p-adic and adelic physics

introduction to p-adic numbers and adeles

p-adic and adelic physics: introductory remarks and resources

p-adic dynamics

p-adic string and brane theories, etc.

p-adic and adelic approaches to quantum mechanics

p-adic probability, statistics and randomness

p-adic and adelic scattering theory

p-adic physics - general

New journal: p-Adic 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 p-adic, 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, non-Archimedean and non-commutative 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 p-adics@mi.ras.ru.

front-matter 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 p-adic number Qp, 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 Qp, 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 Qp. 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 Qp 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 p-adic; 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 non-Archimediean 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) 293-303

V. Varadarajan, from "Some remarks on arithmetic physics"

L. Brekke and P. Freund, "p-adic numbers in physics", Physics Reports 233, (1993) 1-66.

This is a review article related to the achievements in application of p-adic numbers to string theory, quantum field theory and quantum mechanics during the period 1987-1992. The contribution of Freund and his collaborators is emphasised.

Here is an excerpt from pp61-62:

"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 p-adic 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 "even-handed" 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 p-adic 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 p-adics-quantum group connection, which places the ordinary and all the p-adic 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, "Non-Archimedean geometry and physics on adelic spaces" (preprint 06/03)

[abstract:] "This is a brief review article of various applications of non-Archimedean geometry, p-adic numbers and adeles in modern mathematical physics."

V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific Publishing)

I.V. Volovich, "Number theory as the ultimate physical theory", p-Adic Numbers, Ultrametric Analysis and Applications 2 (2010) 77–87

(Abstract) "At the Planck scale doubt is cast on the usual notion of space-time 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 non-Archimedean p-adic or finite geometry."

A. Khrennikov, p-Adic Valued Distributions in Mathematical Physics, (Kluwer, 1994).

R. Rammal, G. Toulouse, and M.A. Virasoro, "Ultrametricity for physicists", Rev. Modern. Physics. 58 (1986) 765-788

p-adic dynamics

M. Nevins and D. Rogers, "Quadratic maps as dynamical systems on the p-adic numbers"

[abstract:] "We describe the trajectories of the successive iterates of the square map and its perturbations on the field of p-adic numbers. We show that the cycles of the square map on Qp arise from cycles of the square map on Fp, and that all nonperiodic trajectories in the unit disk densely define a compact open subset. We find that the perturbed maps x |-> x2 + a, with a inside the unit circle, have similar dynamics to x |-> x2, but that each fundamental cycle arising from Fp can further admit harmonic cycles, for different choices of p and a. In contrast, the cycles of the maps x |-> x2 + 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, "p-Adic dynamics and Sullivan's no wandering domain theorem", Compositio Mathematica 122 (2000) 281-298

[abstract:] "In this paper we study dynamics on the Fatou set of a rational function f(z) defined over a finite extension Qp, the field of p-adic rationals. Using a notion of 'components' of the Fatou set defined in "Hyperbolic maps in p-adic dynamics", we state and prove an analogue of Sullivan's No Wandering Domains Theorem for p-adic rational functions which have no wild recurrent Julia critical points."

R. Benedetto, "Examples of wandering domains in p-adic polynomial dynamics" (Comptes Rendus Mathématique. Académie des Sciences. Paris, 335 (2002), 615--620.

[abstract:] "For any prime p > 0, we construct p-adic polynomial functions in Cp[z] whose Fatou sets have wandering domains."

R. Benedetto, "Non-archimedean holomorphic maps and the Ahlfors Islands Theorem" (American Journal of Mathematics, accepted.) [DVI format]

[abstract:] "We present a p-adic and non-archimedean 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 non-archimedean 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 non-archimedean 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 non-archimedean dynamics" Proceedings of the London Mathematical Society 84 no. 3 (2002) 231-256

[abstract:] "We expand the notion of non-archimedean connected components introduced in "Hyperbolic maps in p-adic 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 non-archimedean 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 "p-Adic 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 p-adic maps with interesting or pathological dynamics."

R. Bendetto, "Hyperbolic maps in p-adic dynamics", Ergodic Theory and Dynamical Systems 21 (2001) 1-11

[abstract:] "In this paper we study the dynamics of a rational function f(z) defined over a finite extension Qp, the field of p-adic 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 p-adic 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 p-adic numbers and p-adic solenoids into Euclidean spaces"

[abstract:] "Explicit formulas are obtained for a family of continuous mappings of p-adic 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 p-adic solenoid is an invariant set of fractional dimension for a dynamic system. Computer drawings of some fractal images are presented."

