Ensemble Of Strings Research Articles

From Homogeneous and Isotropic Universes to Braneworlds with Dynamical Tension Strings

Cosmological solutions are studied in the context of the modified measure formulation of string theory, then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings the values of the tensions can have a certain dispersion. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation. The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and isotropy of the cosmology disappears and the solution turns into an expanding Braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times.

Life of the homogeneous and isotropic universe in dynamical string tension theories

Cosmological solutions are studied in the context of the modified measure formulation of string theory , then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings. The values of the tensions can have a certain dispersion in the ensemble. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation . The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and and isotropy of the cosmological disappears and the solution turns into an expanding braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times. The same principle is applied to the static end of the universe wall solution that lasts a time inversely proportional to the dispersion of string tensions. This suggest a scenario where quantum fluctuations of the cosmological or static solutions induce the evolution towards braneworld scenarios and decoherence between the different string tension states.

Untangling the Galaxy. I. Local Structure and Star Formation History of the Milky Way

Abstract Gaia DR2 provides unprecedented precision in measurements of the distance and kinematics of stars in the solar neighborhood. Through applying unsupervised machine learning on DR2's 5D data set (3D position + 2D velocity), we identify a number of clusters, associations, and comoving groups within 1 kpc and (many of which have not been previously known). We estimate their ages with the precision of ∼0.15 dex. Many of these groups appear to be filamentary or string-like, oriented in parallel to the Galactic plane, and some span hundreds of parsec in length. Most of these string lack a central cluster, indicating that their filamentary structure is primordial, rather than the result of tidal stripping or dynamical processing. The youngest strings (<100 Myr) are orthogonal to the Local Arm. The older ones appear to be remnants of several other arm-like structures that cannot be presently traced by dust and gas. The velocity dispersion measured from the ensemble of groups and strings increase with age, suggesting a timescale for dynamical heating of ∼300 Myr. This timescale is also consistent with the age at which the population of strings begins to decline, while the population in more compact groups continues to increase, suggesting that dynamical processes are disrupting the weakly bound string populations, leaving only individual clusters to be identified at the oldest ages. These data shed a new light on the local galactic structure and a large-scale cloud collapse.

The Canonical Ensemble of Open Self-Avoiding Strings

Statistical models of a single open string avoiding self-intersections in the d-dimensional Euclidean space ℝd, 2 ≤ d < 4, and the ensemble of strings are considered. The presentation of these models is based on the Darwin–Fowler method, used in statistical mechanics to derive the canonical ensemble. The configuration of the string in space ℝd is described by its contour length L and the spatial distance R between its ends. We establish an integral equation for a transformed probability density W(R, L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations. This allows us using the renormalization group method to investigate the asymptotic behavior of this density in the case where R → ∞ and L → ∞. For the model of an ensemble of M open strings with the mean string contour length over the ensemble given by $$ \overline{L} $$ , we obtain the most probable distribution of strings over their lengths in the limit as M → ∞. Averaging the probability density W(R, L) over the canonical ensemble eventually gives the sought density 〈W(R, L)〉.

Evolution of the mean jet shape and dijet asymmetry distribution of an ensemble of holographic jets in strongly coupled plasma

