  The path integral by which quantum field theories are defined is a particular
solution of a set of functional differential equations arising from the
Schwinger action principle. In fact these equations have a multitude of
additional solutions which are described by integrals over a complexified path.
We discuss properties of the additional solutions which, although generally
disregarded, may be physical with known examples including spontaneous symmetry
breaking and theta vacua. We show that a consideration of the full set of
solutions yields a description of phase transitions in quantum field theories
which complements the usual description in terms of the accumulation of
Lee-Yang zeroes. In particular we argue that non-analyticity due to the
accumulation of Lee-Yang zeros is related to Stokes phenomena and the collapse
of the solution set in various limits including but not restricted to, the
thermodynamic limit. A precise demonstration of this relation is given in terms
of a zero dimensional model. Finally, for zero dimensional polynomial actions,
we prove that Borel resummation of perturbative expansions, with several
choices of singularity avoiding contours in the complex Borel plane, yield
inequivalent solutions of the action principle equations.
