Does free will exist in the universe?

One of the major fundamental questions in physics concerns the presence or absence of free will in the universe, or in any physical system, or subset, within it.  

Physics is based on the idea that nature is mechanistic, which means that it works like a machine. A machine is just a system, and therefore, by definition, it is a collection of elements, each of them with a specific, possibly different function, all working together to achieve a specific purpose, general to the whole machine. 

For example, a musical ensemble is a system of people, each playing a different musical score, and all controlled by a director, such that, as a whole, the group can correctly execute a specific melody. In physics we study the systems found in nature by constructing a model that represents them as realistically as desired. 

Let’s now consider the whole universe as a system: according to the best of our knowledge, material particles can never be measured closer to each other than the Planck length (the number 16 preceded by 34 zeroes and a decimal point, or trillionths of trillionths of trillionths of one meter), suggesting that, for matter, states are discrete. 

Even if energy does not seem to suffer from a similar Planck length limitation, we know from quantum mechanics that energy is transferred between physical bodies in discrete amounts, known as quanta. Hence, either Planck length or energy quanta can be considered as the relative sizes of the “pixels” composing the universe. However, this description seems incomplete.

If we take a continuous function, like a straight line – y = a * x + b – where a and b are fixed parameters and x can assume arbitrary values, we know that for arbitrarily, infinitesimally close values of x, we will still be able to compute the respective values of y without confusion. The trajectory of an apple falling from a tree can be modelled by a continuous straight line that approximates an otherwise fractal curve. 

Fractals curves are recursive functions that require an initial state. Fractal curves require a continuous world to evolve, especially if we know that the initial condition must be an irrational number. From this, we can deduce that the behaviour of physical objects seem to be ultimately dictated by continuous functions somehow perceived by humans through a grid of these “pixels”. It is self-evident that we always need to divide such things into discrete sections, anyway.

If we believe in the Big Bang Theory – and the universe’s continuous expansion is a strong indication that such theory must be correct – the initial state of the universe was a single point (known as a singularity) that then expanded to the cosmos we know and perceive today, which, of course, includes us. 

If so, there is a causal relationship between the Big Bang and us. In other words, free will is not allowed, and all of our actions are just a mere consequence of that first event. Such a view is known as “determinism”, or “super-determinism” (if one finds it productive to reinvent the wheel). 
If we believe the initial state of the universe to be quantified by a rational number, we are inferring that it is periodic, non-chaotic and globally predictable in nature. But if the initial state is rather quantified by an irrational number, we are instead inferring that the universe is aperiodic, chaotic and therefore only locally predictable in nature. 
Today, we know that the universe is chaotic.

From such a view, one can be tempted to interject that if free will does not exist, why do we punish criminals? It is not their fault, after all. A counter-argument to that is that punishment is the natural response to crime, such that global equilibrium can be sustained, and therefore punishment is just as unavoidable as the commission of wrongdoing. 

Because the cosmos is clearly chaotic, we can observe time-reversibility only locally, rather than globally. This in turn means that free will is an inevitable illusion for us humans, due to our subjective perception of the universe, rather than its innermost nature.