The idea has been proposed, and often reappears, but is not credible, for two reasons. The first is that the indeterminism of quantum mechanics is governed by a rigorous probabilistic dynamics.  The equations of quantum mechanics do not determine what will happen, but determine strictly the probability of what will happen. In other words, they certify that the violation of determinism is strictly random. This goes in exactly the opposite direction from human freedom to choose. If human freedom to choose was reducible to quantum indeterminism, then we should conclude that human choices are strictly regulated by the chance. Which is the opposite of the idea of freedom of choice. The indeterminism of quantum mechanics is like throwing a coin to see if it falls heads or tails, and act accordingly. This is not this what we call freedom to choose. 

But there is a second, and more important, consideration. If an element of randomness is sufficient to account for free will, there is no need to search it into quantum uncertainty, because in a complex open system such as a human being there are already many sources of uncertainty, entirely independent of quantum mechanics. The microscopic atomic dynamic inside of a man is influenced by countless random events: just consider the fact that it occurs at room temperature, where the thermal motion of the molecules is completely random. The water that fills the molecules of our body and our brain is a source of indeterminism for the simple fact of being hot, and this indeterminism is much higher than the quantum one. If you add to this the fact that quantum indeterminism has a well-known tendency to disappear extremely fast as soon as you consider macroscopic objects (due to "decoherence"), it seems clear that trying to bind human freedom and quantum indeterminism is a very improbable hope. 

This brings us back to the starting point. The problem of the apparent tension between free will and determinism is not relieved by quantum physics. The argument, however, highlights a flaw in the intuition from which the problem itself originates. If the macroscopic dynamics is subjected to the consequences of microscopic indeterminism such as the thermal one, what is the exact nature of the problem of free will?

Clearly the problem is to clarify what it means to be free to choose. Let us get closer to the core of the problem from the other side: not from that of physics, but from our freedom. I can decide whether to declare or not some revenues to the IRS. This is a free choice. What does this mean? First it means that I am not forced to make a choice by external constraints. For example, there is no law that states that I get the money only after I have declared it. If so I would have no choice. Secondly, there is no IRS inspector watching me, in which case I would not have choice either. I am free to choose to be honest or dishonest. We have countless choices of this kind, not only ethical, but also in the daily management of our life.

What happens when I choose? It happens that I evaluate the pros and cons of a choice in my thoughts, all the factors that can determine it. These can be external (if they catch me I'm in trouble), internal (I want to be an honest guy), accidental (now I'm in trouble with money and these fifty dollar more ...), emotional (I just saw a TV show on those not paying taxes and I am disgusted by those people), and so on.

There is therefore a first sense of the expression "be free to choose" which simply refers to the fact that the determining factors are internal and not external. This causes no conflict with determinism. Here is an example, from Daniel Dennett, to clarify this point. The Rover (the machine with wheels) sent to Mars a few months ago is programmed to move autonomously on Mars, and has a complex navigation system that analyzes its surroundings and decides where to move according to a set of assigned priorities. Say to make longer journeys, in order to explore different regions and send the images to Earth. However, the Rover can end up in a situation where it can no longer move, for example because is got stuck between two boulders. Or, scientists from the control center on Earth may decide not to leave the Rover's program to decide by itself, and to intervene and compel the Rover to go back. For example because they have independent observations of a dust storm approaching. In either case, we can say that the Rover is "no longer free" to go where it wants, because it is stuck between two rocks, or because the engineers at NASA have sent a radio control that blocks the freedom of decision of the program on board.  After the sand storm is over or after being freed from the two boulders, the Rover regains its "freedom to decide" and begins to run only on its own "choices" of where to go.

This is a particular sense of the expression "to be free to decide." We often use this expression in this sense. For example: I am not free to decide to go for a walk if I am in prison. This sense of "being free" is the most common, and is not in conflict with physical determinism. After all, the Rover, once freed from the rocks and freed from the radio controls from NASA, becomes free to decide for itself where to go, but the program that runs it is driven by strictly deterministic physics. In this case, to "be free" only refers to the distinction between determinations of behavior that are external (the boulders, the radio controls of NASA, the prison) and determinations of behavior that are internal (the software of the Rover, my intense desire to take a walk). From this point of view, the problem of the conflict between free will and physical determinism dissolves completely, and this is the solution of the problem today proposed by many intellectuals, including, for example, Daniel Dennett.

Is this a complete and satisfactory solution of the problem? Maybe not, because there are issues that remain open. The first one is that the analogy between the rover and a human being does not fully hold. A human being seems to be, and probably is, more "free" than the Rover in the following sense. Both, the Rover and the human being, can be free in the sense that the decision on the behavior is determined by factors internal and not external, but in the case of the Rover we know that there is a precise software that determines this behavior. This software was built (by engineers) in order to be as "deterministic" as possible. Sure, it can break or malfunction, but this causes the behavior of the Rover to be consider abnormal. As long as problems do not occur, and the Rover works well, its behavior is determined in a rigorous manner, by factors within the Rover itself, but still factors that make a strict deterministic structure. Can we say the same of man?

