padic and adelic physics
introduction to padic numbers and adeles
padic and adelic physics: introductory remarks and resources
padic dynamics
padic string and brane theories, etc.
padic and adelic approaches to quantum mechanics
padic probability, statistics and randomness
padic and adelic scattering theory
padic physics  general
New journal: pAdic
Numbers, Ultrametric Analysis, and Applications
"This is a new international interdisciplinary journal which is going to publish original articles, short communications, and reviews on progress in padic, adelic and ultrametric developments in the following research areas: mathematical physics, quantum theory, string theory, cosmology, nanoscience, life sciences; mathematical analysis, number theory, algebraic geometry, nonArchimedean and noncommutative geometry, theory of finite fields and rings, representation theory, functional analysis and graph theory; classical and quantum information, computer science, cryptography, image analysis, cognitive models, neural networks and bioinformatics; complex systems, dynamical systems, stochastic processes, hierarchy structures, modeling, control theory, economics and sociology; mesoscopic and nano systems, disordered and chaotic systems, spin glasses, macromolecules, molecular dynamics, biopolymers, genomics and biology, and other related fields."
The first three issues of the journal are already published.
Papers should be submitted to padics@mi.ras.ru.
frontmatter from Volume 1, Issue 1
"Arithmetic physics, or better, arithmetic quantum theory, is a term that refers to
a collection of ideas and partial results, loosely held together, that suggests that
there are connections between the worlds of quantum physics and number theory and
that one should try to discover and develop these connections. At one extreme is
the modest idea that one should try to formulate some of the mathematical questions
arising in quantum theory over fields and rings other than R, such as the field of
padic number Q_{p}, or the ring of adeles over the rationals A(Q). The point here
is not to try to develop the alternative theories as a substitute for the actual
theory or even look for physical interpretations, but rather to look for results
that would unify what we already know over R.
The basis for this suggestion is the simple fact that all experimental calculations
are essentially discrete and so can be modelled by mathematical structures that
are over Q. The theories over R are thus idealizations that are more convenient
than essential and reflect the fact that the field of real numbers is a completion
of the field of rational numbers. But there are other completions of the reals,
namely the fields Q_{p}, and it is clear that under suitable circumstances a large
finite quantum system may be thought of as an approximation to a system defined
over Q_{p}. If we continue this line of thought further, it becomes necessary to consider
all the completions of Q, which means working over the ring of adeles A(Q).
At the other extreme are bold speculations that push forward the hypothesis that the
exploration of the structure of quantum theories by replacing R by Q_{p} and A(Q) is
not just a pleasant exercise but is essentiaI. I quote the following remarks of Manin
from his beautiful and inspiring paper [1].
"On the fundamental level our world is neither real nor padic; it is adelic. For some
reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that
we are built of massive particles), we tend to project the adelic picture onto its real
side. We can equally well spiritually project it upon its nonArchimediean side and
calculate most important things arithmetically.
The relations between "real" and "arithmetical" pictures of the world is that of
complementarity, like the relation between conjugate observables in quantum mechanics.""
[1] Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293303
V. Varadarajan, from "Some remarks on arithmetic physics"
L. Brekke and P. Freund, "padic numbers in physics", Physics Reports
233, (1993) 166.
This is a review article related to the achievements in application
of padic numbers to string theory, quantum field theory and
quantum mechanics during the period 19871992. The contribution of
Freund and his collaborators is emphasised.
Here is an excerpt from pp6162:
"String theory, the candidate "theory of everything" is expected to
raise fundamental issues both at the level of physics and at the level
of mathematics. The old issue of the nature of continuity in physics
naturally leads to the consideration of padic strings. It is
remarkable that these very simple alternate topologies have not
appeared earlier in physics (ultrametrics have appeared [62]). Yet,
even now it would not be reasonable to actually select a prime and
claim this to be the phenomenologically preferred prime which
"underlies" physics. As we have seen, such a preferred prime could lead
to serious causality problems. But if none of the primes is to be
preferred, then why select a priori the prime at infinity, and deal
exclusively with real numbers? A more "evenhanded" procedure would
envision dealing with all primes at the same time. This naturally leads
to adelic theories. We have seen that this point of view immediately
yields the remarkable adelic product formulae. Could it be that the
adelic string is the "real thing"? This question has been raised by
Manin [41] in the following (somewhat paraphrased) form. Supposing that
the true physics is adelic, then why can we always assume it to be
archimedean, grounded in the real numbers? Maybe this is on account of
some experimental limitations, e.g. low energy. Could it be that once
these limitations get lifted and we reach very high (Planck) energies,
the full adelic structure of the string will reveal itself? This is an
interesting possibility.
Another possibility is that the true theory is archimedean, but that
on account of the product formulae, one could alternatively conceive of
the theory as an Euler product over all padic theories. As we saw,
each such theory puts the strings' world sheet on a Bethe lattice. What
the adelic formulae then tell us is that we should not opt for a
particular Bethe lattice as the discretization of the world sheet, but
rather study absolutely all of them. The cumulative understanding of
all these discretizations is tantamount to understanding the ordinary
archimedean string. Of course, each of these discretizations is far
simpler than the ordinary string.
On the other hand, there is the padicsquantum group connection,
which places the ordinary and all the padic strings at certain special
points in a continuum of theories. It is an important problem to assess
the theoretical consistency of all these "quantum" strings and the
phenomenological possibilities offered by them."
B. Dragovich, "NonArchimedean
geometry and physics on adelic spaces" (preprint 06/03)
[abstract:] "This is a brief review article of various applications of
nonArchimedean geometry, padic numbers and adeles in modern mathematical physics."
V.S. Vladimirov, I.V. Volovich,
E.I. Zelenov,
pAdic Analysis and Mathematical Physics (World Scientific
Publishing)
I.V. Volovich,
"Number theory as the ultimate physical theory", pAdic Numbers, Ultrametric Analysis and Applications 2 (2010) 77–87
(Abstract) "At the Planck scale doubt is cast on the usual notion
of spacetime and one cannot think about elementary particles. Thus,
the fundamental entities of which we consider our Universe to be
composed cannot be particles, fields or strings. In this paper the
numbers are considered as the fundamental entities. We discuss the
construction of the corresponding physical theory.
A hypothesis on the quantum fluctuations of the number field is
advanced for discussion. If these fluctuations actually take place
then instead of the usual quantum mechanics over the complex number
field a new quantum mechanics over an arbitrary field must be
developed. Moreover, it is tempting to speculate that a principle
of invariance of the fundamental physical laws under a change of the
number field does hold.
The fluctuations of the number field could appear on the Planck
length, in particular in the gravitational collapse or near the
cosmological singularity. These fluctuations can lead to the
appearance of domains with nonArchimedean padic or finite
geometry."
A. Khrennikov,
pAdic Valued Distributions in Mathematical Physics,
(Kluwer, 1994).
R. Rammal, G. Toulouse, and M.A. Virasoro, "Ultrametricity for physicists", Rev. Modern. Physics. 58 (1986) 765788
padic dynamics
M. Nevins and
D. Rogers, "Quadratic maps as dynamical systems on
the padic numbers"
[abstract:] "We describe the trajectories of the successive iterates of the square map
and its perturbations on the field of padic numbers. We show that the cycles of the
square map on Q_{p} arise from cycles of the square map on
F_{p}, and that all nonperiodic trajectories in the unit disk densely
define a compact open subset. We find that the perturbed maps x > x^{2} + a, with a inside the unit circle, have similar dynamics to x > x^{2}, but that each fundamental cycle arising from F_{p} can further admit harmonic cycles, for different choices of p and a. In contrast, the cycles of the maps x > x^{2} + a, with a on the boundary of the unit circle, are no longer tied to those of the square map itself. In all cases we give a refined algorithm for computing the finitely many periodic points of the map."
