Transverse Ward-Takahashi Identities and Full Vertices Functions in Different Representation in QED$_3$

We first derive the transverse Ward-Takahashi identities (WTI) of 3-dimensional quantum electrodynamics (QED$_3$) by means of the canonical quantization method and the path integration method, and then prove for the first time that QED$_3$ is strictly solvable based on the transverse WTI and the longitudinal WTI, that is, the full vector and tensor vertices functions can be expressed in term of the fermion propagators in QED$_3$. Further, we discuss the effect of different $\gamma$ matrix representations on the full fermion-boson vertex function. It is found that the full vector vertex function does not depend on the different $\gamma$ matrix representation we use, \textit{i.e.}, it does not depend on whether we use 4 $\times$ 4 representation or 2 $\times$ 2 representation. But the tensor vertex function depends on the representation we use.