A. Deterministic physical mechanics are incompatible with the notion of free will.
As kentox helpfully pointed out, this myth seems to follow from an unwitting conflation of two very different notions. On one hand, we have 'deterministic' as physicists use it to characterize a principle in theories of mechanics. On the other hand, we have 'deterministic' as philosophers use it to juxtapose against the notion of free will. So, if we conflate these two ideas, we certainly compel ourselves to conclude that deterministic physical mechanics contradict the notion of free will. The problem is that this conflation is entirely false. This may not be obvious to the reader, so let's continue:
Let's suppose that there is a human will, in the sense of a free will. Does it make sense to call this situation indeterministic, in the physical sense? It seems not: after all, this will is, by very definition, a cause; that is, a force operating upon the system in question deterministically.
Let us suppose, conversely, that there is a mechanical system which we agree is really -- not just heuristically -- indeterministic. Does it make sense to say that that system is instantiating a free will? No, this does not make any sense; and for the same reason just discussed. If you want to say that the will is a cause like any other cause, and that it is instantiated in your theory, naturally you have to point out the will in your theory and establish its causal efficacy. This is precisely what the indeterministic model is, by definition, not doing. (As an aside: we can suppose that someone will say that the universe is indeterministic, and that the will sits outside the universe and manipulates it within the bounds of this indeterminism; but this position has a whole host of other problems. In any case, it is most certainly not a reductive materialist position, so it seems to be beyond our immediate interests here. If there are any dualists who would like to take up this position, we can discuss its serious problems.)
This point helpfully brings our attention to what may be the central problem in these discussions. While the physical theories purport to be offering explanations for will, they're in fact doing no such thing: they do not explain how or where the will is instantiated in their theory, and they do not explain how its causal efficacy operates. All they do is point out that they describe their theory with a word which philosophers also associate with free will, and this alone is expected to operate as an explanation! But if someone said, 'Hands grasp things and minds grasp things, therefore our thinking is just the movements of our hands', no one would take them seriously!
And, as should now be clear, the relationship between the philosophical notion of the will and the physical notion of determinism is both complicated and unclear, and certainly does not enable the implicit equations required to believe that an indeterministic quantum mechanics explains the will. Again, the point of the above was not to associate will with one physical theory or another, but rather to illustrate just how complicated and fundamentally problematic the very question of such a relationship is. It is not at all clear what implications a statement about one could possibly have about the other.
B. Unless you believe in an immaterial soul, you must believe that consciousness and free will belong to the explanatory domain of physics.
Let's consider the most general case first. Does this claim follow in principle? We can find out by considering some parallel examples. Is it true that unless we believe in an immaterial soul, we must believe that my choice of breakfast cereal this morning belongs to the explanatory domain of physics? What about the result of the first OJ Simpson trial: was that also either a consequence of physical mechanics or of an immaterial soul? What about the identity of 2 and the square root of 4? What about the identity of Mark Twain and Samuel Langhorne Clemens? What about the wrongness of murder? No... I don't think we'd believe all these claims; some of them follow for reasons other than an immaterial soul or physical mechanics. So it's not a principle that things are proper either to one or the other.
OK, but is there something special about consciousness and free will, as specific topics, that validates this claim even though it's not the case in principle? The argument to this effect seems to be as follows: (i) consciousness and free will just are electrochemical reactions in the brain, (ii) electrochemical reactions in the brain just are systems of physical interactions, (iii) therefore, consciousness and free will just are systems of physical interactions and thus in the explanatory domain of physics. The problem with this is that (i) (let's just assume (ii), for sake of discussion) is not obviously true, and in order to pursue it further, we need philosophy. And that answers our question right there. Again: what we need here is not physics, but philosophy.
It may not be clear why we need philosophy to approach this question, so I'm going to explain one of the important underlying principles that establishes this. Bear with me: let's imagine a system of knowledge, called CanMonia, which contains all the facts about the internal relationships of Canadian money. So it contains the fact that four quarters equals a looney, and two looneys equals a twoney, and a twoney equals two hundred pennies, and so forth. And it's only this sort of information that it contains. Now imagine a similar system of knowledge called MexMonia, that is just the same only it relates only to the internal relationships of Mexican money. Now imagine a guy named Bob, who knows about CanMonia, but doesn't know anything else, and a guy named Pedro, who knows only about MexMonia. They're trying to figure out how many Canadian dollars a Mexican peso is worth. Do you think they can do it? They can't: there's no way. Now Pedro is really stubborn, so he keeps learning more and more about MexMonia, thinking that if only he knew everything in that field of knowledge, he could accurately exchange pesos for dollars. But he's wrong, isn't he? What Pedro and Bob need is what we can call a bridging term: they need some knowledge which equates a term in one field with a term in the other. So if Paul comes around, and Paul has access to a bridging term, he can tell them, for example, that a Canadian dime is worth a Mexican peso -- then Pedro and Bob can figure the relevant exchange out.