D. Chistyakov, "Fractal measures, p-adic numbers and continuous transition between dimensions"

[abstract:] "Fractal measures of images of continuous maps from the set of p-adic numbers Qp into complex plane C are analyzed. Examples of 'anomalous' fractals, i.e. the sets where the D-dimensional Hausdorff measures (HM) are trivial, i.e. either zero, or sigma-infinite (D is the Hausdorff dimension (HD) of this set) are presented. Using the Caratheodory construction, the generalized scale-covariant HM (GHM) being non-trivial on such fractals are constructed. In particular, we present an example of 0-fractal, the continuum with HD=0 and nontrivial GHM invariant w.r.t. the group of all diffeomorphisms C. For conformal transformations of domains in Rn, the formula for the change of variables for GHM is obtained. The family of continuous maps Qp 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>Qp is C (real axis in C.); 2) the fractal measures coincide with the images of the Haar measure in Qp, 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 Qp are holomorphic in d, similarly to the dimensional regularization method in QFT."

V. Anashin, "Uniformly distributed sequences of p-adic 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 p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) 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 p-adic unit disk."

E. Thiran, D. Verstegen and J. Weyers, "p-adic dynamics", Journal of Statistical Physics 54 nos. 3-4 (1989) 893-913

D. Verstegen, "p-adic dynamical systems" from Number Theory and Physics (J.-M. Luck, P. Moussa and M. Waldschmidt, eds.), Springer Proceedings in Physics 47 (Springer, 1990) 235-242

L. Hsia, "A weak Néron model with applications to p-adic dynamical systems", Composito Math. 100 (1996) 277-304

Hua-Chieh Li, "p-adic periodic points and Sen's theorem", J. Number Theory 56 no. 2 (1996) 309-318

J. Lubin, "Nonarchimedean dynamical systems", Compositio Math. 94 no,. 3 (1994) 321-346

J. Lubin, "Formal flows on the nonarchimedean open unit disk", Compositio Math. 124 (2000) 123-136

S. Ben-Menahem, "p-Adic iterations", preprint, Tel-Aviv UP (1988) 1627-88

D. Dubischar, V.M. Gundlach, O. Steinkamp, and A. Khrennikov, "Attractors of random dynamical systems over p-adic numbers and a model of noisy cognitive processes", Physica D 130 (1999) 1-12

A. Khrennikov, "p-adic dynamical systems: description of concurrent struggle in biological population with limited growth", Dokl. Akad. Nauk 361 no. 6 (1998) 752-754.

M. de Gosson, B. Dragovich and A. Khrennikov, "Some p-adic differential equations"

"We investigate various properties of p-adic 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 $P_k (n) = n^k + C_{k-1} n^{k-1} + ...+ C_0$ 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 second-order for $F_k (x)$, is shown. In some cases such equations are constructed. For the second-order differential equations there is no other analytic solution of the form $\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous first-order differential equation exists one can construct infinitely many inhomogeneous second-order 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 p-adic dynamics and p-adic information theory."

A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, 1997)

A. Khrennikov and M. Nilsson, "On the number of cycles of p-adic dynamical systems", Journal of Number Theory 90 (2001) 255-264

[abstract:] "We found the asymptotics, p, for the number of cycles for iteration of monomial functions in the fields of p-adic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers."

S. De Smedt, A. Khrennikov, "A p-adic behaviour of dynamical systems", Rev. Mat. Comput. 12 (1999) 301-323

S. Nechaev and O. Vasilyev, "On metric structure of ultrametric spaces", J. Phys. A 37 (2004) 3783-3803

[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 eta-function) to construct the "continuous" analog of the Cayley tree isometrically embedded in the Poincaré upper half-plane. 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 semi-analytically/semi-numerically. The speculation on the new "geometrical" interpretation of replica n -> 0 limit is proposed."

