P-adic String Research Articles

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.

Nonexistence of solutions of the p-adic strings

We discuss mathematical aspects of the nonexistence of continuous (nontrivial) solutions of boundary value problems for equations of p-adic closed and open strings in the one-dimensional case. We find that the number of sign changes of the solution ψ(t) is not equal to the order of zeros of the function ψn(t) and that nonnegative (nonpositive) solutions do not exist. In the case of even n, if a solution ψ exists, then the orders of zeros of the function ψn and the order of its tangency to positive maximums (minimums) are not divisible by four and therefore have the form 2(2r + 1), r ≥ 0.

Tree-like structure of eternal inflation: A solvable model

In this paper we introduce a simple discrete stochastic model of eternal inflation that shares many of the most important features of the continuum theory as it is now understood. The model allows us to construct a multiverse and rigorously analyze its properties. Although simple and easy to solve, it has a rich mathematical structure underlying it. Despite the discreteness of the space-time the theory exhibits an unexpected non-perturbative analog of conformal symmetry that acts on the boundary of the geometry. The symmetry is rooted in the mathematical properties of trees, p-adic numbers, and ultrametric spaces; and in the physical property of detailed balance. We provide self-contained elementary explanations of the unfamiliar mathematical concepts, which have have also appeared in the study of the p-adic string. The symmetry acts on a huge collection of very low dimensional "multiverse fields" that are not associated with the usual perturbative degrees of freedom. They are connected with the late-time statistical distribution of bubble-universes in the multiverse. The conformal symmetry which acts on the multiverse fields is broken by the existence of terminal decays - to hats or crunches - but in a particularly simple way. We interpret this symmetry breaking as giving rise to an arrow of time. The model is used to calculate statistical correlations at late time and to discuss the measure problem. We show that the natural cutoff in the model is the analog of the so-called light-cone-time cutoff. Applying the model to the problem of the cosmological constant, we find agreement with earlier work.

On some exact solutions in p-adic open-closed string theory

The paper is concerned with the construction of exact solutions for nonlinear pseudodifferential equations which describe tachyon dynamics of open-closed p-adic strings. Existence of continuous solutions and their properties are discussed.

Mathematical aspects of nonlinear pseudodifferential equations of p-adic strings

Mathematical description of tachyon dynamics of open, closed and open-closed p-adic strings is considered. Questions on existence and nonexistence of continuous solutions and their properties and also structure of zeros of solutions are discussed. New multidimensional nonlinear equations of ultraparabolic type are obtained. A set of unsolved problems is presented.

Solutions of p-adic string equations

We review several purely mathematical results concerning boundary value problems for nonlinear pseudodifferential equations for p-adic closed and open strings in the tree approximation in the case d = 1. For the solutions of these problems, we present formulas establishing the relations between the numbers of their zeros, the multiplicities of the zeros, and the numbers indicating how many times the solutions change sign.

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.

Finite temperature solitons in nonlocal field theories fromp-adic strings

Non-local field theories which arise from p-adic string theories have vacuum soliton solutions. We find the soliton solutions at finite temperature. These solutions become important for the partition function when the temperature exceeds m_s/g_o^2 where m_s is the string mass scale and g_o is the open string coupling.

Thermodynamics and cosmological constant of non-local field theories from p-adic strings

We develop the thermodynamics of field theories characterized by non-local propagators. We analyze the partition function and main thermodynamic properties arising from perturbative thermal loops. We focus on the p-adic models associated with the tachyon phenomenology in string theories. We reproduce well known features of these theories, but also obtain many new results. In particular, we explain how to maintain consistency of such non-local theories by avoiding the appearance of ghosts at finite temperature. As a consequence of this fact, the vacuum energy in p-adic theories becomes positive. It is also hierarchically suppressed, and we explore the parameter space where it is consistent with the observed value of the cosmological constant.

Nonlocal dynamics of p-adic strings

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.

The p-adic sector of the adelic string

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 d'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.

Thermal Duality and Hagedorn Transition fromp-adic Strings

We develop the finite temperature theory of p-adic string models. We find that the thermal properties of these nonlocal field theories can be interpreted either as contributions of standard thermal modes with energies proportional to the temperature, or inverse thermal modes with energies proportional to the inverse of the temperature, leading to a thermal duality at leading order (genus one) analogous to the well-known T duality of string theory. The p-adic strings also recover the asymptotic limits (high and low temperature) for arbitrary genus that purely stringy calculations have yielded. We also discuss our findings surrounding the nature of the Hagedorn transition.

On nonlinear equations of p-adic strings for scalar tachyon fields

We consider boundary value problems for open and closed p-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems (d = 2).

Dynamics and stability of light-like tachyon condensation

Recently, Hellerman and Schnabl considered the dynamics of unstable D-branes in the background of a linear dilaton. Remarkably, they were able to construct light-like tachyon solutions which interpolate smoothly between the perturbative and nonperturbative vacua, without undergoing the wild oscillations that plague time-like solutions. In their analysis, however, the full structure of the initial value problem for the nonlocal dynamical equations was not considered. In this paper, therefore, we reexamine the nonlinear dynamics of light-like tachyon condensation using a combination of numerical and analytical techniques. We find that for the p-adic string the monotonic behaviour obtained previously relied on a special choice of initial conditions near the unstable maximum. For generic initial conditions the wild oscillations come back to haunt us. Interestingly, we find an ``island of stability'' in initial condition space that leads to sensible evolution at late times. For the string field theory case, on the other hand, we find that the evolution is completely stable for generic choices of initial data. This provides an explicit example of a string theoretic system that admits infinitely many initial data but is nevertheless nonperturbatively stable. Qualitatively similar dynamics are obtained in nonlocal cosmologies where the Hubble damping plays a role very analogous to the dilaton gradient.

The question of the asymptotic behavior as |t| → ∞ of boundary value problem solutions for p-ADIC strings

We obtain lower asymptotic estimates for tachyon fields of open and closed strings as |t| → ∞. They confirm the expressions in the first asymptotic term that were previously found as solutions of linearized equations; this is not confirmed in the second asymptotic term.

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.

Nonlocal inflationThis paper was prsented at the Theory CANADA 4 conference, held at the Centre de Recherches Mathématiques at the Université de Montréal, Québec, Canada on 4–7 June 2008.

We consider the possibility of realizing inflation in nonlocal field theories containing infinitely many derivatives. Such constructions arise naturally in string field theory and also in a number of toy models, such as the p-adic string. After reviewing the complications (ghosts and instabilities) that arise when working with high-derivative theories, we discuss the the initial value problem and perturbative stability of theories with infinitely many derivatives. Next, we examine the inflationary dynamics and phenomenology of such theories. Nonlocal inflation can proceed even when the potential is naively too steep and generically predicts large non-Gaussianity in the cosmic microwave background.

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.