Some of the most important experimentally accessible probes of the quark- gluon plasma (QGP) produced in heavy ion collisions come from the analysis of how the shape and energy of sprays of energetic particles produced within a cone with a specified opening angle (jets) in a hard scattering are modified by their passage through the strongly coupled, liquid, QGP. We model an ensemble of back-to-back dijets for the purpose of gaining a qualitative understanding of how the shapes of the individual jets and the asymmetry in the energy of the pairs of jets in the ensemble are modified by their passage through an expanding cooling droplet of strongly coupled plasma, in the model in a holographic gauge theory that is dual to a 4+1-dimensional black-hole spacetime that is asymptotically anti-de Sitter (AdS). We build our model by constructing an ensemble of strings in the dual gravitational description of the gauge theory. We model QCD jets in vacuum using strings whose endpoints are moving “downward” into the gravitational bulk spacetime with some fixed small angle, an angle that represents the opening angle (ratio of jet mass to jet energy) that the QCD jet would have in vacuum. Such strings must be moving through the gravitational bulk at (close to) the speed of light; they must be (close to) null. This condition does not specify the energy distribution along the string, meaning that it does not specify the shape of the jet being modeled. We study the dynamics of strings that are initially not null and show that strings with a wide range of initial conditions rapidly accelerate and become null and, as they do, develop a similar distribution of their energy density. We use this distribution of the energy density along the string, choose an ensemble of strings whose opening angles and energies are distributed as in perturbative QCD, and show that we can then fix one of the two model parameters such that the mean jet shape for the jets in the ensemble that we have built matches that measured in proton-proton collisions reasonably well. This is a novel way for hybridizing relevant inputs from perturbative QCD and a strongly coupled holographic gauge theory in the service of modeling jets in QGP. We send our ensemble of strings through an expanding cooling droplet of strongly coupled plasma, choosing the second model parameter so as to get a reasonable value for RAAjet, the suppression in the number of jets, and study how the mean jet shape and the dijet asymmetry are modified, comparing both to measurements from heavy ion collisions at the LHC.

Entanglement branes in a two-dimensional string theory

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large N . The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the modular Hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an entanglement brane, or E-brane.

A d-dimensional model of the canonical ensemble of open strings

We propose a d-dimensional model of the canonical ensemble of open self-avoiding strings. We consider the model of a solitary open string in the d-dimensional Euclidean space ℝ d, 2 ≤ d < 4, where the string configuration is described by the arc length L and the distance R between string ends. The distribution of the spatial size of the string is determined only by its internal physical state and interaction with the ambient medium. We establish an equation for a transformed probability density W(R,L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations; this allows using the renormalization group method to investigate the asymptotic behavior of this density in the case where R→∞ and L→∞. We consider the model of an ensemble of M open strings with the mean string length over the ensemble given by $\bar L$ , and we use the Darwin-Fowler method to obtain the most probable distribution of strings over their lengths in the limit as M →∞. Averaging the probability density W(R,L) over the canonical ensemble eventually gives the sought density 〈W(R, $\bar L$ )〉.

Face Recognition Using Ensemble String Matching

In this paper, we present a syntactic string matching approach to solve the frontal face recognition problem. String matching is a powerful partial matching technique, but is not suitable for frontal face recognition due to its requirement of globally sequential representation and the complex nature of human faces, containing discontinuous and non-sequential features. Here, we build a compact syntactic Stringface representation, which is an ensemble of strings. A novel ensemble string matching approach that can perform non-sequential string matching between two Stringfaces is proposed. It is invariant to the sequential order of strings and the direction of each string. The embedded partial matching mechanism enables our method to automatically use every piece of non-occluded region, regardless of shape, in the recognition process. The encouraging results demonstrate the feasibility and effectiveness of using syntactic methods for face recognition from a single exemplar image per person, breaking the barrier that prevents string matching techniques from being used for addressing complex image recognition problems. The proposed method not only achieved significantly better performance in recognizing partially occluded faces, but also showed its ability to perform direct matching between sketch faces and photo faces.

Toward the AdS/CFT gravity dual for high energy collisions. II. The stress tensor on the boundary

In this second paper of the series, we calculate the stress tensor of excited matter, created by debris of high energy collisions at the boundary. The falling open strings, connected to receding charges, produce a nonzero stress tensor which we found analytically from time-dependent linearized Einstein equations in the bulk. It corresponds to exploding nonequilibrium matter: we discuss its behavior in some detail, including its internal energy density in a comoving frame and the 'freeze-out surfaces'. We then discuss what happens for the ensemble of strings.

