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Based on the path integral formalism, we rederive and extend the transverseWard-Takahashi identities (which were first derived by Yasushi Takahashi) forthe vector and the axial vector currents and simultaneously discuss thepossible anomaly for them. Subsequently, we propose a new scheme for writingdown and solving the Schwinger-Dyson equation in which the the transverseWard-Takahashi identity together with the usual (longitudinal) Ward-Takahashiidentity are applied to specify the fermion-boson vertex function. Especially,in two dimensional Abelian gauge theory, we show that this scheme leads to theexact and closed Schwinger-Dyson equation for the fermion propagator in thechiral limit (when the bare fermion mass is zero) and that the Schwinger-Dysonequation can be exactly solved.Comment: 22 pages, latex, no figure

Longitudinal and Transverse Ward–Takahashi Identities, Anomaly and Schwinger–Dyson Equation