Non Trivial Saddle Points and Band Structure of Bound States of the Two Dimensional O(N) Vector Model

We discuss O(N) invariant scalar field theories in 0+1 and 1+1 space-time dimensions. Combining ordinary “Large N” saddle point techniques and simple properties of the diagonal resolvent of one dimensional Schrödinger operators we find exact non-trivial (space dependent) solutions to the saddle point equations of these models in addition to the saddle point describing the ground state of the theory. We interpret these novel saddle points as collective O(N) singlet excitations of the field theory, each embracing a host of finer quantum states arranged in O(N) multiplets, in an analogous manner to the band structure of molecular spectra. We comment on the relation of our results to the classical work of Dashen, Hasslacher and Neveu and to a previous analysis of bound states in the O(N) model by Abbott. ∗ Supported by a post doctoral Rothchild Fellowship and in part by the Robert A. Welch Foundation and NSF Grant PHY 9009850.