CHRONOMETRIC INVARIANCE AND STRING THEORY

The Einstein–Hilbert Lagrangian R is expressed in terms of the chronometrically invariant quantities introduced by Zel'manov for an arbitrary four-dimensional metric gij. The chronometrically invariant three-space is the physical space γαβ = -gαβ+e2ϕ γαγβ, where e 2ϕ = g00 and γα = g0α/g00, and whose determinant is h. The momentum canonically conjugate to γαβ is [Formula: see text], where [Formula: see text] and ∂t≡ e -ϕ∂0 is the chronometrically invariant derivative with respect to time. The Wheeler–DeWitt equation for the wave function Ψ is derived. For a stationary space-time, such as the Kerr metric, παβ vanishes, implying that there is then no dynamics. The most symmetric, chronometrically-invariant space, obtained after setting ϕ = γα = 0, is [Formula: see text], where δαβ is constant and has curvature k. From the Friedmann and Raychaudhuri equations, we find that λ is constant only if k=1 and the source is a perfect fluid of energy-density ρ and pressure p=(γ-1)ρ, with adiabatic index γ=2/3, which is the value for a random ensemble of strings, thus yielding a three-dimensional de Sitter space embedded in four-dimensional space-time. Furthermore, Ψ is only invariant under the time-reversal operator [Formula: see text] if γ=2/(2n-1), where n is a positive integer, the first two values n=1,2 defining the high-temperature and low-temperature limits ρ ~ T±2, respectively, of the heterotic superstring theory, which are thus dual to one another in the sense T↔1/2π2α′T.

THE SUPERSTRING WORLD SHEET IN CURVED SPACE–TIME

The heterotic superstring theory is formulated on the world sheet hab(τ,σ), in a flat space–time background [Formula: see text], by combining a 26-dimensional left-moving sector, consisting only of bosonic fields [Formula: see text], with a 10-dimensional right-moving sector, consisting of bosonic fields [Formula: see text] and Majorana–Weyl fermionic fields ψA(τ - σ), the string coordinates in 10-dimensional space–time being the sum [Formula: see text] and the right-moving sector supersymmetric due to the decoupling of left- and right-moving modes for closed strings. Here, we generalize the background to a curved space–time ĝAB(XC). The equation of motion for XA(τ,σ), given by Short, is then a non-linear modification of the linear wave equation □XA = 0 which yields decoupled left- and right-moving sectors in flat space–time. The linearity of the transverse modes can be maintained, however, if the metric, after reduction to four dimensions, is allowed to depend only upon co-moving time t ∝ X0, although it can be anisotropic, gij = gij(X0). Specializing to the isotropic, Friedmann cosmological space–time ds2 = dt2 - r2(t) dx2, where the radius function of the three-space dx2, assumed flat, is r(t), with d ln r/dt = 2/3γt for a perfect-fluid source whose pressure and energy-density are related by p = (γ - 1)ρ, we find that the string coordinates Xα(τ,σ)(α = 1,2,3), multiplied by the overall prefactor r ≡ t2/3γ, are purely oscillatory only in Minkowski space or for the value γ = 2/3. This result is equivalent to requiring that the Nordström energy-density ρ N ≡ (3γ - 2)ρ vanish. The case ρ = 2/3 corresponds to a space–time generated by an ensemble of cosmic strings, including the superstrings themselves in a self-consistent solution containing no other matter, and defines the Milne universe. The string is only world-sheet-supersymmetric if ρ N = 0, that is if ρ = 0 or γ = 2/3.