S. Matsutani, "p-adic difference-difference Lotka-Volterra equation and ultra-discrete limit", Int. J. Math. and Math. Sci. 27 (2001) 251-260

[abstract:] "We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete 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, "Lotka-Volterra equation over a finite ring $\mathbb{Z}/p^N \mathbb{Z}$", J. Phys. A 34 (2001) 10737-10744

[abstract:] "The discrete Lotka-Volterra equation over $p$-adic space was constructed since $p$-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking the ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean 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, "p-Adic chaos and random numbers", Experimental Mathematics, 7(1) (1998) 333-342.

p-adic 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 space-time 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 p-adic 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 p-adic sector of an adelic open scalar string. Such Lagrangians have their origin in Lagrangian for a single p-adic 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 p-adic 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:] "p-adic strings are important objects of string theory, as well as of p-adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and p-adic 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 p-adic strings exist effective Lagrangians, which are based on real instead of p-adic numbers and describe not only four-point scattering amplitudes but also all higher ones at the tree level. In this work, starting from p-adic Lagrangians, we consider some approaches to construction of effective field Lagrangians for p-adic 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 operator-valued 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 p-adic 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 open-closed 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 p-adic and adelic strings, which takes into account that not only world sheet but also Minkowski space-time and string momenta can be p-adic and adelic, is formulated. p-Adic 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. p-Adic and adelic Moyal products are introduced. In particular, p-adic 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 p-adic string theory" (preprint 06/03)

[abstract:]"In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic 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 space-homogenous 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 q-brane solutions are discussed."

I.Ya. Aref'eva, M.G. Ivanov and I.V. Volovich, "Non-extremal intersecting p-branes in various dimensions", Phys. Lett. B 406 (1997) 44-48

[abstract:] "Non-extremal intersecting p-brane solutions of gravity coupled with several antisymmetric fields and dilatons in various space-time dimensions are constructed. The construction uses the same algebraic method of finding solutions as in the extremal case and a modified "no-force" conditions. We justify the "deformation" prescription. It is shown that the non-extremal intersecting p-brane solutions satisfy harmonic superposition rule and the intersections of non-extremal p-branes are specified by the same characteristic equations for the incidence matrices as for the extremal p-branes. We show that S-duality holds for non-extremal p-brane solutions. Generalized T-duality 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 p-branes in various dimensions", Nucl. Phys. Proc. Suppl. 56B (1997) 52-60

[abstract:] "We review an algebraic method of finding the composite p-brane 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 Fock-De Donder harmonic gauge for the metric and the "no-force" 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 p-branes 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, "p-Brane solutions in diverse dimensions", Phys.Rev. D55 (1997) 4748-4755

[abstract:] "A generic Lagrangian, in arbitrary spacetime dimension, describing the interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An isotropic p-brane 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 p-branes in diverse dimensions", Class. Quant. Grav. 14 (1997) 2991-3000

[abstract:] "We use a simple algebraic method to find a special class of composite p-brane solutions of higher dimensional gravity coupled with matter. These solutions are composed of n constituent p-branes 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, "p-Adic string", Classical Quantum Gravity 4 (1987) 83-87

I.V. Volovich, "From p-adic strings to étale strings", Proc. Steklov Inst. Math. 203 (1995) no. 3, 37–42.

A. Volovich, "Three-block p-branes in various dimensions", Nucl. Phys. B492 (1997) 235-248

[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 p-brane solution consisting of three blocks. Relations with known p-brane solutions are discussed. In particular, in ten-dimensional spacetime the three-block p-brane solution is reduced to the known solution, which recently has been used in the D-brane derivation of the black hole entropy."

C. Castro, "Hints of a new relativity principle from p-branes quantum mechanics", Journal of Chaos, Solitons and Fractals 11 (2000) 1721

C. Castro and A. Granik, "p-loops harmonic oscillators in C-spaces 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), "p-Adic stochastic dynamics, supersymmetry and the Riemann conjecture"

"Supersymmetry, p-adic stochastic dynamics, Brownian motion, Fokker-Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc...are all deeply interconnected in this paper."

C. Castro, "Fractal strings as the basis of Cantorian-Fractal 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 Cantorian-Fractal 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 quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature."

P.H. Frampton and Y. Okada, "p-Adic string N-point function", Phys. Rev. Lett. B 60 (1988) 484-486

J. Minahan, "Mode interactions of the tachyon condensate in p-adic string theory"

"We study the fluctuation modes for lump solutions of the tachyon effective potential in p-adic 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 p-adic string amplitudes for excited string states."

A. Sen, "Tachyon condensation and brane descent relations in p-adic string theory"

"It has been conjectured that an extremum of the tachyon potential of a bosonic D-brane represents the vacuum without any D-brane, and that various tachyonic lump solutions represent D-branes of lower dimension. We show that the tree level effective action of p-adic string theory, the expression for which is known exactly, provides an explicit realisation of these conjectures."

