There is an interesting article in Scientific American, Does Quantum Mechanics Rule Out Free Will?, which is, as is unsurprising, concerned with an argument that quantum mechanics (the many-worlds interpretation of it, to be exact) is deterministic, which at least some people quoted in it take to rule out free will. The matter is actually quite complicated. For one thing, physicists use 'deterministic' in a different sense than it is used when discussing free will. For physicists, 'determinism' is a feature of a model such that from the values of the relevant variables in one state you can determine the values in all the other states in the model. 'Determinism' in free will discussions describes a reality in which the connection of cause to effect is accurately and completely described by a strong modality from a logical system in which the strong modality is implied by a null modality. There are a lot of differences here. For instance, the physics-determinism is not directional (e.g., it doesn't matter whether the states are past or future), whereas causal determinism as we find it in free will discussions is directional (it is the cause to effect direction that is controversial, whereas necessity in the effect to cause direction is quite commonly accepted as a form of conditional necessity). But the most obvious is that physics-determinism is a feature of a model, and causal determinism is a purported feature of the world. The issue, of course, is that you can have a deterministic model of a process that is strictly indeterministic. It will just be sometimes possibly wrong, even if only very rarely; and all our deterministic models, even our best, are in fact sometimes wrong, even if only very rarely, because they are all necessarily idealized and do not in all circumstances account for all variables with perfect precision. Moving from one sense of determinism to another is something one does by a vague and imperfect feeling of analogy between the two, and not because anything in the physics-determinism directly implies causal determinism.
The matter gets even more complicated if we go down a level and consider the many-worlds theory itself. Very crudely and roughly, a central, and unresolved, issue in interpreting the mathematics for quantum mechanics lies in the fact that the Schrodinger wave equation gives us very accurate descriptions of quantum mechanical processes and their outcomes, but that the same equation also gives us answers to which those outcomes do not correspond. In a sense, the wave equation is too generous; it gives us all the possibilities that fit the experimental results and also possibilities that don't. Different interpretations of quantum mechanics try to explain this in different ways, and there are endlessly many different ways. But three core answers keep popping up in different versions, because it seems that one of the three has to be true:
(1) The wave equation is missing something important, and there is some other fact that weeds out the unobtained possibilities.
(2) The wave equation is correct in identifying the possible outcomes, but the real process itself indeterministically 'selects' one of them as the real outcome.
(3) All of the outcomes of the wave equation are in some way real.
This is very crude, but allowing for a lot of nuance in actual development of theories, all of these have had major physicists championing them, and all of them have advantages, and all of them have very serious disadvantages. Historically, the most popular among physicists has been (2), but in recent decades (3), originally dismissed by most physicsts as absurd, has had increasing respect, in part because it makes easier a number of things physicists think important. (3)-based interpretations, of which there are different versions, are generally called 'many worlds' theories, or sometimes 'multiverse' theories.
The whole thing is fascinating from a philosophy of physics point of view. It reminds me in many ways of the problem of the direction of time. The problem of temporal direction is that almost all the major equations of physics make no distinction between past and future, backward or forward in time, but we quite clearly do experience time in a way that makes its direction important. (The second law of thermodynamics is the major exception, but even that is not completely straightforward, since some of the most important explanations of what it means based on probabilities also would not obviously be affected by direction, and you have to get quite precise about what is meant by 'direction' to start seeing why physicists take the increase of entropy to be, in some sense, in one direction.) In both cases, the problem of how to interpret physics can be seen as coming down to figuring out what to do with the failure of fit between the impartiality of the mathematics used by physicists to describe the world and the partiality of the world physicists actually encounter in experiment.
In any case, on a many worlds interpretation, all of the results of the wave equation happen. This raises the obvious next question: Why don't we see them in experiment? And the many worlds interpretation answers this by saying, "They happen in a different universe." There are different ways of making sense of this. The usual way it is described is that at every quantum interaction, the universe 'splits' into one universe for each result from the wave equation. We don't have to puzzle about what we are missing, or how the world selects which result to follow; nothing essential is missing and all of the results follow. Different versions of the many worlds have somewhat different views of this 'splitting' (and also how literally to take the idea that universes split off from each other).
The technicalities are quite extensive, and more than a few of them beyond me. But this is already enough to see that we should be cautious. What exactly is this splitting? How do quantum interactions relate to free will choices or indeed any other apparently macroscopic event within a single universe? We don't know. There are several different versions of many worlds interpretations, and making very small adjustments in the interpretation can give you very different ideas about what it might mean for a universe or world-history to split into two. 'Free will' is obviously not a term in any scientifically respectable many worlds interpretation, and therefore no such interpretation says anything about it at all. It's even been disputed whether the quantum interactions are properly seen as causal interactions. Physicists have been worried from the beginning that these interpretations effectively dead-end experimental inquiry, because any experiment has to occur in a universe, and therefore there is apparently no experiment one could ever do that could confirm or disconfirm the existence of these other universes, or confirm or disconfirm anything about their particular nature or relation to this universe. (So far the only semi-promising attempts of physics to work out what kind of experiment you could even do that would shed even indirect light on the matter all involve things we currently cannot do and in fact do not currently even know to be possible in the way they would need to be done.) Perhaps more immediately relevant to the free will discussion, many worlds interpretations have also tended to struggle with the role of probability in quantum mechanics. In particular, quantum mechanics requires not just any use of probabilities but a very particular kind that makes sense of some very odd quantum experiments and involves principles, like the Born rule, that characterize how they work; but it has always been difficult to find a many worlds interpretation that definitely fits with those principles. Despite the fact that it simplifies a number of things in physics, it's not surprising that physicists have been reluctant to accept it; its two major problems seem to be that it doesn't seem to be something we could establish by any experiment physicists could do and it's still unclear whether it is actually consistent with the experiments they have done. These problems would look very bad for the interpretation under most circumstances; it's only the fact that all other known interpretations have their own very serious problems that make these problems seem less bad here.
The discussion in the article is occasioned by a particular paper, Eddy Chen's Strong Determinism (PDF). It's a fascinating paper. I don't like some of the ways he sets up the account of determinism used in it, which seems like an unholy amalgam of the two different senses of determinism I mentioned at the beginning of this post; it follows from it that most deterministic theories in physics are not deterministic in the sense used here (since they don't specify anything about which possible worlds they apply to), and it also seems that, depending on how your theories are allowed to gerrymander possible worlds, that you could get strong determinism in Chen's sense even dealing with situations that would usually be called indeterministic in free will discussions. I could perhaps just be missing something, but I'm already wary on these grounds. I also don't like shifting back and forth between the actual world and possible worlds; it can be done, but there are endlessly many ways to get something wrong without realizing it. In any case, one (but only one) of the several things that he considers is how his account of strong determinism relates to Everett's version of the many worlds interpretation. (Chen himself does not seem to have any particular commitment to this interpretation.) It is not strongly deterministic in Chen's sense, but he argues that you can have a version of it, with certain additional assumptions beyond the standard Everett interpretation, that is strongly deterministic, and that such a version makes certain things more tractable in such a way that you could have very simple laws covering all the major phenomena physics want to explain. The point of the argument is not to consider whether the world is strongly deterministic but whether strong determinism in his sense requires you to give up on simple laws of nature, and Chen's argument (which, without having gone through every step, seems reasonably plausible) is that it does not, although you need a particular kind of strongly deterministic theory to make it work; and there's a secondary conclusion that, perhaps surprisingly, quantum mechanics is more hospitable to that particular kind of theory than classical mechanics is. That's an interesting, but much less provocative, result than one would have expected from the Scientific American article.
