## Friday, September 23, 2016

### A Copenhagen interpretation of classical mechanics

One can always take an indeterministic theory and turn it deterministic in some way or other while preserving empirical predictions. Bohmian mechanics is an example of doing that with quantum mechanics. It's mildly interesting that one can go the other way: take a deterministic theory and turn it indeterministic. I'm going to sketch how to do that.

Suppose we have classical physics with phase space S and a time evolution operator Tt. If the theory is formulated in terms of a constant finite number n of particles, then S will be a 6n-dimensional vector space (three position and three momentum variables for each particle). The time evolution operator takes a point in phrase space and says where the system will be after time t elapses if it starts at that point. I will assume that there is a beginning to time at time zero. The normal story then is that physical reality is modeled by a trajectory function s from times to points of S, such that Tt(s(u))=s(u+t).

Our indeterministic theory will instead say that physical reality is modeled by a (continuous) sequence of probability measures Pt on the phase space S for times t≥0. These probability measures should be thought of as something like a physical field, akin to the wavefunction of quantum mechanics--they represent physical reality, and not just our state of knowledge of it. Mirroring the consciousness-causes-collapse version of the Copenhagen interpretation of quantum mechanics, we now say this. If from time u (exclusive) to time t+u (inclusive) no observation of the system was made, then Pt+u(A)=Pt(Tu−1[A]). I.e., the probability measure is just given by tracking forward by the time-evolution operator in that case.

On the other hand, suppose that at time t an observation is made. Assume that observations are binary, and correspond to measurable subsets of phase space. Intuitively, when we observe we are checking if reality is in some region A of phase space. (It's easy to generalize this to observations having any countable number of possible outcomes.) Suppose Pt* is the value that Pt would have had there been no observation at t by the no-observation evolution rule. Then I suppose that with objective chance Pt*(A) we observe A and with objective chance 1−Pt*(A) we observe not-A, with the further supposition that if one of these numbers is zero, the corresponding observation physically cannot happen. Then the probability measure Pt equals the conditionalization of Pt* on the observation that does in fact occur. In other words, if we observe A, then Pt(B)=Pt*(B|A) and otherwise Pt(B)=Pt*(B|not-A). And then the deterministic evolution continues as before until the next observation.

As far as I can see, this story generates the same empirical predictions as the original deterministic classical story. Also note that while in this story, collapse was triggered by observation, presumably one can also come up with stories on which collapse is triggered by some other kind of physical process.

So what? Well, here's one thought. Free will is (I and others have argued) incompatible with determinism. One thought experiment that people have raised is this. If you think free will incompatible with determinism, and suddenly the best physics turned out to deterministic, what would you do? Would you deny free will? Or would you become a compatibilist? Well, the above example shows that there is a third option: give an indeterministic but empirically adequate reinterpretation of the physics. (Well, to be honest, this might not entirely solve the problem. For it might be, depending on how the details work out, that past observations narrow down the options for brain states so much that they become deterministic. But at least there would be hope that one wouldn't need to give up on libertarianism.)

The above way of making free will compatible with physical determinism is functionally similar to Kant's idea that our free choices affect the initial conditions of the universe, but without the freaky backwards-like (not exactly backwards, since the noumenal isn't in time) causation.

Here's another thought. Any indeterministic theory can be reinterpreted as a deterministic multiverse theory with traveling minds, while maintaining empirical adequacy. The multiverse traveling minds theory allows for causal closure of a deterministic physics together with robust alternate-possibilities freedom. Combining the two reinterpretations, we could in principle start with a deterministic physics, then reinterpret it in a Copenhagen way, and then impose on top of that the traveling minds interpretation, thereby gaining an empirical equivalent theory with robust alternate-possibilities freedom and no mental-to-physical causation. I bet a lot of people thought this can't be done.

Unknown said...

Shouldn't there be P_u instead of P_t on the right side of second equation?

Alexander R Pruss said...

That's one way to fix it, thank you. But I fixed it differently.

Anonymous said...

"I bet a lot of people thought this can't be done."

I knew it could be done, but doesn't the possibility prove that compatibilism is at least possibly true? If so, then your arguments against compatibilism must fail, since they attempt to show that it cannot be true.

Alexander R Pruss said...

The resulting system is indeterministic overall, but deterministic at the physical level. Hence no help to compatibilists.

Michael Gonzalez said...

This is an interesting insight, that this deterministic/indeterministic re-interpretation can go both ways. I would, of course, raise the concerns I always raise (which are basically John Bell's concerns) that the "observers" (or whatever the physical catalyst for collapse is) are also constructed of those very same sorts of physical parts which cannot attain an actual state until observed, and so you have a very serious incoherence in the whole account.... But that speaks to Copenhagen-style theories in general; it doesn't affect the insight that the re-interpretation goes both ways, which I think is really cool.

As for the issue about Libertarianism and Determinism, I think people are clearly missing the elephant in the room, namely: that there are elephants. A little Aristotle and a pinch of Wittgenstein, and I really think the whole problem dissolves and we are left scratching our heads at what on Earth we were so mystified about in the first place....

Alexander R Pruss said...

I don't think GRW-style collapse theories are subject to the objection. One can give conditions in terms of the uncollapsed wavefunction for the probability that collapse will happen.

And dualist observation-collapse theories are also not subject to the objection.

Michael Gonzalez said...

I thought the point of this particular post hinged on Copenhagen-style (observation-based) interpretations of QM. Does it still work on a GRW style approach?

Dualism also dodges the issue of physical determinism by its very nature, so it is already a free pass out of that problem (no?).

Alexander R Pruss said...

Probably the exact GRW mechanism couldn't work. But I think a lot of collapse mechanisms could be paired with the story. For instance, one could make a collapse happen when there is a high probability of a high density in some macroscopic region.

Michael Gonzalez said...

Fascinating.