We calculate the effective action for Quantum Electrodynamics (QED) in D=2,3
dimensions at the quadratic approximation in the gauge fields. We analyse the
analytic structure of the corresponding nonlocal boson propagators
nonperturbatively in k/m. In two dimensions for any nonzero fermion mass, we
end up with one massless pole for the gauge boson . We also calculate in D=2
the effective potential between two static charges separated by a distance L
and find it to be a linearly increasing function of L in agreement with the
bosonized theory (massive Sine-Gordon model). In three dimensions we find
nonperturbatively in k/m one massive pole in the effective bosonic action
leading to screening. Fitting the numerical results we derive a simple
expression for the functional dependence of the boson mass upon the
dimensionless parameter e^{2}/m .Comment: 10 pages, 2 figure