To a certain extent, also the human behavior is determined by something similar to biological software. There is no doubt that the neuronal structure has numerous aspects similar to the software of the Rover, with subsystems that manage specific behaviors (walking) and other systems that determine complex choices arbitrating between more or less pressing competing demands from other parts of the brain ("I'm hungry, I want to go out to eat a sandwich, but I also want to write this paper"). But granted similarities, a key difference remains in both function and organization: the management of randomness, i.e. indeterminacy. Even the engineers who designed the Rover had to deal with hazard. Electronic equipment ages and deteriorates with time. You cannot predict when a connection will stops working well. But the engineers who designed the Rover have done everything possible to minimize this effect for the Rover. The Rover works well when the randomness of events is kept at best under control.

It does not appear that the functioning of living systems follows the same principle. From biochemistry, living systems are immersed in an environment of randomness. The basic biochemical processes exploit fully the highly random thermal motion of the molecules. For example, our cells build proteins with a molecular mechanisms that combines molecules following the genetic instructions. But these mechanisms are fed using the random thermal motion of the molecules in the vicinity. On the opposite end of the scale, the entire mechanism of Darwinian evolution relies, as Darwin discusses in detail in the first chapters of the "Origin of Species," on the huge variability of individuals and species. This random element, present from the biochemical level to the level of the evolution of species is a primordial ingredient (sometimes a bit neglected) of life on earth. Life is the result of structures that are in balance between the stiffness obtained by shielding casual events, and flexibility achieved by leaving space to the effects of this same randomness. Individuals survive because they have a structure similar enough to that of their parents, and parents have developed that structure as a result of a sufficient number of repeated changes from their parents, which has allowed us to explore the possible space of structures. The variability at the heart of the Darwinian mechanism is our best key to understanding life.

The same balance between rigidity and chance plays an important role in our brain, which functions, in spite of the illuminating similarities with good software, because of the ubiquity of statistics in his working. Actually, there is also software that effectively exploits the generation of random numbers. Examples are the techniques of type Montcarlo used in numerical calculations (eg in particle physics) and so-called techniques of neural networks, which have ended up in the software of our washing machines. But this is an episodic use of randomness. Our brain is a machine, but it is a machine that works in a manner where statistical elements play a continuous and persistent role, next to deterministic functions.

All this brings us back to the original question, the relationship between physical determinism and freedom to decide, but in the light of a new observation: the existence of chance compatible with determinism. How is this possible? It is not difficult to understand. No one doubts that the dynamics of a balloon full of air has nothing indeterministic. But no one is able to predict the motion of a single molecule of air. If you untie the knot that closes the balloon and leave it free, this will empty noisily flitting here and there in a way that no one can predict. How is it possible, in these cases to reconcile physical determinism and unpredictable behavior of the balloon? The answer is simple and well known. Is that in principle we can give two alternative descriptions of the balloon, both correct. The first consists in giving the position of each of its molecules of air, the second in giving simply the radius of the balloon, and for example, the pressure with which it is swollen. These two descriptions are not in contradiction with each other. They are simply two descriptions, one more accurate, the other less, of the same object. There is a key point that links the two descriptions, summarized by a concept that plays an important role in contemporary philosophy: that of "supervenience". If I know the radius and the pressure of the balloon, there are many different configurations in which the molecules can be, so two balloons that appear identical in the sense of having the same radius and the same pressure may in fact differ in the position; but (this is the key point of the definition of supervenience) the reverse is not possible: that is, it is not possible that two balloons with different radius or different pressure have the same configuration of their molecules. In this case, we say that the radius and the pressure are properties of the balloon that "supervene" to the elementary properties of the molecules. Armed with this observation and this definition, we can return to consider the problem of free will. 

To interpret "free will" as the absence of external determinations is possible, but leads us to say that the Rover on Mars has free will, and this does not seem to capture what many intend for free will. The behavior of the Rover is determined by software that, at least as long as it works, acts in a predictable manner. The behavior of our brain, instead, made up of billions of neurons that work in largely probabilistic fashion, fluctuates widely even when the same inputs both external and from its own memory are close. In other words, the brain machine appears to function more as a probabilistic machine than a deterministic machine. This is not in contradiction with a possible physical determinism underlying, or the fact that quantum indeterminism doesn't play any role. Simply, we can give no description of the molecular state of the neurons capable of uniquely determining the future decisions. 

At this point I think that some aspects of the problem are clearing up, and I can go back to the original question: let's say that a human being is free to choose. What does this mean? If we mean that two human beings can behave differently if they are placed in the same "external" conditions and also have the same "internal" state, then we must also specify what do we mean, here, by internal state. If we understand the set of memories, education, emotions, thoughts, and so on, we are still giving a description of the system that is not the positions of the individual atoms, and therefore also with these factors equal, the fact that the same two human beings can decide differently is not in contradiction with the existence of an underlying physical determinism. It is no more surprising than the fact than two balloons that appear identical, with the same pressure, the same radius, the same color, the same plastic ... move in a completely different way when let loose with loose knots.

