Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

The quantization of the relativistic string

The classical field-dependent parametrization covariant Hamiltonian formulation of the open and the closed string is discussed. The formalism is not applicable to the open string. A conformally covariant formalism is developed for the open string. The Rohrlich gauge conditions are justified and applied. The parametrization of classical solutions is not uniquely fixed; the generators of rigid time translation in the parameter space remain first class. The constraints and gauge conditions are taken into account in the quantum theory as conditions on physical states. The required invariance of physical states under rigid displacement of parameter time leads to a mass superselection rule. The set of physical string quantum states is analogous to the set of states constructed by Di Vecchia, Del Guidice, and Fubini. A recursive construction is presented which permits the counting of physical states of any given mass, spin, and parity. Physical states lie on linearly rising Regge trajectories with one universal slope. The intercept of the leading trajectory is constrained only by the requirement that there be no tachyonic physical states. The quantization is carried out in four space-time dimensions.Supported by NSF Grant No. MPS74-15246 and DFG/Az 287/6. A portion of this work has been accepted by Syracuse University in partial fulfillment of the requirements of the doctorate degree.     

The quantization of a massless relativistic string in a time-like gauge

The massless relativistic string is quantized in a time-like orthonormal gauge. The results are completely equivalent to those obtained in the light-like case. In particular, a strictly canonical quantization procedure succeeds only when the dimension of space-time is 26. The discussion is based on Dirac's formalism for contrained systems and the introduction of classical variables analogous to those defined by Del Giudice, Di Vecchia and Fubini in the dual model.     

Infinite relativistic string with a point-like mass

Exact classical solutions of the equations of motion for the infinite relativistic string loaded with a point-like mass are obtained in terms of the Fourier integrals. The gauge when the time evolution parameter is proportional to the proper time of a point-like particle is used. The gauge conditions in this case are the second-class constraints. So in the quantum case the commutators of the dynamical variables have to be defined by the Dirac brackets.     

Solution of the relativistic string problem

The quantum dynamics of the free Nambu-Nielson-Susskind string is known to be consistent only in s + 1 dimensional Minkowski space when s = 25 and the intercept α₀ = 1. Ad hoc modifications are required to allow s < 25. The source of this difficulty is pointed out and the system is solved covariantly and ghost free for any value of s > 1 without ad hoc modifications. The intercept value α₀ = 1 is excluded, but α₀ can be fitted to the experimental value. There are only transverse modes, and linearly rising Regge trajectories result as before.     

Non-archimedean string dynamics

Explicit formulas for the N-point tree amplitudes of the non-archimedean open string are derived. These amplitudes can be generated from a simple non-local lagrangian involving a single scalar field (the tachyon) in ambient space-time. This lagrangian is studied and is found to possess a tachyon free vacuum with no “particles” but with soliton solutions. The question of generalizing the adelic product formular to N-point amplitudes is taken up. The infinite product of 5-point amplitudes is shown to converge in a suitably chosen kinematic region whence it can be analytically continued. Though the precise form of the product formula for the 5-point (and N-point)amplitudes is not found, it is shown that the product is not equal to one as it is for the 4-point amplitudes but rather involves the famous zeros of the Riemann zeta function. Chan-Paton rules for non-archimedean open strings are given. A string over the (global) field of rational numbers is constructed. Other problems that are addressed are the introduction of supersymmetry, the nature of a p-adic string lagrangian, and the possibility of strings over other locally compact fields.     

The synchronization problem in covariant relativistic dynamics

In the classical Stueckelberg-Horwitz-Piron relativistic Hamiltonian mechanics, a significant aspect of evolution of the classical n-body particle system with mutual interaction is the method by which events along distinct particle world lines are put into correspondence as a dynamical state. Approaches to this procedure are discussed in connection with active and passive symmetry principles.     

Treatment of the classical relativistic string in any orthonormal gauge

It is shown that a certain set of gauge invariant functions are, for an appropriate choice of a parameter on which they depend, equal to the Fourier components of the classical relativistic string in any orthonormal gauge. These variables are natural generalizations of the classical DDF operators recently introduced by Goddard, Hanson and Ponzano. The Poisson algebra of the relativistic string in any orthonormal gauge (including the proper time gauge) is written down. Application to quantization is briefly discussed.     

Generalization of the relativistic string model in the geometrical approach

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation – = Hsh. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.     

Generalized relativistic dynamics in a non-inertial reference frame

Generalized relativistic field equations have been derived for dynamics in a non-inertial reference frame interpreted as a Finsler space where events are specified by both spacetime coordinates and corresponding velocities (tangent vectors). The field equations follow in two alternative forms from exact general conservation laws derived through application of Cartan covariant differentiation within the framework of Finsler geometry. Velocity-dependent (curvature) terms in the field equations can account for the anisotropy of the gravitational field, together with the associated acceleration and expansion of the universe.     

Tachyon dynamics in string theory

We summarize the recent developments in the study of time dependent solutions describing the rolling of a tachyon on a non-BPS D-brane system.     

狭义相对论动力学的逻辑体系

阐述了物理逻辑体系的构成,依据马赫动力学逻辑体系的基本思想,给出了狭义相对论动力学的一个严格的逻辑体系.其中给出了一般情况(包括高速和低速情况)下质量和力的操作型定义,并对马赫提出的力的定义———“质量与加速度的乘积”进行了批评和修正.对另外一些相关问题,如牛顿第二定律的实验检验也作了讨论.     

Particle dynamics in a relativistic invariant stochastic medium

The dynamics of particles moving in a medium defined by its relativistically invariant stochastic properties is investigated. For this aim, the force exerted on the particles by the medium is defined by a stationary random variable as a function of the proper time of the particles. The equations of motion for a single one-dimensional particle are obtained and numerically solved. A conservation law for the drift momentum of the particle during its random motion is shown. Moreover, the conservation of the mean value of the total linear momentum for two particles repelling each other according to the Coulomb interaction also follows. Therefore, the results indicate the realization of a kind of stochastic Noether theorem in the system under study.     

Classical geometry of bosonic string dynamics

We develop a treatment of bosonic strings on a general curved background in which the volume element and the coordinates of the worldsheet are related in a similar way as canonically conjugate quantities in mechanics. The resultant formalism is a particular variant of the multi-phase-space approach to classical field theory put forward by Kijowski, Tulczyjew, and others. We study conservation laws within this framework and find that all conserved quantities are related to point symmetries, i.e., isometries of the underlying spacetime. Thus, the symmetries of relativistic mechanics coming from Killing tensors have no analogue here. We furthermore deduce from the present scheme the covariant version of the usual phase space.     

A relativistic quantum dynamical model based on the classical string

Motivated by geometrical considerations of the string model, we construct a dynamical theory of composite particles that is Poincaré invariant in four dimensions, with no tachyons and no ghosts. The resulting physical states lie on indefinitely rising trajectories.