Relativistic String Research Articles

Quantization of a constituent-string model in four dimensions.

In superstring theories, anomalies and mathematical divergences (though they cancel) and the necessity to quantize in ten dimensions are attributable to the strings being extended physical objects, i.e., strings consisting of an infinite number of ``beads.'' Therefore, it is interesting to consider a ``string'' generated by a single ``bead'' which tends toward oscillation with infinite velocity between two spatial points in some Lorentz frame. This picture has been realized classically in earlier work on the constituent model. The Lagrangian is given in terms of a set of generalized harmonic-oscillator normal coordinates and displays a gauge symmetry due to parametric invariance. The model is quantized by going to the Hamiltonian formalism (using constraint theory) and assuming the usual quantum conditions for the normal coordinates and their conjugate momenta. In analogy to the relativistic free string model, a set of orthonormal invariant supplementary conditions are applied. Constituents are defined in terms of the normal coordinates and form two-body composite particles. For any composite (observable) particle of real mass, it is shown that in its rest frame the time oscillations of its constituents vanish and its internal orbital angular momentum is a linear function of the (mass${)}^{2}$. Only composites of equal mass can couple in this model, thereby eliminating couplings between physical and unphysical masses. The invariant scattering amplitude for two-particle (meson) scattering is calculated and is a crossing-symmetric function of a linear trajectory function \ensuremath{\alpha}(z)=\ensuremath{\alpha}'z-${\ensuremath{\alpha}}_{0}$. .AE

Interacting bosonic strings in subcritical dimensions

Interaction theory for relativistic bosonic strings in spacetime dimensions below the critical value 26 is formulated using BRST techniques with an extra scalar field. One-loop zero-point amplitudes for closed strings are modular invariant. For a free scalar field, vertex operators are constructed leading to, e.g., the old dual N-tachyon tree amplitudes in D<26. The N-tachyon one-loop expressions contain closed string poles for open strings, and are modular invariant for closed strings. However, the threshold cuts are wrong in D<25. Only for D=25 do the considered vertex operators lead to consistency.

Quantization of non-Abelian antisymmetric tensor field

We have given a detailed derivation of a covariant expression for the S matrix of an antisymmetric tensor field on the basis of the canonical formulation of the theory. An important technical point in this derivation was the introduction of an additional gauge field with a suitable transformation law. A similar trick can be used in other theories with functionally dependent constraints, in particular, as we hope, in second-quantized models of relativistic strings. Our treatment also shows that the use of the BRST quantization method necessarily requires verification of the fulfillment of all the physical conditions, since otherwise physical unitarity of the theory may be lost.

The spectrum-generating groups program and the string

Schrodinger's approach was analytical, but it is equivalent to an algebraic treatment. We review the evolution of group theory as a physical tool and its application to the Hilbert space of Schrodinger's eigenstates. Special emphasis is put on recent results relating to the relativistic quantized string.

Geometrical method of solving the boundary-value problem in the theory of a relativistic string with masses at its ends

A differential-geometric formulation of the dynamics of a relativistic string with masses at its ends is considered in the Minkowski space E/sub 2//sup 1/. The surface swept out by the string is described by differential forms and is bounded by two curves - the worldlines of its massive ends. These curves have a constant geodesic curvature, and their torsion is determined only up to an arbitrary function on the interval (0, 2/pi/). Equations are obtained that determine the world surface of the string as a function of the curvature and torsion of the trajectories of its massive ends. For the choice of the constant torsions for which the mass points move along helices, the surface of the relativistic string is a helicoid.

On the covariant quantization of the free string:: The conformal structure

We discuss a lagrangian approach to the first quantization of the relativistic string. The conformal properties of the theory are exhibited. We identify the basic fields of the theory, and in particular the Faddeev-Popov ghost fields, as primary conformal fields whose conformal weights are computed.

