Commutation Rules and Spurious Eigenstates in Soluble Field Theories

A study is made of the exactly soluble field theories which are characterized by Hamiltonians quadratic in the field variables. As an example of such a theory, a model describing the electric dipole interaction of photons and a nonrelativistic, harmonically bound electron is studied explicitly. For the cases when the cutoff is sufficiently large to admit the "runway solutions," it is necessary in order to obtain consistency and a sensible physical interpretation that the theory be reformulated by a unique modification of the equaltime commutation rules of the field operators. The problems which arise here in connection with the "run-away solutions" are closely related to the troubles of ghost states and negative transition "probabilities" which have been demonstrated or suggested to exist in other theories. It is hoped, therefore, that the procedure of reformulation required here may be a guide for the eventual resolution of the ghost-state problem should it be demonstrated that such problems actually exist in the physical, relativistic field theories.