Monopole action from vacuum configurations in compact QED

It is possible to derive a monopole action from vacuum configurations obtained in Monte-Carlo simulations extending the method developed by Swendsen. We apply the method to compact QED both in the Villain and in the Wilson forms. The action of the natural monopoles in the Villain case is in fairly good agreement with that derived by the exact dual transformation. Comparing the monopole actions, we find (1) the DeGrand-Toussaint monopole definition may be useful for ${\beta }_V$ larger than about 0.5, (2) the Villain model well approximates the Wilson one for $\beta$ smaller than ${\beta }_c$ and (3) in the Wilson action the monopole condensation occurs in the confinement phase and ${\beta }_c$ may be explained by the energy-entropy balance of monopole loops like in the Villain case.