On non-extensive statistics, chaos and fractal strings

Motivated by the growing evidence of universality and chaos in QFT and string theory, we study the Tsallis non-extensive statistics (with a non-additive q-entropy) of an ensemble of fractal strings and branes of different dimensionalities. Non-equilibrium systems with complex dynamics in stationary states may exhibit large fluctuations of intensive quantities which are described in terms of generalized statistics. Tsallis statistics is a particular representative of such a class. The non-extensive entropy and probability distribution of a canonical ensemble of fractal strings and branes is studied in terms of their dimensional spectrum, which leads to a natural upper cutoff in energy and establishes a direct correlation among dimensions, energy and temperature. The absolute zero temperature (Kelvin) corresponds to zero dimensions (energy) and an infinite temperature corresponds to infinite dimensions. In the concluding remarks some applications of fractal statistics, quasi-particles, knot theory, quantum groups and number theory are briefly discussed within the framework of fractal strings and branes.

A proposal for lowering the coupling unification scale

In the presence of an external electric field an ensemble of short open strings in equilibrium with the heat bath provided by the Euclidean worldvolume of a stack of D-branes undergoes a continuous transition to a long string phase. The transition temperature is at or below the string scale. We point out that this phenomenon provides a simple mechanism within string theory for the lowering of the strong-electro-weak coupling unification scale by up to two orders of magnitude relative to the string scale.

Postmodern string theory: Stochastic formulation.

In this paper we study the dynamics of a statistical ensemble of strings, building on a recently proposed gauge theory of the string geodesic field. We show that this stochastic approach is equivalent to the Carath\'eodory formulation of the Nambu-Goto action, supplemented by an averaging procedure over the family of classical string world-sheets which are solutions of the equation of motion. In this new framework, the string geodesic field is reinterpreted as the Gibbs current density associated with the string statistical ensemble. Next, we show that the classical field equations derived from the string gauge action, can be obtained as the semi-classical limit of the string functional wave equation. For closed strings, the wave equation itself is completely analogous to the Wheeler-DeWitt equation used in quantum cosmology. Thus, in the string case, the wave function has support on the space of all possible spatial loop configurations. Finally, we show that the string distribution induces a multi-phase, or {\it cellular} structure on the spacetime manifold characterized by domains with a purely Riemannian geometry separated by domain walls over which there exists a predominantly Weyl geometry.

BELL FUNCTIONAL INTEGRAL FOR GAS OF STRINGS

We generalize the Bell functional integral representation for a statistical ensemble of strings with two-body interaction.

Dynamical string-parton model for relativistic heavy-ion collisions.

We introduce a dynamical model for the description of hadron-hadron collisions at relativistic energies. The model is based on classical Nambu-Got\ifmmode \bar{o}\else \={o}\fi{} strings. The string motion is performed in unrestricted four-dimensional space-time. The string end points are interpreted as partons which carry energy and momentum. We study ${\mathit{e}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$, e-p, and p-p collisions at various center-of-mass energies. The three basic features of our model are as follows. An ensemble of strings with different end-point dynamics is used to approximately reproduce the valence quark structure functions. We introduce an adiabatic hadronization mechanism for string breakup via qq\ifmmode\bar\else\textasciimacron\fi{} pair production. The interaction between strings is formulated in terms of a quark-quark scattering amplitude and exchange.

Model of evolution of molecular sequences.

A simplified model of the evolution of molecular sequences is proposed. An ensemble of strings is considered that consists of two letters and undergoes random point mutations and natural selections. A set of evolution equations is deduced. From the solution it is found that the first-order (second-order) informational parameters (redundancies) ${\mathit{D}}_{1}$ decrease (${\mathit{D}}_{2}$ increase) in the course of evolution. Furthermore, the statistical correlations of the letters (bases) in the sequences are investigated in detail and the short-distance correlation is demonstrated. These results give a preliminary explanation of some physical aspects in the evolution of nucleic acid sequences.

Strings at the Hagedorn temperature

An ensemble of free strings may be raised to the Hagedorn temperature with a finite energy density. After this, energy added to the system in an attempt to raise its temperature produces long string; the configuration of the rest of the system is determined by an ordinary thermal distribution at the Hagedorn temperature. These conclusions may be altered by new physics at the string scale.