Dynamical Mass in Strongly Coupled QED

We review dynamical mass generation in three and four dimensional Ableian gauge theories. The basic approach analyzed here is to solve Schwinger-Dyson equations in some approximations. In both cases, the coupling has to be larger than a critical value to have dynamical symmetry breaking. In $QED_4$, chiral symmetry is spontaneouly broken if $\alpha>\alpha_c (\simeq \pi/3)$ in ladder approximation. The electron becomes massive, and is realized as Skyrmion in the low energy effective action. In $QED_3$, electron gets dynmical mass in $1/N$ expansion if $N<N_c(\simeq 32/\pi^2)$. With a Chern-Simons term, parity tends to break maximally in $QED_3$. The quntum phase structure of $QED_3$ with or without a Chern-Simons term is peculiar.