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We investigate some aspects of the self-consistency in the Dyson-Schwingerapproach to both the QED and the self-interacting scalar field theories. Weprove that the set of the Dyson-Schwinger equations, together with theGreen-Ward-Takahashi identity, is equivalent to the analogous set of integralequations studied in condensed matter, namely many-body perturbation theory,where it is solved self-consistently and iteratively. In this framework, wecompute the non-perturbative solution of the gap equation for theself-interacting scalar field theory.

On aspects of self-consistency in the Dyson-Schwinger approach to QED and \lambda (\phi^\star \phi)^2 theories