R. Benedetto,
"pAdic dynamics and Sullivan's no wandering domain theorem",
Compositio Mathematica 122 (2000) 281298
[abstract:] "In this paper we study dynamics on the Fatou set of a rational function
f(z) defined over a finite extension Q_{p}, the field of
padic rationals. Using a notion of 'components' of the Fatou set defined in "Hyperbolic
maps in padic dynamics", we state and prove an analogue of Sullivan's No Wandering Domains Theorem for
padic rational functions which have no wild recurrent Julia critical points."
R. Benedetto, "Examples of wandering domains in padic polynomial dynamics"
(Comptes Rendus Mathématique. Académie des Sciences. Paris, 335 (2002), 615620.
[abstract:] "For any prime p > 0, we construct padic polynomial
functions in C_{p}[z] whose Fatou sets have wandering domains."
R. Benedetto, "Nonarchimedean holomorphic maps
and the Ahlfors Islands Theorem" (American Journal of Mathematics, accepted.) [DVI format]
[abstract:] "We present a padic and nonarchimedean version of some classical
complex holomorphic function theory. Our main result is an analogue of the Five Islands
Theorem from Ahlfors' theory of covering surfaces. For nonarchimedean holomorphic maps, our
theorem requires only two islands, with explicit and nearly sharp constants, as opposed to
the three islands without explicit constants in the complex holomorphic theory. We also
present nonarchimedean analogues of other results from the complex theory, including
theorems of Koebe, Bloch, and Landau, with sharp constants."
R. Benedetto, "Components and periodic points in nonarchimedean dynamics"
Proceedings of the London Mathematical Society 84 no. 3 (2002) 231256
[abstract:] "We expand the notion of nonarchimedean connected components introduced
in "Hyperbolic maps in padic dynamics". We define two types of components and
discuss their uses and applications in the study of dynamics of a rational function f
in K(z) defined over a nonarchimedean field K. Using this theory, we
derive several results on the geometry of such components and the existence of periodic
points within them. Furthermore, we demonstrate that for appropriate fields of definition,
the conjectures stated in "pAdic dynamics and Sullivan's No Wandering Domains
Theorem", including the No Wandering Domains conjecture, are equivalent regardless of which
definition of 'component' is used. We also give a number of examples of padic maps
with interesting or pathological dynamics."
R. Bendetto, "Hyperbolic maps in padic dynamics",
Ergodic Theory and Dynamical Systems 21 (2001) 111
[abstract:] "In this paper we study the dynamics of a rational function f(z)
defined over a finite extension Q_{p}, the field of padic
rationals. After proving some basic results, we define a notion of 'components' of the Fatou
set, analogous to the topological components of a complex Fatou set. We define hyperbolic
padic maps and, in our main theorem, characterize hyperbolicity by the location of the
critical set. We use this theorem and our notion of components to state and prove an analogue
of Sullivan's No Wandering Domains Theorem for hyperbolic maps."
D. Chistyakov, "Fractal
geometry for images of continuous map of padic numbers
and padic solenoids into Euclidean spaces"
[abstract:] "Explicit formulas are obtained for a family of continuous mappings of
padic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the
space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for
which the Cantor set and the Sierpinski triangle are images of the unit balls in $\Qn{2}$
and $\Qn{3}$. In each of the families, the subset of the embeddings is found. For these
embeddings, the Hausdorff dimensions are calculated and it is shown that the fractal measure
on the image of $\Qp$ coincides with the Haar measure on $\Qp$. It is proved that under
certain conditions, the image of the padic solenoid is an invariant set of fractional
dimension for a dynamic system. Computer drawings of some fractal images are presented."
D. Chistyakov, "Fractal
measures, padic numbers and continuous transition between dimensions"
[abstract:] "Fractal measures of images of continuous maps from the set of padic
numbers Q_{p} into complex plane C are analyzed. Examples of
'anomalous' fractals, i.e. the sets where the Ddimensional Hausdorff measures (HM)
are trivial, i.e. either zero, or sigmainfinite (D is the Hausdorff dimension (HD)
of this set) are presented. Using the Caratheodory construction, the generalized
scalecovariant HM (GHM) being nontrivial on such fractals are constructed. In particular,
we present an example of 0fractal, the continuum with HD=0 and nontrivial GHM invariant
w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains
in R^{n}, the formula for the change of variables
for GHM is obtained. The family of continuous maps Q_{p} in C
continuously dependent on "complex dimension" d in C is obtained. This family
is such that: 1) if d = 2(1), then the image of b>Q_{p} is C
(real axis in C.); 2) the fractal measures coincide with the images of the
Haar measure in Q_{p}, and at d = 2(1) they also coincide with
the flat (linear) Lebesgue measure; 3) integrals of entire functions over the fractal measures
of images for any compact set in Q_{p} are holomorphic in d,
similarly to the dimensional regularization method in QFT."
V. Anashin, "Uniformly
distributed sequences of padic integers, II"
[abstract:] "The paper describes ergodic (with respect to the Haar measure) functions in
the class of all functions, which are defined on (and take values in) the ring of padic
integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1.
Equiprobable (in particular, measurepreserving) functions of this class are described also.
In some cases (and especially for p = 2) the descriptions are given by explicit
formulae. Some of the results may be viewed as descriptions of ergodic isometric dynamical
systems on padic unit disk."
E. Thiran, D. Verstegen and J. Weyers, "padic dynamics", Journal of Statistical
Physics 54 nos. 34 (1989) 893913
D. Verstegen, "padic dynamical systems" from Number
Theory and Physics (J.M. Luck, P. Moussa and M. Waldschmidt, eds.), Springer
Proceedings in Physics 47 (Springer, 1990) 235242
L. Hsia, "A
weak Néron model with applications to padic dynamical systems",
Composito Math. 100 (1996) 277304
HuaChieh Li, "padic periodic points and Sen's theorem", J. Number Theory
56 no. 2 (1996) 309318
J. Lubin, "Nonarchimedean
dynamical systems", Compositio Math. 94 no,. 3 (1994) 321346
J. Lubin, "Formal flows on the nonarchimedean open unit disk", Compositio Math.
124 (2000) 123136
S. BenMenahem, "pAdic iterations", preprint, TelAviv UP (1988) 162788
D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors
of random dynamical systems over padic numbers and a model of noisy cognitive
processes", Physica D 130 (1999) 112
A. Khrennikov, "padic dynamical systems: description of concurrent
struggle in biological population with limited growth", Dokl. Akad.
Nauk 361 no. 6 (1998) 752754.
M. de Gosson, B. Dragovich and A. Khrennikov, "Some
padic differential equations"
"We investigate various properties of padic differential equations which have as a solution
an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k (n) x^n$, where $P_k (n) = n^k
+ C_{k1} n^{k1} + ...+ C_0$ is a polynomial in n with $C_i\in Z$ (in a more general case
$C_i\in Q$ or $C_i\in C_p$). For some special classes of $P_k (n)$, as well as for the general
case, the existence of the corresponding linear differential equations of the first and
secondorder for $F_k (x)$, is shown. In some cases such equations are constructed. For the
secondorder differential equations there is no other analytic solution of the form
$\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous firstorder differential
equation exists one can construct infinitely many inhomogeneous secondorder equations with the
same analytic solution. Relation to some rational sums with the Bernoulli numbers and to
$F_k (x)$ for some $x\in Z$ is considered. Some of these differential equations can be related
to padic dynamics and padic information theory."
