This is something of a placeholder, a way to cite a construction that I think deserves to be known widely that I put into an otherwise deservedly unpublished paper, https://arxiv.org/abs/1211.2831.
So, three images from that paper, the last being the only bibliographical reference needed (and very general it is):

The point of this is just that it contrasts with the usual way of talking about summing amplitudes over all possible trajectories, forward and backward in time without limit. The Lagrangian can be said to fix a deformation of the differential equation that is satisfied by the free field. Not to discount taking a path-integral understanding of the Lagrangian seriously as well, as a contrasting point of view, but we can take the interacting field to be constructed as a complex of free field operators that is purely contained in the backward light-cone of a given point. We can think of the action of the interacting field as a consequence of interference between a carefully weighted infinite sea of free field components that is isolated in the backward light-cone of x; or we can think of it as just doing what has to be done to make it look like there's something complicated at x; or ... .
Everything here can be said to follow from the action of the interaction Lagrangian on the free quantum field by time-ordered commutation being effectively the same as the action of the same expression on the free classical field by time-ordered Poisson bracket, in the sense that the same differential equation is satisfied by the interacting quantum field and by the interacting classical field --- up to the usual worries about renormalization.