  The zero modes and zero resonances of the Dirac operator $H=\alpha\cdot D +
Q(x)$ are discussed, where
  $\alpha= (\alpha_1, \alpha_2, \alpha_3)$ is the triple of $4 \times 4$ Dirac
matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a
$4\times 4$ Hermitian matrix-valued function with
  $| q_{jk}(x) | \le C < x >^{-\rho} $, $\rho >1$.
  We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$
and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that
$H$ has no zero resonance if $\rho > 3/2$.
