  It is shown that the usual expression for a Feynman diagram in terms of the
Feynman propagator $\Delta_F(x-y)$ can be replaced by an equivalent expression
involving the positive-energy on-shell propagator $\Delta^+(x-y)$, supplemented
by appropriate functions associated with time-ordering. When this alternate way
of expressing a Feynman diagram is Fourier transformed into momentum space, the
momentum associated with each function $\Delta^+(x-y)$ is on-shell, and is only
conserved at each vertex if an energy is attributed to the contributions of the
time-ordering functions. The resulting expression is analogous to what
Kadyshevsky had obtained by deriving an alternate expansion for the
$S$--matrix. A detailed explanation of how this alternate expansion is derived
is given, and it is shown how it provides a straightforward way of determining
the imaginary part of a Feynman diagram, which makes it useful when using
unitarity methods for computing a Feynman diagram. By considering a number of
specific Feynman diagrams in self-interacting scalar models and in QED, we show
how this alternate approach can be related to the old perturbation theory and
can simplify direct calculations of Feynman diagrams.
