Interaction and Evolution in Classical Mechanics and in Quantum Mechanics, and Indefinite Causal Orders

I describe a few theorems that show that, in classical mechanics (Newtonian and Lagrangian mechanics), the intrinsic structure of the dynamics naturally distinguishes the concept of interaction from that of evolution, and, correlatively, position from momentum.

One gets for free, moreover, a characterization of "free" evolution (or "isolation"). This is not the case in Hamiltonian and quantum mechanics.  Along the way, I point out that the theorems allow one to entirely reconstruct the full 4-dimensional spacetime structure of Newtonian physics from classical dynamics, again in a way not possible in quantum theory. Thus, it is also the case that the dynamics of quantum theory needs to be hooked up to background spacetime structure "by hand" in a way not required in classical mechanics. I conclude by discussing what I take to be the lessons for all this with regard to the Measurement Problem: not only is the idea of "measurement" problematic in quantum theory, but the entire idea of "interaction" per se in quantum theory is more deeply problematic than has been recognized. I argue that the conceptual situation as I have laid it out provides a natural and possibly fruitful way to formulate and work with the idea of indefinite causal orders, and possibly to hook it up in a natural way with the idea of quantum reference frames.