P-adic String Theory Research Articles

THE SOLVABILITY OF AN INFINITE SYSTEM OF NONLINEAR ALGEBRAIC EQUATIONS WITH TOEPLITZ MATRIX

The work is focused on studying of the existence, uniqueness, and various qualitative properties of the constructive solution of an infinite system of algebraic equations with a concave nonlinearity property, which are generated by Toeplitz matrices. In addition to its independent mathematical interest, such systems have a significant application in several branches of mathematical physics and mathematical biology. Those particularly appear in discrete problems within radiative transfer theory, kinetic theory of gases, dynamic theory of p-adic strings, and the mathematical theory of epidemic propagation. We establish the existence of a positive solution for the system in the class of bounded sequences, as well as provide an iterative method to approximate to the solution. We also study the asymptotic behavior of the solution at infinity and the uniqueness of the nontrivial solution with non-negative elements in the class of bounded sequences. The last section of the paper provides examples of applications of the corresponding Toeplitz matrix and the function that describes the nonlinearity.

Towards Unification of General Relativity and Quantum Theory: Dendrogram Representation of the Event-Universe.

Following Smolin, we proceed to unification of general relativity and quantum theory by operating solely with events, i.e., without appealing to physical systems and space-time. The universe is modelled as a dendrogram (finite tree) expressing the hierarchic relations between events. This is the observational (epistemic) model; the ontic model is based on p-adic numbers (infinite trees). Hence, we use novel mathematics: not only space-time but even real numbers are not in use. Here, the p-adic space (which is zero-dimensional) serves as the base for the holographic image of the universe. In this way our theory is connected with p-adic physics; in particular, p-adic string theory and complex disordered systems (p-adic representation of the Parisi matrix for spin glasses). Our Dendrogramic-Holographic (DH) theory matches perfectly with the Mach’s principle and Brans–Dicke theory. We found a surprising informational interrelation between the fundamental constants, h, c, G, and their DH analogues, h(D), c(D), G(D). DH theory is part of Wheeler’s project on the information restructuring of physics. It is also a step towards the Unified Field theory. The universal potential V is nonlocal, but this is relational DH nonlocality. V can be coupled to the Bohm quantum potential by moving to the real representation. This coupling enhances the role of the Bohm potential.

Solvability of a Certain System of Singular Integral Equations with Convex Nonlinearity on the Positive Half-Line

We study a system of nonlinear singular integral equations with a sum-difference kernel on the positive half-line. Such a system occurs (in various representations) in many branches of mathematical physics and applied mathematics. In particular, one encounters a system of equations with a kernel representing a Gaussian distribution and with a power nonlinearity in the dynamic theory of p-adic open-closed strings; in the case when the nonlinearity has a certain exponential structure, such a system occurs in mathematical biology, namely, in the theory of the spatio-temporal distribution of an epidemic. We prove constructive theorems on the existence of nonnegative nontrivial continuous and bounded solutions. We study the uniqueness and the asymptotic behavior of constructed solutions at infinity. In conclusion, we give concrete applied examples of the mentioned equations that satisfy all assumptions of the proved theorems.

О разрешимости одного класса нелинейных интегральных уравнений Урысона на положительной полупрямой

The paper investigates the Urysohn’s nonlinear integral equation on the positive half-line. Some special cases of this equation have specific applications in different areas of modern natural science. In particular, such equations arise in the kinetic theory of gases, in the theory of 𝑝-adic open-closed strings, in mathematical theory of the spatiotemporal spread of the epidemic, and in theory of radiative transfer in spectral lines. The existence theorem for nonnegative nontrivial and bounded solutions is proved. Some qualitative properties of the constructed solution are studied. Specific applied examples of the Urysohn’s kernel satisfying all the conditions of the approved theorem are provided.

On the qualitative properties of the solution of a nonlinear boundary value problem in the dynamic theory of p-adic strings

The article considers a boundary value problem for a class of singular integral equations with an almost total-difference kernel and convex nonlinearity on the positive half-line. This problem arises in the dynamic theory of p-adic open-closed strings. It is proved that any nonnegative and bounded solution of a given boundary value problem is a continuous function and the difference between the limit and the solution is itself an integrable function on the positive half-line. For a particular case, it is proved that the solution is a monotonically non-decreasing function. A uniqueness theorem is established in the class of nonnegative and bounded functions. At the conclusion of the article, a specific applied example of this boundary problem is given.

Solvability of Some Classes of Nonlinear Singular Boundary Value Problems in the Theory of p-Adic Open-Closed Strings

We investigate boundary value problems for a singular integral equation of the convolution type with a power-law nonlinearity. Such problems arise in the theory of p-adic open-closed strings. We prove constructive theorems on the existence of nontrivial solutions and also prove a uniqueness theorem in a certain weight class of functions. We study asymptotic properties of the constructed solutions and obtain the Vladimirov theorem on tachyon Gelds for open-closed strings as a particular case of the proved results.

Cosmology of non-local f(R) gravity

We consider a modification of GR with a special type of a non-local f (R). The structure of the non-local operators is motivated by the string field theory and p-adic string theory. The spectrum is derived explicitly and the ghost-free condition for the model is formulated. We pay special attention to the classical stability of the de Sitter solution in our model and formulate the conditions on the model parameters to have a stable configuration. Relevance of unstable configurations for the description of the de Sitter phase during inflation is specifically discussed.

