  We discuss the case of histories labelled by a continuous time parameter in
the {\em History Projection Operator} consistent-histories quantum theory. We
describe how the appropriate representation of the history algebra may be
chosen by requiring the existence of projection operators that represent
propositions about time averages of the energy. We define the action operator
for the consistent histories formalism, as the quantum analogue of the
classical action functional, for the simple harmonic oscillator case. We show
that the action operator is the generator of two types of time transformations
that may be related to the two laws of time-evolution of the standard quantum
theory: the `state-vector reduction' and the unitary time-evolution. We
construct the corresponding classical histories and demonstrate the relevance
with the quantum histories; we demonstrate how the requirement of the temporal
logic structure of the theory is sufficient for the definition of classical
histories. Furthermore, we show the relation of the action operator to the
decoherence functional which describes the dynamics of the system. Finally, the
discussion is extended to give a preliminary account of quantum field theory in
this approach to the consistent histories formalism.