But is this a satisfactory answer? Is it true that all mental properties supervene to a physical description? Imagine an extreme case, a bit artificial, perhaps, but significant. Imagine a page with printed characters. By a strange coincidence, these characters closely resemble Chinese characters but you can also see them as Latin characters. In fact, a Chinese would see a line from the great poet Li Po, while an English speaker reads an English verse by Shakespeare. The position of the atoms of the page is one and only one, but the content of the poem is perceived differently. Two different contents correspond to the same microscopic configuration. The example shows that not necessarily the content of the text supervene to its physical configuration.

But the same example also provides the solution.  It is possible to say that the page does not contains neither the poetry of Li Po nor that of Shakespeare. The meaning is not determined just by the medium (the page shown), but also by the external context, the cultural context, by a framework involving other external systems. But this does not prevent the conclusion to be the same as before, only widened: meaning does not supervene to the paper and the ink, but to the ink, the paper, and the physical state of those reading the paper.

In either case, the main point remains: mental states, whatever we mean by that and whatever the amount of information they imply, contain immensely less information than the information necessary to determine the full physical state of the brain, which, let us not forget, has about a million billion synapses, but is composed of a number of molecules still much larger. So the fact remains that, however you look at it, to the same mental state correspond a large number of molecular states.  And in any case, the relationship between the former and the latter is statistics.  There is therefore no reason a physical determinism should determine psychic determinism.  Physical determinism is perfectly consistent with psychic indeterminism.

There remains one last question, the crucial one, and the main reason for me for writing this piece. Trying to force the meaning of "free will" beyond the simple meaning of freedom from "exterior" constraints, is an enterprise doomed to failure anyway.  Is our "free" decision completely determined by internal factors? Let's assume for moment that it is not, and we see that we are in trouble. Suppose to be able to do an experiment where we can put a person in exactly the same mental situation (with the same memories, values, character, mood ...) and suppose we repeat the experiment many times, always with the same initial conditions. What would observe? There are two extreme possibilities: the first is that we see that the person will decide entirely at random. In this case the results will be just governed by chance. Half the time he will make a choice, the other half he will make the other choice. The second extreme possibility is that instead the person will always make the same choice.

In which of these two cases, is there free will?

Both answers are meaningless. If we answer in the first case, we are saying that free will is manifested when we decide completely at random, throwing a coin. I do not think that this is what people believing in free will mean.  If so, we must conclude that we go to heaven or hell by pure chance.  But the second answer is even worse: in this case free will is to be determined by our own internal mental states! That is, it means the absence of free will! In either case, we are in trouble, and this shows that the idea that free will may have to do with the ability to make different choices on equal internal states is an absurdity.

All this brings us back to the only possible solution of the problem of free will, which is a classical solution, the one that was put forward in the Ethics by Spinoza. We humans are complex systems and develop a picture of the world and of ourselves. We look for causal connections in the world and then construct a number of interpretative representations, which allow us to predict to some extent the behavior of the world. We do this also in reference to ourselves. We have a representation of ourselves and this allows us to know how we will act, or how another person will act in this or that situation. But this representation we have of ourselves is extremely crude and approximate compared to the complex details of our own real being, and thus we find ourselves acting continuously in manners that we are not able to predict, neither for others nor for ourselves. When we observe behavior unpredictable in us or in others, we call this "free choice" and "free will".  There is nothing wrong or illusory in this use of the term; it is a reasonable name, suitable for an approximate description. There is no contradiction between the use of this concept and the fact that our behavior arises from the movement of the molecules of our body, and that this can be, at the molecular level, perfectly deterministic. This is the only reasonable solution of the apparent tension between determinism and free will.  Quantum theory, statistical physics, neural biology, cognitive sciences, and the rest of the knowledge that we have

Any attempt to link this discussion to moral, ethical or legal issues, as is often been done, is pure nonsense. The fact that it is possible to say that a criminal has been driven to kill because of the ways in which Newton's laws have acted on the molecules of his body has nothing to do either with the opportunity of punishment, nor with the moral condemnation. It is respecting those same laws by Newton that putting criminals in jail reduces the murders, and it is respecting those same laws by Newton that society as a whole functions, including its moral structure, which in turn determines behavior.  There is no contradiction between saying that a stone flew into the sky because a force pushed it, or because a volcano exploded.  In the same manner, there is no contradiction in saying we do not commit murder because something is encoded in the decision-making structure of our brain or because we are bound by a moral belief.

Free will has nothing to do with quantum mechanics. We are deeply unpredictable beings, like most macroscopic systems. There is no incompatibility between free will and microscopic determinism.  The significance of free will is that behavior is not determined by external constraints, nor by the psychological description of our neural states to which we access. The idea that free will may have to do with the ability to make different choices on equal internal states is an absurdity, as the ideal experiment I have described above shows.  The issue has no bearing on questions of a moral or legal nature. Our idea of being free is correct, but it is just a way to say that we are ignorant on why we make choices.