Schrodinger makes a good argument in the book “Nature and the Greek and Science and Humanism” that we should actually just abandon the idea that there even is a trajectory.

Our sciences are derived from inductive reasoning. You drop a ball, it falls to the ground, you repeat it, it falls again, and eventually, you come up with a mathematical law to describe this. You assume from that point if you drop it an infinite number of times, it will always fall to the ground, but this is just an assumption that cannot be proven.

When the members of the Frontiers of Science discussed physics, they often used the abbreviation “SF.” They didn’t mean “science fiction,” but the two words “shooter” and “farmer.” This was a reference to two hypotheses, both involving the fundamental nature of the laws of the universe. In the shooter hypothesis, a good marksman shoots at a target, creating a hole every ten centimeters. Now suppose the surface of the target is inhabited by intelligent, two-dimensional creatures. Their scientists, after observing the universe, discover a great law: “There exists a hole in the universe every ten centimeters.” They have mistaken the result of the marksman’s momentary whim for an unalterable law of the universe. The farmer hypothesis, on the other hand, has the flavor of a horror story: Every morning on a turkey farm, the farmer comes to feed the turkeys. A scientist turkey, having observed this pattern to hold without change for almost a year, makes the following discovery: “Every morning at eleven, food arrives.” On the morning of Thanksgiving, the scientist announces this law to the other turkeys. But that morning at eleven, food doesn’t arrive; instead, the farmer comes and kills the entire flock.

— Cixin Liu

We also do this to derive our concept of trajectories. We can measure something a x(0) and x(t), then repeat the experiment and measure it at x(0.5t), then repeat it again and measure it at x(0.25t) and x(0.75t), so on and so forth, measuring many many in-between points. From that, we assume that if we continue to cut the intervals in half and measuring in between, our predictions will continue to hold, making us conclude that there is a completely continuous transition between x(0) and x(t) exactly as described by our mathematics, which we can fit to unambiguous mathematical equations.

Yet, this is just an assumption. We cannot actually know that this continuous transition exists, and what Schrodinger argued is that there is in fact good reason to think it doesn’t. This is because, in various particle experiments, you cannot actually try to reconstruct this path in a way that is unambiguous and would be consistent with every experiment. It is much simpler just to treat it as if the particle was over there at x(0), and now it is over here at x(t), with a time delay of t. Rovelli describes it as nature evolving through succession of events, rather than nature being made up of “stones bouncing around,” nature flows according to these succession of events whereby things manifest their properties to one another during an interaction, but there is no trajectory the particle actually took in between interactions.

These trajectories are entirely metaphysical and could never actually be experimentally verified, since verification requires observation, and observation is an interaction, so to posit that there is any path in between interactions is to posit that there exists something in between observations, and by definition you could not observe that. It would always have to be something assumed a priori. This is what I meant when I said most people approach quantum mechanical interpretation seem to have a desire to assume quantum theory can tell us about things beyond what is even possible to observe, and much of the confusion around the theory is trying to philosophically understand this unobservable realm of what is going on in between observations.

I tend to agree with physicists like Schrodinger, Rovelli, and Francois Igor Pris that what makes the most sense is to just abandon this because it is entirely metaphysical and ultimately faith-based and cannot actually be experimentally verified. We should just stick to what we can actually confirm through observational evidence, and observations are discrete, so any continuity we assume about nature is ultimately metaphysical and could not be derived from observation. That is why it makes more sense to consider reality not as autonomous stones bouncing around, but as a succession of discrete events, and the physical sciences allows us to predict what properties of systems will be realized during those events.