Now, this is the situation that physics finds itself in with respect to consciousness and free will. The latter are not -- at least immediately -- in the language of physics. We need a bridging term to translate them into that language. And this term, by definition, cannot come from physics -- just like Pedro could not get his bridging term from MonMonia. By definition, this bridging term is about something that is, at least in the first place, external to physics. (NB: this doesn't mean that such a bridging term necessarily exists.) So, again, we need something other than physics; something that can speak the native language of consciousness and free will, and something that can step back from physics and understand the meanings of its claims as well. We need philosophy.
Now, someone might say: 'Ok, right, there are some philosophical problems here, and I'm a physicist, not a philosopher, so, frankly, I haven't studied them; I don't understand them. But who cares? There's lots of things I don't understand. Let's just assume that the philosophical work has been done, let's assume consciousness and free will just are electrochemical brain processes, and then my argument will follow naturally!' The problem with this is that it's not science, it's not even reason or argument in general: it's wishful thinking, it's fantasy, it's pure faith. We can't just arbitrarily assume that highly technical problems will just happen to work out exactly the way our pet theories need them to. Again, this is not argument; it's not going to help us here.
Now, someone might say: 'Fine, but I have said from the beginning that the other alternative is an immaterial soul, and that's exactly what you're advocating, so what's the problem?' But that's not what's being advocated at all. In fact, there are a number of positions on the relationship between the mind and the brain that differ profoundly from both the assumption of an immaterial soul (substance dualism) and that of mind-brain identity. For example, eliminativism and supervenience are fundamentally important contemporary positions which strongly oppose both of these alternatives. We are again into highly technical philosophical territory. Again, the conclusion being: what we need to resolve these questions is not physics, but philosophy.
C. Human behavior is caused by electrochemical brain processes, which just are systems of physical interactions, so if systems of physical interactions are strictly deterministic, then human behavior is strictly deterministic and so there is no will.
There are two problems here. Again, 'will' has not been defined in any way that is relevant to the question of physical determinism. We don't know what will is in physical terms, so it's impossible to say, on the basis of physical terms, whether or not will is there. But I have covered this point extensively under section A, so I will not say more here.
Secondly, there is the question of whether the first premise is true: whether we can strictly reduce human behavior to electrochemical brain processes. On this question, there are two observations to make: (i) Again, it's not physics that we must turn to to proceed. You can't derive a position on this axiomatically from a theory of mechanics. (Again, I have said much already relating to this point.) Rather, we need to go look at human behavior and see what it is like as behavior; that is, we need psychology. (ii) If we were to consult psychology, it would turn out that this claim is false: human behavior cannot be reduced to electrochemical brain processes. We need, at least, to look to the principles of behavior: to the relationship between the organism and the environment. And, at least if we take psychology at its word, we need also to look at cognitions; that is, beliefs. And so also to the meanings established socially and culturally.
Now, again, someone might say: 'Very well, suppose I refine my position and say human behavior is not reducible to neuroscience, bur rather to the entire physical world. Now, you must either grant my argument, then, that human behavior is just a system of physical interactions, unless, of course, you want to posit an immaterial soul.' But, again, this argument is false. It is the same argument I have already replied to in section B.
And again, someone might say: 'Very well, but suppose that all of the difficult philosophical work surrounding all these concepts is, at some point in the future, worked out, and it happens to turn out that this work just proved I was right all along; certainly then you must grant my case!' But, again, this is just the appeal to fantasy which I have also addressed above.
D. Notwithstanding all of the above, I still want to believe my argument about the relationship between physics and free will, even if I cannot defend its validity now, I can believe that something will happen in the future which we just can't forsee but which will make it valid. Let's say, for example, that I cannot offer any defense against the idea that free will is a philosophical or metaphysical problem rather than a physical or empirical one -- maybe in the future things will change!
This is another argument from wishful thinking, against which I have already commented. If you want to claim that free will is an empirical or physical problem, you actually have to show how this could be the case, not just arbitrarily state it.
E. But aren't you just arbitrarily stating that it is a metaphysical problem? Couldn't I just turn this argument around and use it against you?
No. To say that it's a metaphysical problem is simply to acknowledge that it's not an empirical one. Now, perhaps someone will say: 'If that is the case, reason demands that you demonstrate for us all the ways in which it is not an empirical problem!' But, of course, reason makes no such demand: reason does not proceed by trying to prove negatives. For if we claimed that unicorns did not exist, and someone said: 'If that is the case, reason demands that you demonstrate all the ways in which unicorns are not existing, otherwise it is natural to suppose that they exist!' we would not take them seriously.