A. Khrennikov, NonArchimedean Analysis: Quantum Paradoxes, Dynamical Systems and
Biological Models (Kluwer, 1997)
A. Khrennikov and M. Nilsson, "On
the number of cycles of padic dynamical systems",
Journal of Number Theory 90 (2001) 255264
[abstract:] "We found the asymptotics, p, for the number of cycles for iteration of monomial functions in the fields of
padic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers."
S. De Smedt, A. Khrennikov, "A padic behaviour of dynamical systems",
Rev. Mat. Comput. 12 (1999) 301323
S. Nechaev and O. Vasilyev, "On
metric structure of ultrametric spaces", J. Phys. A 37 (2004) 37833803
[abstract:] "In our work we have reconsidered the old problem of diffusion at the boundary of
ultrametric tree from a "number theoretic" point of view. Namely, we use the modular functions
(in particular, the Dedekind etafunction) to construct the "continuous" analog of the Cayley tree
isometrically embedded in the Poincaré upper halfplane. Later we work with this continuous Cayley
tree as with a standard function of a complex variable. In the frameworks of our approach the results
of Ogielsky and Stein on dynamics on ultrametric spaces are reproduced semianalytically/seminumerically.
The speculation on the new "geometrical" interpretation of replica n > 0 limit is proposed."
S. Matsutani, "padic differencedifference LotkaVolterra equation
and ultradiscrete limit", Int. J. Math. and Math. Sci. 27 (2001) 251260
[abstract:] "We study the differencedifference LotkaVolterra equations in padic number space and its
padic valuation version. We point out that the structure of the space given by taking the ultradiscrete limit is
the same as that of the padic valuation space. Since ultradiscrete limit can be regarded as a classical limit of
a quantum object, it implies that a correspondence between classical and quantum objects might be associated with
valuation theory."
S. Matsutani, "LotkaVolterra equation over a finite ring $\mathbb{Z}/p^N
\mathbb{Z}$", J. Phys. A 34 (2001) 1073710744
[abstract:] "The discrete LotkaVolterra equation over $p$adic space was constructed since $p$adic space is a
prototype of spaces with nonArchimedean valuations and the space given by taking the ultradiscrete limit studied in
soliton theory should be regarded as a space with the nonArchimedean valuations given in my previous paper
(Matsutani, S 2001 Int. J. Math. Math. Sci.). In this paper, using the natural projection from a $p$adic integer
to a ring $\mathbb{Z}/p^N \mathbb{Z}$, a soliton equation is defined over the ring. Numerical computations show that
it behaves regularly."
N. Smart and C. Woodcock,
"pAdic chaos and random numbers",
Experimental Mathematics, 7(1) (1998) 333342.
padic string and brane theories, etc.
B. Dragovich, "Nonlocal dynamics of $p$adic strings" (preprint 11/2010)
[abstract:] "We consider the construction of Lagrangians that might be suitable for describing the entire $p$adic sector of an adelic open scalar string. These Lagrangians are constructed using the Lagrangian for $p$adic strings with an arbitrary prime number $p$. They contain spacetime nonlocality because of the d'Alembertian in argument of the Riemann zeta function. We present a brief review and some new results."
B. Dragovich, "On padic sector of adelic string" (Presented at the 2nd Conference on SFT and Related Topics, Moscow, April 2009. Submitted to Theor. Math. Phys.)
[abstract:] "We consider construction of Lagrangians which are candidates for padic sector of an adelic open scalar string. Such Lagrangians have their origin in Lagrangian for a single padic string and contain the Riemann zeta function with the 'Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach which takes into account all padic Lagrangians. The very attractive feature of this new Lagrangian is that it is an analytic function of the d'Alembertian. Investigation of the field theory with Riemann zeta function is interesting in itself as
well."
B. Dragovich, "Towards effective Lagrangians for adelic strings" (preprint 02/2009)
[abstract:] "padic strings are important objects of string theory, as well as of padic mathematical physics and nonlocal cosmology. By a
concept of adelic string one can unify and simultaneously study various aspects of ordinary and padic strings. By this way, one can consider
adelic strings as a very useful instrument in the further investigation of modern string theory. It is remarkable that for some scalar padic strings
exist effective Lagrangians, which are based on real instead of padic numbers and describe not only fourpoint scattering amplitudes but also
all higher ones at the tree level. In this work, starting from padic Lagrangians, we consider some approaches to construction of effective field
Lagrangians for padic sector of adelic strings. It yields Lagrangians for nonlinear and nonlocal scalar field theory, where spacetime nonlocality
is determined by an infinite number of derivatives contained in the operatorvalued Riemann zeta function. Owing to the Riemann zeta function
in the dynamics of these scalar field theories, obtained Lagrangians are also interesting in themselves."
B. Dragovich, "Some Lagrangians with zeta function nonlocality" (preprint, 05/2008)
[abstract:] "Some nonlocal and nonpolynomial scalar field models originated from $p$adic string theory are considered. Infinite number of
spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding
Lagrangians begins with the exact Lagrangian for effective field of $p$adic tachyon string, which is generalized replacing $p$ by arbitrary natural number $n$
and then taken a sum of over all $n$. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the
equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right."
B. Dragovich, "Zeta nonlocal scalar fields" (preprint, 04/2008)
[abstract:] "We consider some nonlocal and nonpolynomial scalar field models originated from padic string theory. Infinite number
of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction
of the corresponding Lagrangians $L$ starts with the exact Lagrangian $\mathcal{L}_p$ for effective field of $p$adic tachyon string, which is
generalized replacing $p$ by arbitrary natural number $n$ and then taken a sum of $\mathcal{L}_n$ over all $n$. The corresponding new objects we call
zeta scalar strings. Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the
equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well."
B. Dragovich, "Zeta strings" (preprint 03/2007)
[abstract:] "We introduce nonlinear scalar field models for open and openclosed strings with spacetime derivatives encoded
in the operator valued Riemann zeta function. The corresponding two Lagrangians are derived in an adelic approach starting from
the exact Lagrangians for effective fields of $p$adic tachyon strings. As a result tachyons are absent in these models. These
new strings we propose to call zeta strings. Some basic classical properties of the zeta strings are obtained and presented in
this paper."
B. Dragovich, "On adelic
strings" (preprint 05/00)
[abstract:] "New approach to padic and adelic strings, which takes into account
that not only world sheet but also Minkowski spacetime and string momenta can be padic
and adelic, is formulated. pAdic and adelic string amplitudes are considered within
Feynman's path integral formalism. The adelic Veneziano amplitude is calculated. Some
discreteness of string momenta is obtained. Also, adelic coupling constant is equal
to unity."
B. Dragovich, "Adelic
strings and noncommutativity"
"We consider adelic approach to strings and spatial noncommutativity. Path integral method
to string amplitudes is emphasized. Uncertainties in spatial measurements in quantum gravity
are related to noncommutativity between coordinates. pAdic and adelic Moyal products are
introduced. In particular, padic and adelic counterparts of some real noncommutative scalar
solitons are constructed."
B. Dragovich, "Lagrangians with Riemann zeta function" (preprint 08/2008)
[abstract:] "We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is $p$adic string theory.
These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main
motivation for this research is intention to find an effective Lagrangian for adelic scalar strings."
