Stability of Various Perturbative and Nonperturbative Solutions Appearing in the Theory of Spontaneous Breakdown of Symmetries

The stability of various solutions appearing in the theory of spontaneous breakdown of symmetries is investigated. It is shown for simple models that we can choose a unique solution which is physically acceptable if we require a usual analyticity of the two particle scattering matrix. 1 )- 5 ) have been proposed to interprete the diversity and the complexity of the elementary particle phenomena from a simple dynamical equation. Each solution defines a corresponding vacuum and a Hilbert space. Hilbert spaces thus obtained are mutually orthogonal and we have a vanishing transition probability between any vectors in mutually different spaces. In these theories one usually encounters the problem of how to discriminate a meaningful solution (or Hilbert spaces) from others which is stable against any small perturbation. It has been sometimes argued that this stable solution must have the lowest vacuum energy}) )'his condition does not seem to us conclusive, due to the orthogonality of the different spaces. In the present paper we investigate the stability of various solutions 1n terms of the scattering matrix. In the simple models considered below the scattering matrix has a pole corresponding to a resonance or a bound state which we assign to a physical boson. The behavior of the mass of the bosonic excitation as the function of the coupling constant is of essential importance for the following consideration. We require for the meaningful solution that the mass of the above boson should not be purely imaginary, which is nothing but a usual spectrum condition for