Problem of Ghost States in Field Theories

Abstract

It is shown that the ghost problem in the Lee model cannot be circumvented by using a renormalized integral equation as a basis instead of the Hamiltonian formulation. This equation (the Low equation) possesses no solutions for coupling constants larger than the critical value, and an infinite family of solutions for coupling constants smaller than the critical value. The Low equation for the Lee model is also mathematically equivalent to an exact integral equation for the boson propagator in relativistic field theory, and permits one therefore to describe the properties of possible ghosts in a relativistic theory in terms of already known properties of Lee-model ghosts. There exists an interesting symmetry between this integral equation and its solution, both of which have the form of dispersion relations. The excellent experimental tests of quantum electrodynamics provide no evidence that the theory is free of the ghost problem.