The action operator for continuous-time histories

We define the action operator in the consistent histories formalism, as the quantum analogue of the classical action functional, for the simple harmonic oscillator case. The action operator is shown to be the generator of time transformations, and to be associated with the two types of time-evolution of the standard quantum theory: the wave-packet reduction and the unitary time-evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories. Finally, we show the relation of the action operator to the decoherence functional.