  We investigate the quantum structure of Witten's cubic open bosonic string
field theory by computing the one-loop contribution to the open string tadpole
using both oscillator and conformal field theory methods. We find divergences
and a breakdown of BRST invariance in the tadpole diagram arising from
tachyonic and massless closed string states, and we discuss ways of treating
these problems. For a Dp-brane with sufficiently many transverse dimensions,
the tadpole can be rendered finite by analytically continuing the closed string
tachyon by hand; this diagram then naturally incorporates the (linearized)
shift of the closed string background due to the presence of the brane. We
observe that divergences at higher loops will doom any straightforward attempt
at analyzing general quantum effects in bosonic open string field theory on a
Dp-brane of any dimension, but our analysis does not uncover any potential
obstacles to the existence of a sensible quantum open string field theory in
the supersymmetric case.
