Complete Set of Dispersion Relations for a Class of Fixed-Source Meson Theories

The structure of the transition matrices for all processes that can occur for a class of fixed-source meson theories is studied. The model consists of a scalar meson field coupled to an extended source in such a way that any finite number of quanta can be emitted or absorbed at a given time (multiple vertices), but that all interactions are restricted to be $S$ wave in nature. The general reaction matrix element for $m$ incident and $n$ emergent particles contains many terms describing sequences of independent processes, which must be removed before one obtains a proper object for studies of a dispersion-theoretic nature. It is shown that the ratio of the residual matrix element to a suitable product of source functions possesses those analytic properties, as a function of the total energy of the system, which permit dispersion relations to be stated. Other than for the elastic scattering amplitude the latter make reference to values of the amplitude in a nonphysical energy region. In conjunction with a suitably generalized unitarity condition, however, the scheme, when viewed as a set of coupled integral equations, can be solved by successive approximations in terms of a number of arbitrary constants, essentially equal to the number of coupling constants in the original Hamiltonian (actually one fewer). Nevertheless, it is pointed out that the scheme does not admit a unique solution, and this is illustrated physically by exhibiting an extended class of Hamiltonians which yield the same dispersion relations, but which, as a class, contain more coupling constants than make their appearance in the dispersion relations. Physically, these are connected with the occurrence of resonance scattering.