Nonunique solution to the Schwinger-Dyson equations

In principle, a path-integral representation for a quantum field theory uniquely determines all of the Green’s functions of the theory. One possible way to calculate the Green’s functions is to derive from the path-integral representation an infinite set of coupled partial differential equations for the Green’s functions known as the Schwinger-Dyson equations. One might think that all nonperturbative information about the Green’s functions is contained in the Schwinger-Dyson equations. However, we show that while the Schwinger-Dyson equations do determine the weak-coupling perturbation expansions of the Green’s functions, the solution to the Schwinger-Dyson equations is not unique and therefore the nonperturbative content of the Green’s functions remains undetermined. In particular, one cannot use the Schwinger-Dyson equations to compute high-temperature or strong-coupling expansions.