Massive Ends Research Articles

Analogs of Fourier series for a relativistic string model with massive ends

A description of the motions of a relativistic string with massive ends (for the model with a finite mass on the first end and an infinite mass on the second end) is proposed. This approach uses the expansion of the string world surface into a series that is not reduced to the ordinary Fourier series due to the nonlinearity of the problem. The state equation of the string is derived from the mass shell condition for its end. The string motions are classified, allowing linearization of the boundary condition by a natural parametrization of the trajectory of the moving end. The set of such world surfaces is shown to be limited; for the special cases of 2+1 - and 3+1-dimensional Minkowski spaces, all of them reduce to a helicoid.

Initial?boundary-value problem for the relativistic string with massive ends

It is shown that the initial-boundary-value problem for a relativistic string with masses at the ends can be solved for the most general form of specification of the initial position and initial velocities of the points of the string. An investigation is made of the connection between the freedom in the parametrization of the initial curve and the reparametrization invariance preserving linearity of the equations of motion of the string. The posed problem is solved by extending the solution determined by the initial conditions from the restricted initial region to the entire world surface of the string by means of boundary conditions of various types: the mass at a given end is equal to zero, is infinitely large, or is finite. In the last case it is shown that the problem of the extension reduces to the solution of a normal system of ordinary differential equations. Specific examples are considered.

The Casimir energy of the rigid string with massive ends

The Casimir energy in the rigid string with massive ends is investigated. This system models the flux tube of finite thickness that connects the massive quarks. The string stiffness and quark mass give additive contributions to the Casimir energy, the second contribution being the same as in the Nambu-Goto string with massive ends. The string stiffness results in an additional positive contribution to the Casimir energy, however it alone does not compensate the negative Casimir energy in the Nambu-Goto string. Only common consideration of the string thickness and the quark mass enables one to get the positive Casimir energy, i.e. to remove the tachyon from the string spectrum.

Is there a tachyon in the Nambu-Goto string with massive ends?

The dependence of Casimir energy on quark mass is investigated in the model of relativistic strings with massive quarks attached to the ends. The quark dynamics are treated in the nonrelativistic approximation, and the equations of motion and boundary conditions are linearized. The Casimir energyE as a function of quark massm is found by two methods (numerically and analytically). Different subtraction procedures for both approaches result in different functional dependences ofE onm. But both cases have values ofm for which the Casimir energy is definitely positive. The sign of this energy is known to coincide with the sign of the squared mass of the ground state in the string spectrum. Hence, the obtained result indicates that it is possible at least in principle to solve the tachyon problem in the model of relativistic strings with massive ends.

New formulation of the classical dynamics of the relativistic string with massive ends

Dynamic equations in the theory of a relativistic string with point masses at the ends are formulated only in terms of geometric invariants of the world trajectories of the massive ends of the string (curvature ka and torsion kappa a of the trajectories). These characteristics allow the authors to reproduce the string world surface up to its position in Minkowski space E1. The torsions kappa a( tau ), a=1, 2 obey a system of second-order differential equations with delay describing the retardation effects of the interaction of masses through the string, the ka being constants. A new particular solution to these equations is found that corresponds to periodic torsions.

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 Bi-Local Model and String Model

The mechanical models of the hi-local fields and the string models are discussed in the formal point of view. Since the formulation of the extended particle pictures has no unique way, there are various formulations even in the simplest hilocal models. We confine ourselves mainly in the formulations by following which the interactions have been investigated. The scattering amplitudes and the form factors are discussed qualitatively. Some comments on the subsidiary conditions suppressing the relative time freedom are given by following Dirac's formalism. A short discussion of the hi-local model with spin is also shown by introducing Grassmann variables as classical variables corresponding to spin freedom. As to the string model, we have discussed various interesting but unsolved problems; namely, the electromagnetic interactions, the string with massive ends, the spinning string and so on. We have proposed tentative prescriptions for treating some of these problems. We also discuss the different point of view concerning the string model. That is, we show that the difference equation furnishes us the same result as that given by the string model.

Some solutions of boundary conditions for relativistic string with massive ends

A string model of meson with massive quarks at ends is studied. As the action of the quark masses, we take (i) the length of world trajectories of the string ends (Model I), and also (ii) the integral of the geodesic curvature of world trajectories of the string ends (Model II). For these models, particular solutions of nonlinear boundary conditions are derived. For Model II the nonrelativistic limit is analysed.

Some solutions of the equations of motion of the relativistic string with massive ends

The classical theory is discussed for the relativistic string with point masses at its ends. The dynamical equations are solved for the class of motions of this system when the time evolution parameter τ is the proper time of both massive string ends. In this case the solution of the boundary equations is given by the almost periodic functions. Constraints on the normal modes resulting from the orthonormal gauge conditions differ essentially from the Virasoro ones. Incidentally one obtains an exact solution for the half-infinite string with mass at one end. It is also proved that the exact solution for the string with massive ends cannot be a periodic function.

Relativistic string with massive ends

Relativistic string with massive ends