  The Bethe-Salpeter equation (BSE) for bound states in scalar theories is
reformulated and solved in terms of a generalized spectral representation
directly in Minkowski space. This differs from the conventional approach, where
the BSE is solved in Euclidean space after a Wick rotation. For all but the
lowest-order (i.e., ladder) approximation to the scattering kernel, the {\it
naive} Wick rotation is invalid. Our approach generates the vertex function and
Bethe-Salpeter amplitude for the entire allowed range of momenta, whereas these
cannot be directly obtained from the Euclidean space solution. Our method is
quite general and can be applied even in cases where the Wick rotation is not
possible.
