  In this work we apply Thompson's method (of the dimensions) to study the
quantum electrodynamics (QED). This method can be considered as a simple and
alternative way to the renormalisation group (R.G) approach and when applied to
QED lagrangian is able to obtain the running coupling constant behavior $\alpha
(\mu)$, namely the dependence of $\alpha$ on the energy scale. We also obtain
the dependence of the mass on the energy scale. The calculations are evaluated
just at $d_c=4$, where $d_c$ is the upper critical dimension of the problem, so
that we obtain logarithmic behavior both for the coupling $\alpha$ and the mass
$m$ on the energy scale $\mu$.