Real classical strings

A general solution is constructed for the equations of motion of the classical relativistic string. All allowable string configurations are encompassed, apart from a special set of measure zero characterized by the vanishing of a certain invariant for which the motion of the string is restricted to a time-like hyperplane. The solution is given in terms of an essentially freely specifiable curve lying in the hypersurface PN, the space of ‘null twistors’.

Classical strings in ten dimensions

This article analyses the motion of a classical relativistic string in flat complex ten-dimensional space-time. A general solution is presented for the equations of motion. The solution is given in terms of essentially freely specifiable functions. In deriving these results extensive use is made of spinors for ten dimensions, the basic properties of which are described in some detail. In particular, a significant role is played by those spinors in ten dimensions that satisfy the ‘purity’ property of E. Cartan. These constitute an eleven-dimensional algebraic variety V in the sixteen-dimensional linear space of reduced (Weyl) spinors for the group SO(10). A general classical relativistic string in ten dimensions can be represented by means of a set of arbitrarily specifiable twice-differentiable curves in V . As a by-product of the investigation a general solution is also given for the equations of motion of a classical relativistic string in eight­-dimensional space-time. Seldom have thinkers become so absorbed in revery, or so far estranged from reality, as to imagine for our space a number of dimensions exceeding the three of the given space of sense ...(Ernst Mach, 1906)

A note on Fujikawa's method of determining the critical dimension of the relativistic string

Abstract We reconsider Fujikawa's derivation of the critical dimension of the relativistic string in the path-integral formulation. It is shown that the correct prescription for choosing the functional measure is obtained by requiring standard BRST invariance without modifications and that Fujikawa's choice of measure is not unique. We find a one-parameter family of BRST-invariant measures even after fixing a gauge for local Weyl rescalings. Gauge independence of the resulting theory is demonstrated in the critical number of dimensions.

Investigations On the Full and Reduced Salpeter Equation for the Quark-Anti Quark Bound States

The solutions for both the full Salpeter equation and the reduced Salpeter equation for the quark-anti quark bound states are investigated. General solutions and the solutions in the nonrelativistic approximation are discussed. Differences between two kinds of Salpeter equations are emphasized. It is especially noted that the observed mass splitting could be accounted for by using the full salpeter equation with a smaller vector Coulomb potential -α/r, where , which was obtained in very recent lattice gauge calculations and might be associated with the zero-point transverse fluctuations of the relativistic string.

Calculation of static interquark potential in a string model in a timelike gauge

The quantum theory of a relativistic string with fixed ends is constructed using a timelike gauge. A gauge condition is introduced into the reparametrization-invariant string action. In the theory, there are no restrictions on the space-time dimension and there are no tachyon states. The constraints and gauge conditions are satisfied in the quantum case only on the average, at the level of the matrix elements between the physical state vectors. The string generates a static potential V(R) = ..sqrt gamma../sup 2/R/sup 2/ + epsilon /sub 0/, where the constant epsilon/sub 0/ is a free parameter of the theory which can be determined from experiment.

Quantum strings

The theory of the classical string and the method of canonical quantization are reviewed. The nonrelativistic string is then quantized using the canonical quantization procedure. A number of properties of quantized strings, such as quantized normal modes of vibration and ground state fluctuations, are discussed. The quantized nonrelativistic string is compared to quantized fields and to the strings in the recent superstring theories. The theory of relativistic strings is reviewed.

BRST quantization of a bilocal model inspired by the relativistic string

A bilocal model derived projecting out a particular state of motion of the relativistic string is considered. In principle the model describes integer spin massless states. However, its BRST quantization is shown to be consistent only for three values for the spacetime dimensions, and the corresponding solutions describe a rank-2 symmetric tensor ( D = 2), a vector ( D = 4), and a scalar ( D = 6).

A new model for hadronization and e +e − annihilation

A new string/cluster model for e +e − annihilation is presented. Event evolution is described using a simple, 3-stage formalism: (i) coherent parton shower generation, (ii) evolution of elementary strings formed from the parton final state, and (iii) parameterized hadronization of small mass clusters formed during string evolution. Significant improvements over previous cluster models arise from the use of full, massless relativistic string motion in the second stage of the model, so that the formalism is naturally insensitive to soft perturbative radiation. Parameter fitting and sensitivity of model predictions to variations in the cluster formation parameters are discussed in detail. The model is shown to reproduce a variety of e +e − annihilation data over a wide range of E CM. Specific suggestions are made for future improvements in the perturbative QCD and cluster decay stages of the model.

