  Feynman's formulation of quantum theory is remarkable in its combination of
formal simplicity and computational power. However, as a formulation of the
abstract structure of quantum theory, it is incomplete as it does not account
for most of the fundamental mathematical structure of the standard von
Neumann-Dirac formalism such as the unitary evolution of quantum states. In
this paper, we show how to reconstruct the entirety of the finite-dimensional
quantum formalism starting from Feynman's rules with the aid of a single new
physical postulate, the no-disturbance postulate. This postulate states that a
particular class of measurements have no effect on the outcome probabilities of
subsequent measurements performed. We also show how it is possible to derive
both the amplitude rule for composite systems of distinguishable subsystems and
Dirac's amplitude-action rule, each from a single elementary and natural
assumption, by making use of the fact that these assumptions must be consistent
with Feynman's rules.
