Schrödinger's plates

The Schrödinger's plates image has been around since at least 2016, as here, but it's lately been having still another renaissance. Philip Ball, on Twitter, suggests that it's "Possibly the worst Schrödinger's cat analogy ever," but I think it has some merit as a way into seeing how classical physics can ask the same kinds of noncommutative questions as quantum physics asks. That's enough to make an unconstrained classical physics as good for constructing models as quantum physics.

Here it is:

Classical physics might perhaps ask just one question, such as "Do the plates break when you open the door?", but it can also ask a multitude of questions such as "Can the plates be saved?", or, with much more detail, "If with a few friends we make sure the door doesn't vibrate when we remove the adjacent pane of glass, how likely is it that we can save the plates?"

There is a continuum of such yes/no questions that can all be asked by a classical physicist, any of which can be turned into a question about likelihood. We can ask whether we can save the plates if we smash the adjacent plane of glass with a stone of a given size or shape, or we can ask how likely it is that we can save the plates if we use a glass cutter to cut holes of particular sizes and positions in one or more of the panes of glass.

One tricky part is that it looks like we can't run the experiment more than once, because it looks difficult to get the plates into that position many times to find out what approach works best. At the heart of statistical physics, however, is the act of creating very elaborate initial conditions, making as sure as we can that each time is as close to the same as we can engineer them to be, and seeing what happens when we ask different questions. So imagine that we set up the conditions in the Schrödinger's plates image over and over again as multiple trials and record the results of many different questions, each many, many times. In physics experiments there may be as many as millions of trials per second, each engineered to be as close as possible to every other.

Crucially, for each trial we can ask many yes/no questions, but very often we can ask only one and not another for any one trial: we can ask "Do the plates break if we open the door?" but we cannot for the same trial also ask "Do the plates break if we smash the adjacent pane of glass?" On the other hand, we can ask a multitude of questions jointly of the same trial: "Does more than one plate break?" is compatible with "Can we find any shards further than 3 meters away?" We can imagine a continuum of both incompatible and compatible questions even though we only actually ask a small number of them.

With many trials, we can ask the same question over and over and discover that for some fraction the answer is yes: we can call that fraction the observed likelihood. When we ask only slightly different questions of different sets of trials, we expect that the observed likelihood will also only change slightly, whether the different questions are compatible or incompatible. We can, incompatibly, use a glass cutter to cut holes of different particular sizes and positions in one of the panes of glass, which we would not expect to give different observed likelihoods if the two sets of holes differed only by a millimeter in size and hardly at all in position.

When we ask compatible questions, we can ask what the joint observed likelihood is for the answers to different questions: we can ask for the same trial both "Does more than one plate break?" and "Can we find any shards further than 3 meters away?" because they can be jointly observed. Crucially, there is no necessity for the observed likelihoods for incompatible questions to allow a joint observed likelihood for the answers to questions that are not and cannot be asked jointly: if we ask "Do the plates break if we open the door?" we cannot jointly ask "Do the plates break if we smash the adjacent pane of glass?" In incompatible cases, we can still assign numbers, which it is best not to call jointly observed likelihoods because (1) they are not jointly observed and (2) they may for the sake of consistency and continuity have to be not fractions between 0 and 1, but this mathematically natural but apparently strange behavior as much happens for classical mechanics as for quantum mechanics.

This is no more than to picture what happens in physics experiments, or indeed to reimagine Schrödinger's cat. One can invent many examples of this kind of thought experiment, and there have been academic papers on contextuality for decades (see "Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity," for example). More such examples will not compel anybody to think about the relationship between classical and quantum mechanics any differently. The mathematics of An algebraic approach to Koopman classical mechanics (or, in Annals of Physics 2020), however, and the Hilbert space mathematics it allows us to construct, supports an idea that classical mechanics can be as capable as quantum mechanics for modeling any physical situation. Classical models are significantly different, but they are as capable. Understanding the relationship between classical mechanics and quantum mechanics frees us from the distraction of thinking they are so very different.

EDIT: I use Facebook rather extensively as a whiteboard. It's quite public, so the embarrassment of making a mistake can be great, but I tell myself that it's worth it because one remembers mistakes that one is embarrassed about. One never knows what flights of fancy or of folly some comment will lead to. In any case, a comment on a post on the Facebook Group Quantum Mechanics & Theoretical Physics led me to look more carefully for how the Schrödinger's cat thought experiment is like the Schrödinger's plates thought experiment:

"There is one thing that is the same, if we can open the box at a time of our own choosing. In that case, the probability that the cat is alive begins at 1, then the probability at some rate decreases. If we more want the cat to live, we should open the box more quickly. The measurement we make determines the probability of different results.

If we open the box at some early time, we cannot also open the box at some later time, without starting a new trial. Those are incompatible measurements. On the other hand, we can on the same trial measure whether the cat died and whether the cat died of panic and a heart attack or of failure of the lungs, ..., which are compatible measurements that can be jointly measured."