H. Furusho, "p-adic multiple zeta values I - p-adic multiple polylogarithms and the p-adic KZ equation"

[abstract:] "Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Coleman's p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the p-adic KZ equation and introduce the p-adic Drinfel'd associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear on coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear on coefficients of the p-adic Drinfel'd associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation."

M. Nardelli, "On the possible mathematical connections between the Hartle-Hawking no boundary proposal concerning the Randall-Sundrum cosmological scenario, Hartle-Hawking wave-function in the mini-superspace of physical superstring theory, p-adic Hartle-Hawkind wave function and some sectors of number theory" (preprint, 2007)

M. Nardelli, "On the possible mathematical connections concerning the relation between three-dimensional gravity related to Chern-Simons gauge theory, p-adic Hartle-Hawking wave function, Ramanujan's modular functions and some equations describign the Riemann zeta-function" (preprint, 2007)

M. Nardelli, "On the link between the structure of A-branes observed in homological mirror symmetry and the classical theory of automorphic forms. Mathematical connections with the modular elliptic curves, p-adic and adelic numbers and p-adic 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 A-branes 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 three-dimensional pure quantum gravity with negative cosmological constant, the Selberg zeta-function, the ten-dimensional 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 three-dimensions, in the Selberg zeta function, in the vanishing of cosmological constant and in the ten-dimensional anomaly cancellations. In the Section 1, we have described some equations concerning the pure three-dimensional quantum gravity with a negative cosmological constant and the pure three-dimensional supergravity partition functions. In the Section 2, we have described some equations concerning the Selberg super-trace formula for Super-Riemann surfaces, some analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic strings. In the Section 3, we have described some equations concerning the ten-dimensional anomaly cancellations and the vanishing of cosmological constant. In the Section 4, we have described some equations concerning p-adic strings, p-adic 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 super-zeta 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, p-adic strings, zeta strings and p-adic 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 (p-adic and adelic strings, p-adic 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 p-adic strings, p-adic and adelic zeta functions, zeta strings and p-adic cosmology (with regard the p-adic 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 Thomas-Fermi method, some sectors of Number Theory, the modes corresponding to the physical vibrations of superstrings, p-adic and Adelic free relativistic particle and p-adic 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 Thomas-Fermi 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 p-adic and Adelic free relativistic particle and p-adic and adelic strings (Sections 3 and 4)."

M. Nardelli, "On some mathematical connections between the cyclic universe, inflationary universe, p-adic Inflation, p-adic 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 M-theory" (Cyclic Universe), p-adic inflation and p-adic 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 space-time, the M-theory 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 five-dimensional theory of the strongly coupled heterotic string as a gauged version of $N = 1$ five dimensional supergravity with four-dimensional 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 p-adic string theory. In the Section 6, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In the Section 7, we have showed various and interesting mathematical connections between some equations concerning the p-adic Inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. 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 p-adic 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 mean-convergence 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 non-monotonicity of the Euler phi function and the related Riemann zeta function.

In Section 5, we have showed some equations concerning the p-adic 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 p-adic and adelic strings. Principally, in Section 3, where is frequently used the Hardy-Ramanujan 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 p-adic 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 M-theory."

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 multi-term 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 dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium 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 p-Adic models in Hartle-Hawking proposal and p-Adic 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 p-adic and adelic cosmology."

M. Nardelli, "On some mathematical equations concerning the functions $\zeta(s)$ and $\zeta(s,w)$ and some Ramanujan-type 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 Ramanujan-type 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 Palumbo--Nardelli model and the Ramanujan's modular equations that are related to the physical vibrations of bosonic strings and of superstrings."

p-adic and adelic approaches to quantum mechanics

G. Djordjevic, B. Dragovich and L. Nesic, "p-Adic and adelic free relativistic particle", Modern Physics Letters A, 14, no. 5 (1999) 317-325.

"We consider spectral problem for a free relativistic particle in p-adic and adelic quantum mechanics. In particular, we found p-adic and adelic eigenfunctions. Within adelic approach there exist quantum states that exhibit discrete structure of space-time 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 non-Archimedean and noncommutative aspects. In particular, p-adic path integral and adelic quantum cosmology are considered. Some similarities between p-adic analysis and q-analysis are noted. The p-adic 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 one-dimensional 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, "p-Adic and adelic minisuperspace quantum cosmology", International Journal of Modern Physics A 17 (2002) 1413-1434.

