  We examine the fluctuations around a Dp-brane solution in an unstable D-brane
system using boundary states and also boundary string field theory. We show
that the fluctuations correctly reproduce the fields on the Dp-brane. Plugging
these into the action of the unstable D-brane system, we recover not only the
tension and RR charge, but also full effective action of the Dp-brane exactly.
Our method works for general unstable D-brane systems and provides a simple
proof of D-brane descent/ascent relations under the tachyon condensation. In
the lowest dimensional unstable D-brane system, called K-matrix theory,
D-branes are described in terms of operator algebra. We show the equivalence of
the geometric and algebraic descriptions of a D-brane world-volume manifold
using the equivalence between path integral and operator formulation of the
boundary quantum mechanics. As a corollary, the Atiyah-Singer index theorem is
naturally obtained by looking at the coupling to RR-fields. We also generalize
the argument to type I string theory.