V.S. Vladimirov and Ya.I. Volovich, "On
the nonlinear dynamical equation in the padic string theory" (preprint 06/03)
[abstract:]"In this work nonlinear pseudodifferential equations with the infinite
number of derivatives are studied. These equations form a new class of equations which
initially appeared in padic string theory. These equations are of much interest in
mathematical physics and its applications in particular in string theory and cosmology.
In the present work a systematical mathematical investigation of the properties of these
equations is performed. The main theorem of uniqueness in some algebra of tempered
distributions is proved. Boundary problems for bounded solutions are studied, the
existence of a spacehomogenous solution for odd p is proved. For even p it is proved
that there is no continuous solutions and it is pointed to the possibility of existence
of discontinuous solutions. Multidimensional equation is also considered and its soliton
and qbrane solutions are discussed."
I.Ya. Aref'eva, M.G. Ivanov and I.V. Volovich,
"Nonextremal
intersecting pbranes in various dimensions", Phys. Lett. B
406 (1997) 4448
[abstract:] "Nonextremal intersecting pbrane solutions of gravity coupled
with several antisymmetric fields and dilatons in various spacetime dimensions
are constructed. The construction uses the same algebraic method of finding
solutions as in the extremal case and a modified "noforce" conditions. We justify
the "deformation" prescription. It is shown that the nonextremal intersecting
pbrane solutions satisfy harmonic superposition rule and the intersections of
nonextremal pbranes are specified by the same characteristic equations for the
incidence matrices as for the extremal pbranes. We show that Sduality holds for
nonextremal pbrane solutions. Generalized Tduality takes place under additional
restrictions to the parameters of the theory which are the same as in the extremal
case."
I.Ya.Arefeva, K.S.Viswanathan, A.I.Volovich and I.V.Volovich,
"Composite
pbranes in various dimensions", Nucl. Phys. Proc. Suppl.
56B (1997) 5260
[abstract:] "We review an algebraic method of finding the composite pbrane
solutions for a generic Lagrangian, in arbitrary spacetime dimension, describing an
interaction of a graviton, a dilaton and one or two antisymmetric tensors. We set the
FockDe Donder harmonic gauge for the metric and the "noforce" condition for the
matter fields. Then equations for the antisymmetric field are reduced to the Laplace
equation and the equation of motion for the dilaton and the Einstein equations for
the metric are reduced to an algebraic equation. Solutions composed of n constituent
pbranes with n independent harmonic functions are given. The form of the solutions
demonstrates the harmonic functions superposition rule in diverse dimensions. Relations
with known solutions in D = 10 and D = 11 dimensions are discussed."
I.Ya. Aref'eva, K.S. Viswanathan and I.V. Volovich,
"pBrane solutions in diverse
dimensions", Phys.Rev. D55 (1997) 47484755
[abstract:] "A generic Lagrangian, in arbitrary spacetime dimension, describing the
interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An
isotropic pbrane solution consisting of three blocks and depending on four parameters in
the Lagrangian and two arbitrary harmonic functions is obtained. For specific values of
parameters in the Lagrangian the solution may be identified with previously known
superstring solutions."
I.Ya.Arefeva, K.S.Viswanathan, A.I.Volovich and I.V.Volovich,
"Composite pbranes
in diverse dimensions", Class. Quant. Grav. 14 (1997) 29913000
[abstract:] "We use a simple algebraic method to find a special class of composite
pbrane solutions of higher dimensional gravity coupled with matter. These solutions are
composed of n constituent pbranes corresponding n independent harmonic functions. A
simple algebraic criteria of existence of such solutions is presented. Relations with
D = 11, 10 known solutions are discussed."
I.V. Volovich, "pAdic string", Classical Quantum
Gravity 4 (1987) 8387
I.V. Volovich, "From padic strings to étale strings", Proc. Steklov Inst. Math. 203 (1995) no. 3, 37–42.
A. Volovich,
"Threeblock pbranes in various
dimensions", Nucl. Phys. B492 (1997) 235248
[abstract:] "It is shown that a Lagrangian, describing the interaction of the
gravitation field with the dilaton and the antisymmetric tensor in arbitrary dimension
spacetime, admits an isotropic pbrane solution consisting of three blocks. Relations
with known pbrane solutions are discussed. In particular, in tendimensional spacetime
the threeblock pbrane solution is reduced to the known solution, which recently has
been used in the Dbrane derivation of the black hole entropy."
C. Castro, "Hints of a new
relativity principle from pbranes quantum mechanics", Journal
of Chaos, Solitons and Fractals 11 (2000) 1721
C. Castro and A. Granik,
"ploops harmonic
oscillators in Cspaces and the explicit derivation of the black
hole entropy"
C. Castro and J. Mahecha,
"Comments on the Riemann
conjecture and index theory on Cantorian fractal spacetime"
C. Castro (Perelman),
"pAdic stochastic
dynamics, supersymmetry and the Riemann conjecture"
"Supersymmetry, padic stochastic dynamics, Brownian motion,
FokkerPlanck equation, Langevin equation, prime number random distribution,
random matrices, padic fractal strings, the adelic condition, etc...are
all deeply interconnected in this paper."
C. Castro, "Fractal
strings as the basis of CantorianFractal spacetime and the fine structure
constant"
[abstract:] "Beginning with the most general fractal strings/sprays construction
recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the
complexified extension of El Naschie's CantorianFractal spacetime model belongs to
a very special class of families of fractal strings/sprays whose scaling ratios are
given by suitable pinary (pinary, p prime) powers of the Golden Mean. We then proceed
to show why the logarithmic periodicity laws in Nature are direct physical consequences
of the complex dimensions associated with these fractal strings/sprays. We proceed with
a discussion on quasicrystals with padic internal symmetries, von Neumann's Continuous
Geometry, the role of wild topology in fractal strings/sprays, the BanachTarski paradox,
tesselations of the hyperbolic plane, quark confinement and the Mersenneprime hierarchy
of bitstring physics in determining the fundamental physical constants in Nature."
P.H. Frampton and Y. Okada, "pAdic
string Npoint function", Phys. Rev. Lett. B 60 (1988) 484486
J. Minahan,
"Mode interactions of the tachyon condensate in padic string
theory"
"We study the fluctuation modes for lump solutions of the tachyon
effective potential in padic open string theory. We find a
discrete spectrum with equally spaced mass squared levels. We also
find that the interactions derived from this field theory are
consistent with padic string amplitudes for excited string
states."
A. Sen,
"Tachyon condensation and brane descent relations in padic
string theory"
"It has been conjectured that an extremum of the tachyon potential of
a bosonic Dbrane represents the vacuum without any Dbrane, and that
various tachyonic lump solutions represent Dbranes of lower dimension.
We show that the tree level effective action of padic string
theory, the expression for which is known exactly, provides an
explicit realisation of these conjectures."
H. Furusho, "padic
multiple zeta values I  padic multiple polylogarithms and the padic KZ
equation"
[abstract:] "Our main aim in this paper is to give a foundation of the theory of
padic multiple zeta values. We introduce (one variable) padic multiple
polylogarithms by Coleman's padic iterated integration theory. We define
padic multiple zeta values to be special values of padic multiple
polylogarithms. We consider the padic KZ equation and introduce the padic
Drinfel'd associator by using certain two fundamental solutions of the padic KZ
equation. We show that our padic multiple polylogarithms appear on coefficients of
a certain fundamental solution of the padic KZ equation and our padic
multiple zeta values appear on coefficients of the padic Drinfel'd associator. We
show various properties of padic multiple zeta values, which are sometimes
analogous to the complex case and are sometimes peculiar to the padic case, via
the padic KZ equation."
