  We explore the sign problem in strongly coupled lattice QED with one flavor
of Wilson fermions in four dimensions using the fermion bag formulation. We
construct rules to compute the weight of a fermion bag and show that even
though the fermions are confined into bosons, fermion bags with negative
weights do exist. By classifying fermion bags as either simple or complex, we
find numerical evidence that complex bags with positive and negative weights
come with almost equal probabilities and this leads to a severe sign problem.
On the other hand simple bags mostly have a positive weight. Since the complex
bags almost cancel each other, we suggest that eliminating them from the
partition function may be a good approximation. This modified partition
function suffers only from a mild sign problem. We also find a simpler model
which does not suffer from any sign problem and may still be a good
approximation at small and intermediate values of the hopping parameter. We
also prove that when the hopping parameter is strictly infinite all fermion
bags are non-negative.