"We consider the formulation and some elaboration of p-adic and adelic quantum cosmology. The adelic generalization of the Hartle-Hawking proposal does not work in models with matter fields. p-Adic and adelic minisuperspace quantum cosmology is well defined as an ordinary application of p-adic and adelic quantum mechanics. It is illustrated by a few minisuperspace cosmological models in one, two and three minisuperspace dimensions. As a result of p-adic 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, "p-Adic and adelic generalization of quantum cosmology", Grav. Cosmol. 5 (1999) 222.

"p-Adic and adelic generalization of ordinary quantum cosmology is considered. In [1], we have calculated p-adic wave functions for some minisuperspace cosmological models according to the "no-boundary" Hartle-Hawking proposal. In this article, applying p-adic and adelic quantum mechanics, we show existence of the corresponding vacuum eigenstates. Adelic wave function contains some information on discrete structure of space-time at the Planck scale."

G. Djordjevic and B. Dragovich, "p-Adic and adelic harmonic oscillator with time-dependent frequency", Theor. Math. Phys. 124 (2000) 1059-1067.

"The classical and quantum formalism for a p-adic and adelic harmonic oscillator with time-dependent frequency is developed, and general formulae for main theoretical quantities are obtained. In particular, the p-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. Space discreteness and p-adic quantum-mechanical phase are noted."

G. Djordjevic and B. Dragovich, "p-Adic path integrals for quadratic actions", Modern Physics Letters 12 (1997) 1455-1463.

"The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude ${\cal K}_p (x^{\prime\prime},t^{\prime\prime}; x^\prime,t^\prime)$ for one-dimensional 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 p-adic functional integration"

"p-Adic 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 p-adic space have the same form as those over R."

G. Djordjevic, "On p-adic path integral"

"Feynman's path integral is generalized to quantum mechanics on p-adic space and time. Such p-adic 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) 2349-2365

[abstract:] "Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic 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 Schwartz-Bruhat 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) 197-203.

[abstract:] "Some aspects of adelic generalized functions, as linear continuous functionals on the space of Schwartz-Bruhat 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 p-adic analysis and physics

V. Varadarajan, "Path integrals for a class of p-adic Schrodinger equations"

V. Varadarajan, "Some remarks on arithmetic physics"

D. Bump, Kwok-Kwong Choi, P. Kurlberg, and J. Vaaler, "A Local Riemann Hypothesis, I", Mathematische Zeitschrift 233 (1) (2000), 1-18. (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 p-adic 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 Grande-De Kimpe, A. Khrennikov, "p-Adic probability and an interpretation of negative probabilities in quantum mechanics", Russian J. Math. Phys. 6 (1999) 3-19.

B. Grossman, "The adelic components of the index of the Dirac operator", Journal of Physics A 21 (1988) L1051-1054.

[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 non-zero rational numbers Q* into the completions Qp of Q with respect to the prime valuations |.|p."

T.N. Palmer, "p-Adic distance, finite precision and emergent superdeterminism: A number-theoretic consistent-histories approach to local quantum realism" (preprint, 09/2016)

[abstract:] "Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative p-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of 'consistent histories' based on number-theoretic properties of the transcendental cosine function. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory – neither conspiratorial, retrocausal, fine tuned nor nonlocal – superdeterminism is not invoked by fiat but is emergent from these 'consistent histories' number-theoretic constraints. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model."

C. Castro, "Hints of a new relativity principle from p-branes quantum mechanics", Journal of Chaos, Solitons and Fractals 11 (2000) 1721

p-adic probability, statistics and randomness

A.N. Kochubei, Pseudo-differential Equations and Stochastics over non-Archimedian Fields (Marcel Dekker, 2001)

(from publisher's description) "This state-of-the-art reference provides comprehensive coverage of the most recent developments in the theory of non-Archimedean pseudo-differential equations and its application to stochastics and mathematical physics offering current methods of construction for stochastic processes on the field of p-adic numbers and related structures. Develops a new theory for parabolic equations over non-Archimedean fields in relation to Markov processes!"

S. Albeverio and W. Karwowski, "A Random Walk on p-Adics - the Generator and Its Spectrum", Stochastic Processes and their Applications 53 (1994) 1-22.

W. Karwowski and R. Vilela Mendes, "Hierarchical Structures and Asymmetric Stochastic Processes on p-Adics and Adeles", Journal of Mathematical Physics 35 (1994) 4637-4650.

S. Albeverio, R. Cianci, N. De Grande-De Kimpe, A. Khrennikov, "p-Adic probability and an interpretation of negative probabilities in quantum mechanics", Russian J. Math. Phys. 6 (1999) 3-19.

