is
shown that degeneration of the genetic code is a p-adic phenomenon. We have also put forward a hypothesis on evolution of the genetic code assuming that primitive
code was based on single nucleotides and chronologically first four amino acids. This formalism of p-adic genomic information systems can be implemented in computer
programs and applied to various concrete cases."
V. Bezgin, M. Endo, A. Khrennikov, and M. Yuoko, "Statistical biological
models with p-adic stabilization", Dokl. Akad. Nauk 334,
no.1 (1994) 5-8.
A. Khrennikov, "p-adic model for population growth", from
Fractals in Biology and Medicine, 2, Eds. G.A. Losa, et. al.
"Learning of p-adic neural
networks" (preprint, 1999)
[abstract:] "A p-adic model which describes a large class of neural
networks is presented. In this model the staes of neurons are described by
digits in the canonical expansion of a p-adic number. Thus each p-adic
number represents a configuration of firing and non-firing neurons. We present the
algorithm of learning for p-adic neural networks based on the minimization
of the error-functional (here we use a random search procedure in the space of
p-adic weights). This algorithm (or its more advanced versions) could be
applied for image recognition."
numerous articles published by A. Khrennikov
relating p-adic analysis to various branches of physics, biology,
neural networks, etc.
The Third International Conference on p-adic Mathematical Physics:
From Planck scale physics to complex systems to biology, Steklov Mathematical Institute
Moscow, Russia, October 1-6, 2007
"p-adic mathematical physics is a rapidly developing area with numerous
applications in different fields ranging from quantum theory to chaotic
and nano systems to molecular biology and to information science.
The aim of this conference is to present recent results in p-adic
mathematical physics, related fields, and applications, as well as to
discuss earlier results and possible future directions of investigation."
geometro-dynamics and p-adic numbers
An astonishingly comprehensive work-in-progress by Finnish physicist
Matti Pitkänen. Individual
chapters can be downloaded the form of pdf files. In a recent communication he
stated "...quantum criticality, fractality and spin glass property are
basic elements of TGD universe. In fact, I have a hunch that physics (I
hope quantum TGD) could be essentially number theory in some generalized
sense". The chapter "p-adicization of quantum TGD" presents a sharpened
form of the Riemann Hypothesis. This ties in with Michael Berry's work,
and Pitkänen appeals for number theorists to examine it, stating that he
is "just a poor physicist without the needed skills".
Recently (7 January, 2001), Pitkännen submitted the following notes:
"Intuitive arguments in favour of a sharpened
form of the Riemann Hypothesis"
A p-adic version of the Riemann zeta function is considered, leading
to physical interpretations of the zeros of the classical Riemann zeta
function. This is part of Pitkänen's ultimate program of reducing all of
physics to number theory.
"Further ideas about the Riemann hypothesis leading
to a further sharpening of the Riemann hypothesis and to a p-adic
particle physicist's articulation of what it is to be a zero of the Riemann
Quantum TGD and how to prove Riemann hypothesis (3/2/2001)
"During last month further ideas about Rieman hypothesis have emerged
and have led to further sharpening of Riemann hypothesis and to p-adic
particle physicist's articulation for what it is to be zero of Riemann Zeta
and to the idea that Riemann hypothesis reduces to superconformal invariance
of the physical system involved. One can verify Hilbert-Pólya hypothesis on
basis of the physical picture obtained. This means an explicit construction
of the differential operator having the moduli squared of the zeros of
Riemann Zeta as eigenvalues. This operator is product of two operators which
are Hermitian conjugates of each other and have zeros of Riemann Zeta as
their eigenvalues. The facts that x corresponds to the real part of
conformal weight in this model and that one has x = n/2 for the
operators appearing in the representations of Super Virasoro, suggest that
x = n/2 is indeed the only possible value of x for the
zeros of Riemann zeta both in real and p-adic context. Hence Riemann
hypothesis would indeed reduce to superconformal invariance."
hypothesis and super-conformal invariance"
notes on p-adic zeta functions
and a sharpened form of the RH (20/03/01)
"Quantum criticality and 1/f noise"
(submitted for publication in Fluctuation and Noise Letters)
M. Pitkänen's homepage