Scale holography

We present a new correspondence between a d-dimensional dynamical system and a family of higher-order derivative (d+1)-dimensional systems. This new scale-holographic relation is built by the explicit introduction of a dimensionful constant which determines the size of the additional dimension. Scale holography is particularly useful for studying non-local theories, since the equivalent dual system on the higher dimensional manifold can be made to be local, as we illustrate with the specific examples of the p-adic string theory and the free particle.

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.

Solvability of a Class of Nonlinear Pseudo-Differential Equations in ℝ n

In the present work the existence of continuous and bounded solutions for a class of nonlinear pseudo-differential equations on ℝ n is proved. The monotonicity, asymptotic behavior and other properties for obtained solutions are also presented. Mentioned class of equations arises in p-adic string theory.

On finite-temperature string field theory and p-adic string

Covariant string field theory at finite temperature using quantum double (thermofield dynamics) formalism is considered. Thermal p-adic string theory amplitudes and a natural generalization of p-adic string effective action to the case on non-zero temperature are proposed.

Nonlocal gravity and the diffusion equation

We propose a nonlocal scalar-tensor model of gravity with pseudodifferential operators inspired by the effective action of p-adic string and string field theory on flat spacetime. An infinite number of derivatives act both on the metric and scalar field sector. The system is localized via the diffusion equation approach and its cosmology is studied. We find several exact dynamical solutions, also in the presence of a barotropic fluid, which are stationary in the diffusion flow. In particular, and contrary to standard general relativity, there exist solutions with exponential and power-law scale factor also in an open universe, as well as solutions with sudden future singularities or a bounce. Also, from the point of view of quantum field theory, spontaneous symmetry breaking can be naturally realized in the class of actions we consider.

Zeta-nonlocal scalar fields

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument. The construction of the corresponding Lagrangians L starts with the exact Lagrangian \( \mathcal{L}_p \) for the effective field of the p-adic tachyon string, which is generalized by replacing p with an arbitrary natural number n and then summing \( \mathcal{L}_n \) over all n. We obtain several basic classical properties of these fields. In particular, we study some solutions of the equations of motion and their tachyon spectra. The field theory with Riemann zeta-function dynamics is also interesting in itself.

NON-LOCAL INFLATION AROUND A LOCAL MAXIMUM

It is shown that non-local, higher-derivative operators, which arise generically in string field theory, can act as additional sources of friction on the inflaton field as it rolls away from a maximum in its potential. Moreover, the cosmic dynamics can be quantified in terms of a local field theory, where the curvature of an effective potential has been suppressed. A prolonged phase of quasi-exponential expansion can therefore be realised with steep potentials that typically arise in particle physics models. We illustrate this effect within the context of p-adic string theory.

Dynamics with infinitely many derivatives: the initial value problem

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off. Our results place certain recent attempts to study inflation in the context of nonlocal field theories on a much firmer mathematical footing.

QUANTIZATION OF THE RIEMANN ZETA-FUNCTION AND COSMOLOGY

Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein–Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat–Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

P-adic inflation

We construct approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory, a nonlocal theory with derivatives of all orders. Novel features include the existence of slow-roll solutions even when the slow-roll parameters, as usually defined, are much greater than unity, as well as the need for the Hubble parameter to exceed the string mass scale m_s. We show that the theory can be compatible with CMB observations if g_s / \sqrt{p} ~ 10^{-7}, where g_s is the string coupling, and if m_s < 10^{-6} M_p. A red-tilted spectrum is predicted, and the scalar-to-tensor ratio is bounded from above as r < 0.006. The p-adic theory is shown to have identical inflationary predictions to a local theory with superPlanckian parameter values, but with the advantage that the p-adic theory is ultraviolet complete.

TACHYON DYNAMICS IN OPEN STRING THEORY

In this review we describe our current understanding of the properties of open string tachyons on an unstable D-brane or brane–antibrane system in string theory. The various string theoretic methods used for this study include techniques of two-dimensional conformal field theory, open string field theory, boundary string field theory, noncommutative solitons, etc. We also describe various attempts to understand these results using field theoretic methods. These field theory models include toy models like singular potential models and p-adic string theory, as well as more realistic version of the tachyon effective action based on Dirac–Born–Infeld type action. Finally we study closed string background produced by the "decaying" unstable D-branes, both in the critical string theory and in the two-dimensional string theory, and describe the open string completeness conjecture that emerges out of this study. According to this conjecture the quantum dynamics of an unstable D-brane system is described by an internally consistent quantum open string field theory without any need to couple the system to closed strings. Each such system can be regarded as a part of the "hologram" describing the full string theory.

Deformation of p-adic string amplitudes in a magnetic field

A new term in the p-adic world-sheet action is proposed, which couples a constant B-field to the boundary of the world-sheet at disk level. The induced deformation of tachyon scattering amplitudes by star-products is derived. This is in agreement with the deformation of effective action postulated in recent investigations of non-commutative solitons in p-adic string theory.

Towards p-adic string in constant B-field

Spacetime properties of the tachyon of the p-adic string theory can be derived from a (non-local) action on the p-adic line Q p , thought of as the boundary of the ‘worldsheet’. We show that a term corresponding to the background of the antisymmetric second rank tensor field B can be added to this action. We examine the consequences of this term, in particular, its relation to a non-commutative deformation of the effective theory of the p-adic tachyon in spacetime.