M. Nardelli, "On the possible mathematical connections between
the HartleHawking no boundary proposal concerning the RandallSundrum cosmological scenario, HartleHawking
wavefunction in the minisuperspace of physical superstring theory, padic HartleHawkind wave function
and some sectors of number theory" (preprint, 2007)
M. Nardelli, "On the possible mathematical connections concerning
the relation between threedimensional gravity related to ChernSimons gauge theory, padic HartleHawking
wave function, Ramanujan's modular functions and some equations describign the Riemann zetafunction" (preprint, 2007)
M. Nardelli, "On the link between the structure of Abranes
observed in homological mirror symmetry and the classical theory of automorphic forms. Mathematical
connections with the modular elliptic curves, padic and adelic numbers and padic and adelic strings"
(preprint 03/2008)
[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the
categories of Abranes on the dual Hitchin fibers and some interesting phenomena associated with the
endoscopy in the geometric Langlands correspondence of various authoritative theoretical physicists and
mathematicians."
M. Nardelli, "On some mathematical connections
concerning the threedimensional pure quantum gravity with negative cosmological constant,
the Selberg zetafunction, the tendimensional anomaly cancellations, the vanishing of
cosmological constant, and some sectors of string theory and number theory" (preprint 06/2008)
[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the
quantum gravity partition functions in threedimensions, in the Selberg zeta function, in the vanishing of cosmological
constant and in the tendimensional anomaly cancellations. In the Section 1, we have described some equations
concerning the pure threedimensional quantum gravity with a negative cosmological constant and the pure
threedimensional supergravity partition functions. In the Section 2, we have described some equations concerning the
Selberg supertrace formula for SuperRiemann surfaces, some analytic properties of Selberg super zetafunctions
and multiloop contributions for the fermionic strings. In the Section 3, we have described some equations concerning
the tendimensional anomaly cancellations and the vanishing of cosmological constant. In the Section 4, we have
described some equations concerning padic strings, padic and adelic zeta functions and zeta strings. In conclusion, in
the Section 5, we have described the possible and very interesting mathematical connections obtained between some
equations regarding the various sections and some sectors of number t heory (Riemann zeta functions, Ramanujan
modular equations, etc...) and some interesting mathematical applications concerning the Selberg superzeta functions
and some equations regarding the Section 1."
M. Nardelli, "On the physical interpretation of the Riemann zeta function, the Rigid Surface Operators in Gauge Theory, the adeles and
ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula, padic strings, zeta strings and
padic cosmology and mathematical connections with some sectors of String Theory and Number Theory" (preprint 10/2008)
[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the physical interpretation
of the Riemann zeta function as a FZZT Brane Partition Function associated with a matrix/gravity correspondence and some aspects
of the Rigid Surface Operators in Gauge Theory. Furthermore, we describe the mathematical connections with some sectors of String
Theory (padic and adelic strings, padic cosmology) and Number Theory.
In the Section 1 we have described various mathematical aspects of the Riemann Hypothesis, matrix/gravity correspondence and
master matrix for FZZT brane partition functions. In the Section 2, we have described some mathematical aspects of the rigid surface
operators in gauge theory and some mathematical connections with various sectors of Number Theory, principally with the
Ramanujan's modular equations (thence, prime numbers, prime natural numbers, Fibonacci's numbers, partitions of numbers,
Euler's functions, etc...) and various numbers and equations related to the Lie Groups. In the Section 3, we have described some
very recent mathematical results concerning the adeles and ideles groups applied to various formulae regarding the Riemann zeta
function and the Selberg trace formula (connected with the Selberg zeta function), hence, we have obtained some new connections
applying these results to the adelic strings and zeta strings. In the Section 4 we have described some equations concerning padic
strings, padic and adelic zeta functions, zeta strings and padic cosmology (with regard the padic cosmology, some equations
concerning a general class of cosmological models driven by a nonlocal scalar field inspired by string field theories). In conclusion,
in the Section 5, we have showed various and interesting mathematical connections between some equations concerning the
Section 1, 3 and 4."
M. Nardelli, "On the mathematical connections between some equations concerning
the calculation of all the eigenfunctions of atoms with the ThomasFermi method, some sectors of Number Theory, the modes
corresponding to the physical vibrations of superstrings, padic and Adelic free relativistic particle and padic strings"
(preprint 12/2008)
[abstract:] "According to quantum mechanics, the properties of an atom can be calculated easily if we known the eigenfunctions
and eigenvalues of quantum states in which the atom can be found. The eigenfunctions depend, in general, by the coordinates of all
the electrons. However, a diagram effective and enough in many cases, we can get considering the individual eigenfunctions for
individual electrons, imagining that each of them is isolated in an appropriate potential field that represent the action of the
nucleus and of other electrons. From these individual eigenfunctions we can to obtain the eigenfunction of the quantum state of
the atom, forming the antisymmetrical products of eigenfunctions of the individual quantum states involved in the configuration
considered. The problem, with this diagram, is the calculation of the eigenfunctions and eigenvalues of individual electrons of
each atomic species. To solve this problem we must find solutions to the Schroedinger's equation where explicitly there is the
potential acting on the electron in question, due to the action of the nucleus and of all the other electrons of the atom. To
research of potential it is possible proceed with varying degrees of approximation: a first degree is obtained by the statistical
method of ThomasFermi in which electrons are considered as a degenerate gas in balance as a result of nuclear attraction. This
method has the advantage of a great simplicity as that, through a single function numerically calculated once and for all, it is
possible to represent the behaviour of all atoms. In this work (Sections 1 and 2) we give the preference to the statistical method,
because in any case it provides the basis for more approximate numerical calculations. Furthermore, we describe the mathematical
connections that we have obtained between certain solutions concerning the calculation of any eigenfunctions of atoms with this
method, the Aurea ratio, the Fibonacci's numbers, the Ramanujan modular equations, the modes corresponding to the physical vibrations
of strings, the padic and Adelic free relativistic particle and padic and adelic strings (Sections 3 and 4)."
M. Nardelli, "On some mathematical connections between the cyclic universe, inflationary universe,
padic Inflation, padic cosmology and various sectors of number theory" (preprint 02/2009)
[abstract:] "This paper is a review, a thesis, of some interesting results that has been obtained in
various researches concerning the "brane collisions in string and Mtheory" (Cyclic Universe), padic
inflation and padic cosmology.
In Section 1 we have described some equations concerning cosmic evolution in a Cyclic Universe. In
the Section 2, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang
spacetime, the Mtheory model of a Big Crunch/Big Bang transition and some equations concerning the solution of
a braneworld Big Crunch/Big Bang Cosmology. In the Section 3, we have described some equations concerning the
generating Ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective
fivedimensional theory of the strongly coupled heterotic string as a gauged version of $N = 1$ five dimensional
supergravity with fourdimensional boundaries, and some equations concerning the colliding branes and the
origin of the Hot Big Bang. In the Section 4, we have described some equations regarding the "null energy
condition" violation concerning the inflationary models and some equations concerning the evolution to a
smooth universe in an ekpyrotic contracting phase with $w > 1$. In the Section 5, we have described some
equations concerning the approximateinflationary solutions rolling away from the unstable maximum of padic
string theory. In the Section 6, we have described various equations concerning the padic minisuperspace model,
zeta strings, zeta nonlocal scalar fields and padic and adelic quantum cosmology. In the Section 7, we have
showed various and interesting mathematical connections between some equations concerning the padic Inflation,
the padic quantum cosmology, the zeta strings and the brane collisions in string and Mtheory. Furthermore, in
each section, we have showed the mathematical connections with various sectors of number theory, principally the
Ramanujan's modular equations, the Aurea Ratio and the Fibonacci numbers."
