This is an idea I just got today. I might expand upon it later.
Basically, the processes by which our brains function must either be classical in nature, or envolve a combination of classical and quantum mechanics.
Let us for the moment suppose it is the former.
The very nature of classical physics is determinacy - if you know the positions and momenta of a group of particles that act classically, with enough computing power you can determine precisely what condition the system will be in at any point in the future. Sure, even a simple scenario (eg three planets in orbit around each other) can get very complex indeed. But the point is, it's possible, in principle, to predict with utter precision, because all the particles follow determinable ballistic trajectories.
Therefore, if the brain functions in a purely classical manner, it would be theoretically possible to predict how any individual's brain will respond to any particular set of sensory input.
Therefore, a classical brain means that free will doesn't exist. With the state your brain is in right now, the next thing you think is predictable. The next action you take is inevitable.
Specifically, the information needed to theoretically predict what a particular brain will do next, is the layout of the neural network of neurons and axons, the chemical composition of the fluid that exists between the individual synapses, the current state of electrical signals being sent down all the individual axons at this present moment in time, and the sensory information that is being fed into the brain.
However, these are mere details: the central point remains - a wholy classical brain has no free will, no freedom of thought, and no freedom of action. Only the illusion of these is present, due to some feedback loop of the neural network.
Now lets consider a quantum scenario:
Simply put, if an individual neuron can be considered to be a quantum system, the state of this individual neuron may be described as a|A> + b|B> where "A" corresponds to the neuron doing nothing, and "B" corresponds to the neuron firing. "a" and "b" are associated probability amplitudes, and are complex numbers (i.e. of the form c + id).
The point is, that here nothing is certain - by definition, quantum mechanics envolves probabilites. While these probabilies can be calculated with utter precision, the outcome of measurements cannot, of course, be determined precisly beforehand.
The neuron doing nothing has a probability equal to the complex conjugate of "a" (a*a), that of it firing is b*b.
Obviously, our brain are not entirely random. However, the point is that evoking possible quantum mechanical effects in the brain, as some (eg Roger Penrose) have, yields randomness, to a varying degree.
Inherent randomness, however, also kills the idea of free will. It means that our minds are by definition unpredictable, within certain bounds, but also removes any supposed power of choice that we have, at least up to a point.
However, I will say this also: the computer I'm writing this on is reasonably impressive. It boasts 2.8 GHz of clock speed, and 512 MB of RAM. It is utterly predictable. Knowing the source code of a particular program, how the particular compiler used works, and the internal workings of the CPU etc, will tell you exactly how a particular program will function. Computers are classical in nature.
I strongly suspect classical mechanics holds sway in our brains also. Quantum mechanical effects (for example Heisenberg uncertainity of position of molecules in the synapses) are undoubtably present, but probably cancel each other out on average, and have zero net effect on the thought processes.
Either way, free will simply doesn't exist. Our brains are nothing but biological computers, classical or not.