Anomalous BRST quantization

A quantum gauge theory is anomalous within the BRST quantization if there does not exist any conserved, nilpotent BRST charge. What is to be called the BRST charge is then either a nilpotent of a conserved charge. It is proposed that a particular conserved charge may be used to consistently quantize an anomalous gauge theory. Such a charge always requires a consistent anomalous constraints algebra, which also makes the formalism applicable to systems with second class constraints. Some simple general properties of this approach is demonstrated. The example of the previously solved free anomalous, relativistic string theory is treated in detail.

The Lund model

The Lund model, inspired by quantum chromodynamics, has provided a very promising approach to the dynamics of quark and gluon interactions. Starting with a brief reprise of basic concepts in relativity, quantum mechanics of fields and particle physics, this 1998 book discusses: the dynamics of the massless relativistic string; confinement; causality and relativistic covariance; Lund fragmentation processes; QED and QCD Bremsstrahlung; multiplicities and particle-parton distributions. Throughout the book, theory is confronted with current experimental data, and implications for future experiments are also considered. The book also explores the relationships between the Lund model and other models based on field theory (the Schwinger model, S-matrix models, light-cone algebra physics and variations of the parton model), and models based on statistical mechanics (the Feynman-Wilson gas, scaling, iterative cascade models).

World sheets for world sheets

In this paper I suggest that the two-dimensional world sheet underlying the dynamics of relativistic strings may itself emerge from an approximation to a two-dimensional string theory. Coordinates that represent space-time and internal symmetries should emerge as wave functions of the massless states of that theory. This is motivated by studying various string theories with two-dimensional target spaces and internal symmetry which are related to generalized nonlinear sigma models.

A framework for the fragmentation of the massless relativistic string

The massless relativistic string is used to model QCD flux tubes in the process of hadronization. The solutions to the equations of motion of strings are interpreted as superpositions of momentum currents which circulate on domains with the topology of a circle. An equivalence between string fragmentation and cutting and pasting circular domains is demonstrated.

Superstrings: a new approach to a unified theory of fundamental interactions

The basic ideas of the superstring approach in the physics of elementary particles are presented at a rather simple theoretical level. The theory of interacting superstrings with dimensions of the order of the Planck dimension is presently regarded as a fundamentally new and extremely promising approach to a unification of all the fundamental interactions (strong, electromagnetic, weak, and gravitational) in a common theoretical framework. Although this field is still in a stage of rapid development, several results of definite interest have been obtained. The classical dynamics and the quantum dynamics of boson and spin strings are described. Such strings arise primarily in hadron physics. How a critical dimensionality of space-time and tachyon states arise upon the quantization of string models is demonstrated. A superstring is a supersymmetric generalization of very simple models of a relativistic string. The total action of a superstring is examined. The field theory of superstrings is discussed, as are various methods for introducing internal symmetries in string theories. The low-energy (field) limit in superstring theory is studied. The cancellation of anomalies and the problem of divergences are discussed. Compactification in superstring theories is discussed, as is a phenomenology which arises here for the modern physics of elementary particles. Some cosmological consequences of the superstring approach are outlined.

On the Prolongation Structure Approach to the Relativistic String in Curved Space-Time

We have deduced the inverse problem associated with the eqation of motion of a relativistic string in a space-time of constant curvature. We have shown that it is possible to deduce the required Lax pair by the usual approach of Wahlquist and Estabrook, and also by the method of C. C. ideal as suggested by Harrison. Lastly we discuss and approach which does not at all use the prolongation variables but is helpful in obtaining the closure of the Lie algebra throuh the use of Holonomy algebra, and hence fixes up the proper representation of the Lax pair.