M. Nardelli, "On the possible applications of some theorems concerning the Number Theory to the various mathematical aspects and sectors of String Theory I" (preprint 04/2009)
[abstract:] "The aim of this paper is that of show the further and possible connections between the padic and adelic strings and Lagrangians with Riemann zeta function with some problems, equations and theorems in number theory.
In Section 1, we have described some equations and theorems concerning the quadrature and meanconvergence in the Lagrange interpolation. In Section 2, we have described some equations and theorems concerning the difference sets of sequences of integers. In Section 3, we have showed some equations and theorems regarding some problems of a statistical group theory (symmetric groups) and in Section 4, we have showed some equations and theorems concerning the measure of the nonmonotonicity of the Euler phi function and the related Riemann zeta function.
In Section 5, we have showed some equations concerning the padic and adelic strings, the zeta strings and the Lagrangians for adelic strings
In conclusion, in Section 6, we have described the mathematical connections concerning the various sections previously analyzed. Indeed, in the Section 1, 2 and 3, where are described also various theorems on the prime numbers, we have obtained some mathematical connections with Ramanujan's modular equations, thence with the modes corresponding to the physical vibrations of the bosonic and supersymmetric strings and also with padic and adelic strings. Principally, in Section 3, where is frequently used the HardyRamanujan stronger asymptotic formula and are described some theorems concerning the prime numbers. With regard Section 4, we have obtained some mathematical connections between some equations concerning the Euler phi function, the related Riemann zeta function and the zeta strings and field Lagrangians for padic sector of adelic string (Section 5). Furthermore, in Sections 1, 2, 3 and 4, we have described also various mathematical expressions regarding some frequency connected with the exponents of the Aurea ratio, i.e. with the exponents of the number phi = 1.61803399. We consider important remember that the number 7 of the various exponents is related to the compactified dimensions of Mtheory."
M. Nardelli, "On the Boltzmann equation applied in various sectors of string theory and the black hole entropy in canonical quantum gravity and superstring theory" (preprint 09/2009)
[abstract:] "In this paper we have showed the various applications of the Boltzmann equation in string theory and related topics. In Section 1, we have described some equations concerning the time dependent multiterm solution of Boltzmann's equation for charged particles in gases under the influence of electric and magnetic fields, the Planck's blackbody radiation law, the Boltzmann's thermodynamic derivation and the connections with the superstring theory. In Section 2, we have described some equations concerning the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipativesource and noncriticalstring terms in dilatondriven nonequilibrium string cosmologies. In Section 3, we have described some equations concerning the entropy of an eternal Schwarzschild black hole in the limit of infinite black hole mass, from the point of view of both canonical quantum gravity and superstring theory. We have described some equations regarding the quantum corrections to black hole entropy in string theory. Furthermore, in this section, we have described some equations concerning the thesis "Can the Universe create itself?" and the adapted Rindler vacuum in Misner space. In Section 4, we have described some equations concerning pAdic models in HartleHawking proposal and pAdic and Adelic wave functions of the Universe. Furthermore, we have described in the various sections the various possible mathematical connections that we've obtained with some sectors of number theory and, in the Section 5, we have showed some mathematical connections between some equations of arguments above described and padic and adelic cosmology."
M. Nardelli, "On some mathematical equations concerning the functions $\zeta(s)$ and $\zeta(s,w)$ and some Ramanujantype series for $1/\pi$. Mathematical connections with some equations concerning the $p$adic open string for the scalar tachyon field and the zeta strings" (preprint 10/2010)
[abstract:] "In this paper, in Section 1, we have described some equations concerning the functions $\zeta(s)$ and $zeta(s,w)$. In this Section, we have described also some equations concerning a transformation formula involving the gamma and Riemann zeta functions of Ramanujan. Furthermore, we have described also some mathematical connections with various theorems concerning the incomplete elliptic integrals described in "Ramanujan's lost notebook". In Section 2, we have described some Ramanujantype series for $1/\pi$ and some equations concerning the $p$adic open string for the scalar tachyon field. In this section, we have described also some possible and interesting mathematical connections with some Ramanujan's Theorems, contained in the first letter of Ramanujan to G.H. Hardy. In Section 3, we have described some equations concerning the zeta strings and the zeta nonlocal scalar fields. In conclusion, in Section 4, we have showed some possible mathematical connections between the arguments above mentioned, the PalumboNardelli model and the Ramanujan's modular equations that are related to the physical vibrations of bosonic strings and of superstrings."
padic and adelic approaches to quantum mechanics
G. Djordjevic, B. Dragovich and L. Nesic, "pAdic
and adelic free relativistic particle", Modern Physics Letters A, 14, no. 5 (1999) 317325.
"We consider spectral problem for a free relativistic particle in
padic and adelic quantum mechanics. In particular, we found
padic and adelic eigenfunctions. Within adelic approach there exist
quantum states that exhibit discrete structure of spacetime at the Planck
scale."
G. Djordjevic, B. Dragovich and Lj. Nesic, "Adelic
quantum mechanics: Nonarchimedean and noncommutative aspects"
"We present a short review of adelic quantum mechanics pointing out its nonArchimedean and
noncommutative aspects. In particular, padic path integral and adelic quantum cosmology are
considered. Some similarities between padic analysis and qanalysis are noted.
The padic Moyal product is introduced."
G. Djordjevic, B. Dragovich and Lj. Nesic, "Adelic
path integrals for quadratic lagrangians"
"Feynman's path integral in adelic quantum mechanics is considered. The propagator
K(x'',t'';x',t') for onedimensional adelic systems with quadratic Lagrangians is analytically
evaluated. Obtained exact general formula has the form which is invariant under interchange of
the number fields R and Q_p."
G. Djordjevic, B. Dragovich and Lj. Nesic, "pAdic
and adelic minisuperspace quantum cosmology", International Journal of Modern Physics A 17 (2002) 14131434.
"We consider the formulation and some elaboration of padic and adelic quantum cosmology.
The adelic generalization of the HartleHawking proposal does not work in models with matter
fields. pAdic and adelic minisuperspace quantum cosmology is well defined as an ordinary
application of padic and adelic quantum mechanics. It is illustrated by a few minisuperspace
cosmological models in one, two and three minisuperspace dimensions. As a result of padic
quantum effects and the adelic approach, these models exhibit some discreteness of the
minisuperspace and cosmological constant. In particular, discreteness of the de Sitter space
and its cosmological constant is emphasized. "
B. Dragovich and Lj. Nesic, "pAdic
and adelic generalization of quantum cosmology", Grav. Cosmol. 5 (1999) 222.
"pAdic and adelic generalization of ordinary quantum cosmology is considered. In [1], we
have calculated padic wave functions for some minisuperspace cosmological models according to
the "noboundary" HartleHawking proposal. In this article, applying padic and adelic quantum
mechanics, we show existence of the corresponding vacuum eigenstates. Adelic wave function
contains some information on discrete structure of spacetime at the Planck scale."
G. Djordjevic and B. Dragovich, "pAdic
and adelic harmonic oscillator with timedependent frequency", Theor. Math. Phys. 124 (2000) 10591067.
"The classical and quantum formalism for a padic and adelic harmonic oscillator with
timedependent frequency is developed, and general formulae for main theoretical quantities are
obtained. In particular, the padic propagator is calculated, and the existence of a simple
vacuum state as well as adelic quantum dynamics is shown. Space discreteness and padic
quantummechanical phase are noted."
G. Djordjevic and B. Dragovich, "pAdic
path integrals for quadratic actions", Modern Physics Letters 12 (1997) 14551463.