A. Khrennikov, "Limit behaviour of sums of independent random variables with respect to the uniform p-adic distribution", Statistics and Probability Letters 51 (2001) 269-276

[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 p-adic metric (where p is a prime number). We found that (despite of rather unusual features of a p-adic metric) limits of classical probabilities exist in a field of p-adic 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 p-adic probability distribution, namely the uniform distribution."

N. Smart and C. Woodcock, "p-Adic chaos and random numbers", Experimental Mathematics, 7(1) (1998) 333-342.

J.L. Lucio and Y. Meurice, "Asymptotic properties of random walks on p-adic spaces", University of Iowa Preprint 90-33, 1-5 (1990)

C. Castro (Perelman), "p-Adic stochastic dynamics, supersymmetry and the Riemann conjecture"

"Supersymmetry, p-adic stochastic dynamics, Brownian motion, Fokker-Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc...are all deeply interconnected in this paper."

p-adic and adelic scattering theory

J.-F. Burnol, "Scattering on the p-adic field and a trace formula"

[abstract] "I apply the set-up of Lax-Phillips Scattering Theory to a non-archimedean 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 Lax-Phillips 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 semi-group 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 (non-negative) spectral function of T, which is also the restriction to the diagonal of the kernel of K. The contraction semi-group Z is related to S (and T) through a trace formula. Introducing an odd-even 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 p-adic and adelic symmetric spaces", Physics Letters B 257 (1991) 119-124

[abstract:] "Explicit S-matrices are constructed for scattering on p-adic hyperbolic planes. Combining these with the known S-matrix on the real hyperbolic plane, an adelic S-matrix is obtained. It has poles at the nontrivial zeros of the Riemann zeta-function, 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 p-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we find a wide class of integrable evolutions which respect the number-theoretic properties of the zsf problem. This means that at all times these real and p-adic systems can be unified into an adelic system with an S-matrix which involves (Dirichlet, Langlands, Shimura...) L-functions."

[introduction:] "Scattering theory on real [1] and p-adic [2] symmetric spaces can be unified in an adelic context. This had the virtue of producing S-matrices involving the Riemann zeta function and of throwing new light on earlier work [4] concerning scattering on the noncompact finite-area 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 p-adic hyperbolic planes Hp are discrete spaces (trees), and the corresponding Schrodinger equations are second order difference equations. The Jost functions, and therefore the S-matrices from all these local problems combine in adelic products, which then involve the Riemann zeta function [2].

At a given time consider all these ("S-wave") scattering problems and then let all of them undergo an integrable time evolution. In general such an evolution need not respect the number-theoretic endowment of the initial problem. In other words, even though at the initial time the real and p-adic 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 Zakharov-Shabat (ZS) system and follow its integrable evolution. For the p-adic 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,...) L-functions [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, "L-functions in scattering on p-adic 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 L-Functions (Academic, 1988)

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) 1231-1235.;

(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 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 (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.

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, 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 (aoo;a2,a3,a5,a7, 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 spectral analysis."

C. Consani and M. Marcolli, "Non-commutative geometry, dynamics, and infinity-adic 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) 779-784

[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, "p-Adic modelling of the genome and the genetic code" (preprint 07/2007)

[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."

F. Murtagh, Hierarchical matching and regression with application to photometric redshift estimation" (preprint, 12/2016)

[abstract:] "This work emphasizes that heterogeneity, diversity, discontinuity, and discreteness in data is to be exploited in classification and regression problems. A global a priori model may not be desirable. For data analytics in cosmology, this is motivated by the variety of cosmological objects such as elliptical, spiral, active, and merging galaxies at a wide range of redshifts. Our aim is matching and similarity-based analytics that takes account of discrete relationships in the data. The information structure of the data is represented by a hierarchy or tree where the branch structure, rather than just the proximity, is important. The representation is related to p-adic number theory. The clustering or binning of the data values, related to the precision of the measurements, has a central role in this methodology. If used for regression, our approach is a method of cluster-wise regression, generalizing nearest neighbour regression. Both to exemplify this analytics approach, and to demonstrate computational benefits, we address the well-known photometric redshift or 'photo-z' problem, seeking to match Sloan Digital Sky Survey (SDSS) spectroscopic and photometric redshifts."

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. (Birkhauser, 1998).

A. Khrennikov, "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."

Topological 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 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 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."

"Riemann 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

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