  The path decomposition expansion is a path integral technique for decomposing
sums over paths in configuration space into sums over paths in different
spatial regions. It leads to a decomposition of the configuration space
propagator across arbitrary surfaces in configuration space. It may be used,
for example, in calculations of the distribution of first crossing times. The
original proof relied heavily on the position representation and in particular
on the properties of path integrals. In this paper, an elementary proof of the
path decomposition expansion is given using projection operators. This leads to
a version of the path decomposition expansion more general than the
configuration space form previously given. The path decomposition expansion in
momentum space is given as an example.
