Theta vacua and boundary conditions of the Schwinger-Dyson equations

Quantum field theories and Matrix models have a far richer solution set than is normally considered, due to the many boundary conditions which must be set to specify a solution of the Schwinger-Dyson equations. The complete set of solutions of these equations is obtained by generalizing the path integral to include sums over various inequivalent contours of integration in the complex plane. We discuss the importance of these exotic solutions. While naively the complex contours seem perverse, they are relevant to the study of theta vacua and critical phenomena. Furthermore, it can be shown that within certain phases of many theories, the physical vacuum does not correspond to an integration over a real contour. We discuss the solution set for the special case of one component zero dimensional scalar field theories, and make remarks about matrix models and higher dimensional field theories that will be developed in more detail elsewhere. Even the zero dimensional examples have much structure, and show some analogues of phenomena which are usually attributed to the effects of taking a thermodynamic limit.