The Ruijgrok-Van Hove model of field theory in terms of "dressed" operators

Abstract

A “dressing” transformation for the Ruijgrok-Van Hove model of field theory is given. By means of this transformation the Hamiltonian is expressed in terms of “dressed” operators only. It is shown that the Hamiltonian depends explicitly on the cut-off. The solution of the eigenvalue problem is given for one meson present in addition to the dressed nucleon. It also depends on the cut-off. A ghost state with negative norm as well as a bound state with positive norm are found, both belonging to the same energy value. The solution for one nucleon and several mesons is outlined. A detailed discussion of the metric in Hilbert space is given. It is shown that the “bare” particle representation is not adequate to describe the system considered. The most suitable representation, from physical and mathematical point of view, seems to be the asymptotic stationary states representation, introduced by Van Hove. A test as to the merits of these two representations is given in the case of indefinite metric; here the “bare” particle representation can no longer be used to describe the system, in contrast to the asymptotic stationary states representation which can be employed with success.