  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.