"The Feynman path integral in padic quantum mechanics is considered. The probability
amplitude ${\cal K}_p (x^{\prime\prime},t^{\prime\prime}; x^\prime,t^\prime)$
for onedimensional systems with quadratic actions is calculated in an exact form, which is the
same as that in ordinary quantum mechanics."
G. Djordjevic and B. Dragovich, "On padic
functional integration"
"pAdic generalization of the Feynman path integrals in quantum mechanics is considered. The
probability amplitude for a particle in a constant field is calculated. Path integrals over
padic space have the same form as those over R."
G. Djordjevic, "On padic
path integral"
"Feynman's path integral is generalized to quantum mechanics on padic space and time. Such
padic path integral is analytically evaluated for quadratic Lagrangians. Obtained result has
the same form as that one in ordinary quantum mechanics."
B. Dragovich, "Adelic Model of Harmonic
Oscillator", Theoretical and Mathematical Physics 101 (1994)
B. Dragovich, "Adelic harmonic
oscillator", Int. J. Mod. Phys. A 10 (1995) 23492365
[abstract:] "Using the Weyl quantization we formulate onedimensional adelic quantum
mechanics, which unifies and treats ordinary and padic quantum mechanics on an equal
footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple,
exact and instructive adelic model. Eigenstates are SchwartzBruhat functions. The Mellin
transform of a simplest vacuum state leads to the well known functional relation for the Riemann
zeta function. Some expectation values are calculated. The existence of adelic matter at very
high energies is suggested."
B. Dragovich, "On generalized functions in
adelic quantum mechanics", Integral Transforms and Special Functions 6 (1998) 197203.
[abstract:] "Some aspects of adelic generalized functions, as linear continuous functionals on the space of
SchwartzBruhat functions, are considered. The importance of adelic generalized functions in adelic quantum
mechanics is demonstrated. In particular, adelic product formula for Gauss integrals is derived, and the
connection between the functional relation for the Riemann zeta function and quantum states of the harmonic
oscillator is stated."
Numerous other published articles by Dragovich on padic
analysis and physics
V. Varadarajan,
"Path integrals for
a class of padic Schrodinger equations"
V. Varadarajan, "Some remarks on arithmetic physics"
D. Bump, KwokKwong Choi,
P. Kurlberg, and J. Vaaler,
"A Local Riemann Hypothesis, I",
Mathematische Zeitschrift 233 (1) (2000), 118.
(A subscription to Mathematische Zeitschrift is required if you wish
to download this.)
"[This paper describes] how local Tate
integrals formed with eigenfunctions of the quantum mechanical harmonic
oscillator, and its padic analogs, have their zeros on the line
Re(s) = 1/2. This...incorporates new material on the harmonic oscillator in n dimensions, Mellin transforms
of the Laguerre functions, and a reciprocity law for their values at negative integers."
S. Albeverio, R. Cianci, N. De GrandeDe Kimpe, A. Khrennikov, "pAdic
probability and an interpretation of negative probabilities in quantum mechanics",
Russian J. Math. Phys. 6 (1999) 319.
B. Grossman, "The
adelic components of the index of the Dirac operator", Journal
of Physics A 21 (1988) L10511054.
[Abstract:] "The authors demonstrate that the index of the Dirac
operator can be expressed as an adelic product over the primes"
P. Cohen "Dedekind zeta functions and quantum statistical
mechanics"
[excerpt:] "The interaction detected in the phase transition comes
about from the interaction between primes coming from considering at
once all the embeddings of the nonzero rational numbers Q* into the
completions Q_{p} of Q with respect to the prime
valuations ._{p}."
C. Castro, "Hints of a new
relativity principle from pbranes quantum mechanics", Journal
of Chaos, Solitons and Fractals 11 (2000) 1721
padic probability, statistics and randomness
A.N. Kochubei, Pseudodifferential
Equations and Stochastics over nonArchimedian Fields
(Marcel Dekker, 2001)
(from publisher's description) "This stateoftheart reference
provides comprehensive coverage of the most recent developments in the
theory of nonArchimedean pseudodifferential equations and its
application to stochastics and mathematical physics offering current
methods of construction for stochastic processes on the field of
padic numbers and related structures. Develops a new theory for
parabolic equations over nonArchimedean fields in relation to
Markov processes!"
S. Albeverio and W. Karwowski,
"A Random Walk on pAdics  the Generator and Its Spectrum",
Stochastic Processes and their Applications 53 (1994) 122.
W. Karwowski and R. Vilela Mendes, "Hierarchical Structures and Asymmetric
Stochastic Processes on pAdics and Adeles", Journal of
Mathematical Physics 35 (1994) 46374650.
S. Albeverio, R. Cianci, N. De GrandeDe Kimpe, A. Khrennikov, "pAdic
probability and an interpretation of negative probabilities in quantum mechanics",
Russian J. Math. Phys. 6 (1999) 319.
A. Khrennikov, "Limit
behaviour of sums of independent random variables with respect to the uniform padic
distribution", Statistics and Probability Letters 51 (2001) 269276
[abstract:] "We investigate (as usual) limit behaviour of sums $S_{n}(\omega)$ of independent
equally distributed random variables. However, limits of probabilities are studied with respect
to a padic metric (where p is a prime number). We found that (despite of rather
unusual features of a padic metric) limits of classical probabilities exist in a field
of padic numbers. These probabilities are rational numbers (which can be calculated by
using simple combinatorial considerations). Limit theorems are related to divisibility of sums
$S_{n}(\omega) by p. In fact, limits depend on choices of subsequences ${S_{n_k}(\omega)}. We
obtain two limit theorems which describe all possible limit behaviours. All considerations are
based on one special padic probability distribution, namely the uniform distribution."
N. Smart and C. Woodcock,
"pAdic chaos and random numbers",
Experimental Mathematics, 7(1) (1998) 333342.
J.L. Lucio and Y. Meurice, "Asymptotic properties of random walks
on padic spaces", University of Iowa Preprint 9033, 15 (1990)
C. Castro (Perelman),
"pAdic stochastic
dynamics, supersymmetry and the Riemann conjecture"
"Supersymmetry, padic stochastic dynamics, Brownian motion,
FokkerPlanck equation, Langevin equation, prime number random distribution,
random matrices, padic fractal strings, the adelic condition, etc...are
all deeply interconnected in this paper."
padic and adelic scattering theory
J.F. Burnol,
"Scattering on the padic field and a trace formula"
[abstract] "I apply the setup of LaxPhillips Scattering Theory to a nonarchimedean local
field. It is possible to choose the outgoing space and the incoming space to be Fourier
transforms of each other. Key elements of the LaxPhillips theory are seen to make sense and to
have the expected interrelations: the scattering matrix S, the projection K to the interacting
space, the contraction semigroup Z and the time delay operator T. The scattering matrix is
causal, its analytic continuation has the expected poles and zeros, and its phase derivative is
the (nonnegative) spectral function of T, which is also the restriction to the diagonal of the
kernel of K. The contraction semigroup Z is related to S (and T) through a trace formula.
Introducing an oddeven grading on the interacting space allows to express the Weil local
explicit formula in terms of a "supertrace". I also apply my methods to the evaluation of a
trace considered by Connes."
P.G.O. Freund, "Scattering
on padic and adelic symmetric spaces", Physics Letters B 257 (1991)
119124
[abstract:] "Explicit Smatrices are constructed for scattering on padic hyperbolic
planes. Combining these with the known Smatrix on the real hyperbolic plane, an adelic
Smatrix is obtained. It has poles at the nontrivial zeros of the Riemann zetafunction, and is
closely related to scattering on the modular domain of the real hyperbolic plane.
