Wednesday, June 06, 2007

Bohm and Lewis

So I've been thinking and reading a bit about quantum theory recently (originally in connection with work on ontic vagueness). One thing that's been intriguing me is the Bohmian interpretation of non-relativistic quantum theory. The usual caveats apply: I'm no expert in this area, on a steep learning curve, wouldn't be terribly surprised if there's some technical error in here somewhere.

What is Bohmianism? Well, to start with it's quite a familiar picture. There are a bunch of particles, each supplied with non-dynamical properties (like charge and mass) and definite positions, which move around in a familiar three-dimensional space. The actual trajectories of those particles, though, are not what you'd expect from a classical point of view: they don't trace straight lines through the space, but rather wobbly ones, if if they were bobbing around on some wave.

The other part of the Bohmian picture, I gather, is that one appeals to a wavefunction that lives in a space of far higher dimension: configuration space. As mentioned in a previous post I'm thinking of this as a set of (temporal slices of) possible worlds. The actual world is a point in configuration space, just as one would expect given this identification.

The first part of the Bohmian picture sounds all very safe from the metaphysician's perspective: the sort of world at which, for example, Lewis's project of Humean supervenience could get off and running, the sort of thing to give us the old-school worries about determinism and freedom (the evolution of a Bohmian world is totally deterministic). And so on and so forth.

But the second part is all a bit unexpected. What is a wave in modal space? Is that a physical thing (after all, it's invoked in fundamental physical theory)? How can a wave in modal space push around particles in physical space? Etc.

I'm sure there's lots of interesting phil physics and metaphysics to be done that takes the wave function seriously (I've started reading some of it). But I want to sketch a metaphysical interpretation of the above that treats it unseriously, for those of us with weak bellies.

The inspiration is Lewis's treatment of objective chance (as explained, for example, in his "Humean supervenience debugged"). The picture of chance he there sketches has some affinities to frequentism: when we describe what there is and how it is in fundamental terms, we never mention chances. Rather, we just describe patterns of instantiation: radioactive decay here, now, another radioactive decay there, then (for example). What one then has to work with is certain statistically regularities that emerge from the mosaic of non-chancy facts.

Now, it's very informative to be told about these regularities, but it's not obvious how to capture that information within a simple theory (we could just write down the actual frequencies, but that'd be pretty ugly, and wouldn't allow us to to capture underlying patterns among the frequencies). So Lewis suggests, when we're writing down the laws, we should avail ourselves of a new notion "P", assigning numbers to proposition-time pairs, obeying the usual probability axioms. We'll count a P-theory as "fitting" with facts (roughly) to the extent that the P-values it assigns to propositions match up, overall, to the statistically regularities we mentioned earlier. Thus, if we're told that a certain P-theory is "best", we're given some (cp) information on what the statistical regularities are. At not much gain in complexity, therefore, our theory gains enormously in informativeness.

The proposal, then, is that the chance of p at t is n, iff overall best theory assigns n to (p,t).

That's very rough, but the I hope the overall idea is clear: we can be "selectively instrumentalist" about some of the vocabulary that appears in fundamental physical theory. Though many of the physical primitives will also be treated as metaphysically basic (as expressing "natural properties") some bits that by the lights of independently motivated metaphysics are "too scary" can be regarded as just reflections of best theory, rather than part of the furniture of the world.

The question relevant here is: why stop at chance? If we've been able to get rid of one function over the space of possible worlds (the chance measure), why not do the same with another metaphysically troubling piece of theory: the wavefunction field.

Recall the first part of the Bohmian picture: particles moving through 3-space, in rather odd paths "as if guided by a wave". Suppose this was all there (fundamentally) was. Well then, we're going to be in a lot of trouble finding a decent way of encapsulating all this data about the trajectories of particles: the theory would be terribly unwieldy if we had to write out in longhand the exact trajectory. As before, there's much to be gained in informativeness if we allow ourselves a new notion in the formulation of overall theory, L, say. L will assign scalar values (complex numbers) to proposition-time pairs, and we can then use L in writing down the wavefunction equations of quantum mechanics which elegantly predicts the future positions of particles on the basis of their present positions. The "best" L-theory, of course will be that one whose predictions of the future positions of particles fits with the actual future-facts. The idea is that wavefunction talk is thereby allowed for: the wave function takes value z at region R of configuration space at time t iff Best L-theory assigns z to L(R,t).

So that's the proposal: we're selectively instrumentalist about the wavefunction, just as Lewis is selectively instrumentalist about objective chance (I'm using "instrumentalist" in a somewhat picturesque sense, by the way: I'm certainly not denying that chance or wavefunction talk has robust, objective truth-conditions.) There are, of course, ways of being unhappy with this sort of treatment of basic physical notions in general (e.g. one might complain that the explanatory force has been sucked from notions of chance, or the wavefunction). But I can't see anything that Humeans such as Lewis should be unhappy with here.

(There's a really nice paper by Barry Loewer on Lewisian treatments of objective chance which I think is the thing to read on this stuff. Interestingly, at the end of that paper he canvasses the possibility of extending the account to the "chances" one (allegedly) finds in Bohmianism. It might be that he has in mind something that is, in effect, exactly the position sketched above. But there are also reasons for thinking there might be differences between the two ideas. Loewer's idea turns on the idea that one can have something that deserves the name objective chance, even in a world for which there are deterministic laws underpinning what happens (as is the case for both Bohmianism, and for the chancy laws of statistically mechanics in a chancy world). I'm inclined to agree with Loewer on this, but even if that were given up, and one thought that the measure induced by the wavefunction isn't a chance-measure, the position I've sketched is still a runner: the fundamental idea is to use the Lewisian tactics to remove ideological commitment, not to use the Lewisian tactics to remove ideological commitment to chance specifically. [Update: it turns out that Barry definitely wasn't thinking of getting rid of the wavefunction in the way I canvass in this post: the suggestion in the cited paper is just to deal with the Bohmian (deterministic) chances in the Lewisian way])

[Update: I've just read through Jonathan Schaffer's BJPS paper which (inter alia) attacks the Loewer treatment of chance in Stat Mechanics and Bohm Mechanics (though I think some of his arguments are more problematic in the Bohmian case than the stat case.) But anyway, if Jonathan is right, it still wouldn't matter for the purposes of the theory presented here, which doesn't need to make the claim that the measure determined by the wavefunction is anything to do with chance: it has a theoretical role, in formulating the deterministic dynamical laws, that's quite independent of the issues Jonathan raises.]

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