Detailed Wave Equation and Dual Amplitude: Relativistic Quantum Dynamics of String Model of Hadrons

Abstract

Relativistic quantum dynamics of a one-dimensional elastic continuum (string) is represented by a “detailed wave equation” which is defined at every point of the string and has local coupling to an external field at the end of string. The actual form of this wave equation is severely determined from its self-consistency, which is guaranteed by the existence of a closed algebra for the “invariant hamiltonian density operator”. The algabra in turn directly represents the “gauge invariance” of the theory. The invariant hamiltonian density operator consists of real and imaginary parts, of which the latter represents nonlocal interaction inside the string and exhibits a remarkable Hilbert transform relation. The detailed wave equation is brought to the set of the global wave equation corresponding to the wave equation in the usual sense and an infinite number of subsidiary conditions which work to suppress unphysical states. The unconventional definition of normal model operators {Cµr} enables a uniform treatment of external and internal motions and the factorization of all wave operators into convolutions. The theory leads to dual amplitude for a particular value of intercept. A completely equivalent formulation of theory with the introduction of a proper-time like parameter is also given, which supplies further insights. Finally speculations are made for generalizing the model to more realistic cases.