Generalizations of this work and their possible arithmetic relevance are outlined."
M. Pigli, "Adelic
integrable systems" (preprint 07/95)
[abstract:] "Incorporating the zonal spherical function (zsf) problems on
real and padic hyperbolic planes into a ZakharovShabat integrable system
setting, we find a wide class of integrable evolutions which respect the numbertheoretic
properties of the zsf problem. This means that at all times these real and
padic systems can be unified into an adelic system with an Smatrix which
involves (Dirichlet, Langlands, Shimura...) Lfunctions."
[introduction:] "Scattering theory on real [1] and padic [2] symmetric spaces can be
unified in an adelic context. This had the virtue of producing Smatrices
involving the Riemann zeta function and of throwing new light on earlier work [4]
concerning scattering on the noncompact finitearea fundamental domain of
SL(2,Z) on the Real hyperbolic plane $H_{\infty}$.
The real hyperbolic plane is a smooth manifold and as such quantum
mechanics on $H_{\infty}$ involves a second order Schrodinger differential equation. By contrast the padic hyperbolic planes H_{p} are discrete spaces
(trees), and the corresponding Schrodinger equations are second order difference equations. The Jost functions, and therefore the Smatrices from all these local problems combine in
adelic products, which then involve the Riemann zeta function [2].
At a given time consider all these ("Swave") scattering
problems and then let all of them undergo an integrable
time evolution. In general such an evolution need not respect
the numbertheoretic endowment of the initial problem. In other words,
even though at the initial time the real and padic scattering
problems assembled into an interesting adelic scattering problem, at later
times this need no longer be so. We want to explore here the conditions
under which the integrable evolution respects adelizability and to see
what kind of scattering problems can be obtained this way at later times.
Specifically, we will incorporate the initial scattering problem into a
ZakharovShabat (ZS) system and follow its integrable evolution. For the
padic problems, time has to be discrete and for adelic purposes time then has to be discrete in the real problem as well. We will see that
along with the Riemann zeta function involved in the adelic problem at the
initial time, various (Dirichlet, Langlands, Shimura,...)
Lfunctions [5] appear at later times."
[1] M.A. Olshanetsky and A.M. Perelomov, Phys. Rep.
94 (1984) 313; R.F. Wehrhahn, Rev. Math. Phys.
6 (1994) 1339
[2] P.G.O. Freund, Phys. Lett. B 257 (1919) 119; L. Brekke and P.G.O. Freund, Phys. Rep. 233 (1993) 1
[3] L.O. Chekhov, "Lfunctions
in scattering on padic multiloop surfaces", J. Math. Phys. 36 (1995) 414
[4] L.D. Faddeev and B.S. Pavlov, Sem. LOMI 27 (1972) 161; P.D.
Lax and R.S. Phillips, Scattering Theory for Automorphic Functions (Princeton Univ. Press, 1976)
[5] S. Gelbart and F. Shahidi, Analytic Properties of
Automorphic LFunctions (Academic, 1988)
padic 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)
12311235.;
(Abstract) "We reduce the Riemann hypothesis for Lfunctions 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
k."
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 (PerronFrobenius) operator of a classical transformation.
Such classical operators formally resemble quantum Hamiltonians, but
usually have complicated nondiscrete 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 padic 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.
S. Haran,
Mysteries
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, qseries, Elliptic
curves, the Heisenberg group, quantum groups, and special functions (such as the Gamma,
Beta, Zeta, theta, Bessel functions, the AskeyWilson and the classical orthagonal
polynomials) The text discusses real numbers from a padic 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 padic 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
padic valuations) on the rationals  the one which completes Q to give R
rather than some Q_{p}. An adele is an object of the form
(a_{oo};a_{2},a_{3},a_{5},a_{7},
a_{11},a_{13},...).The first term is a real number, corresponds to
'the real prime' or the prime at infinity. The other a_{p} are padic
numbers  they correspond to the 'finite' primes p. Adeles are central to Connes'
approach to the Riemann Hypothesis (his allimportant trace formula is defined on a noncommutative space
of Adele classes).
R. Pearson, "Number theory and critical exponents",
Phys. Rev. B 22 (1980) 34653470
[abstract:] "The consequences of assuming padic 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 Qstate 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. Phaselocking
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, padic
and adelic noncommutative scalar solitons" (preprint 06/03)
[abstract:] "We considered real, padic and adelic noncommutative scalar solitons and obtained
some new results."
M.V. Altaisky and B.G. Sidharth, "pAdic
physics below and above Planck scales"
[abstract:] "We present a rewiew and also new possible applications of padic
numbers to prespacetime physics. It is shown that instead of the extension
R^{n} to Q_{p}^{n}, which is usually implied in padic quantum field theory, it is possible
to build a model based on the R^{n} to Q_{p},
where p = n + 2 extension and get rid of loop
divergences. It is also shown that the concept of mass naturally arises in padic
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 nonGaussian 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 infinitedimensional 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 selfcontained 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, zetafunction and fine structure constant are also considered."
S. Kozyrev, "Wavelets
analysis as padic harmonic analysis"
[abstract:] "New orthonormal basis of eigenfunctions for the Vladimirov operator of
padic fractional derivation is constructed. The map of padic numbers
onto real numbers (padic 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 L^{2}(R) generated from the Haar
wavelet. This means that the wavelet analysis can be considered as a padic
spectral analysis."
C. Consani
and M. Marcolli, "Noncommutative geometry,
dynamics, and infinityadic Arakelov geometry" (to appear in
Selecta Mathematica)
[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) 779784
[abstract:] "This note is a brief overview of the results of
math.AG/0205306.
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
monodromy."
B. Dragovich and A. Dragovich, "pAdic modelling of the genome and the genetic code"
(preprint 07/2007)
[abstract:] "The present paper is devoted to foundations of padic modelling in genomics. Considering nucleotides, codons, DNA and RNA sequences,
amino acids, and proteins as information systems, we have formulated the corresponding padic 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 padic 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 padic 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 padic stabilization", Dokl. Akad. Nauk 334,
no.1 (1994) 58.
A. Khrennikov, "padic model for population growth", from
Fractals in Biology and Medicine, 2, Eds. G.A. Losa, et. al.
(Birkhauser, 1998).
A. Khrennikov,
"Learning of padic neural
networks" (preprint, 1999)
[abstract:] "A padic 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 padic number. Thus each padic
number represents a configuration of firing and nonfiring neurons. We present the
algorithm of learning for padic neural networks based on the minimization
of the errorfunctional (here we use a random search procedure in the space of
padic weights). This algorithm (or its more advanced versions) could be
applied for image recognition."
numerous articles published by A. Khrennikov
relating padic analysis to various branches of physics, biology,
neural networks, etc.
The Third International Conference on padic Mathematical Physics:
From Planck scale physics to complex systems to biology, Steklov Mathematical Institute
Moscow, Russia, October 16, 2007
"padic 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 padic
mathematical physics, related fields, and applications, as well as to
discuss earlier results and possible future directions of investigation."
Topological
geometrodynamics and padic numbers
An astonishingly comprehensive workinprogress 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 "padicization 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 padic 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 padic
particle physicist's articulation of what it is to be a zero of the Riemann
zeta function"
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 padic
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 HilbertPólya 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 padic context. Hence Riemann
hypothesis would indeed reduce to superconformal invariance."
"Riemann
hypothesis and superconformal invariance"
notes on padic 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
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