Getting back to free will, a natural first question is then: when and how does the 1^st person perspective arise? We know we have it, but at what level does this perspective emerge? Does a single particle have it? An atom? A molecule? A large configuration of molecules? Only a certain configuration of molecules? A cell? Only a special configuration of cells? All of these seem somewhat arbitrary and it feels implausible to suggest that the 1^st person perspective is not present in any level of matter until the final, critical particle or molecule is added, and then, suddenly, a whole new 1^st person perspective emerges. There are formal arguments against free will as an emergent phenomenon from the field of philosophy detailed here, for example, supervenience, but these don’t seem any less baffling than emergence itself. Besides, sleep can be induced in humans with certain anesthetics, radically altering their 1^st person experience temporarily. This suggests that whatever does give rise to the 1^st person perspective, it is dynamic, transitory, and depends on the environment. So, where to start?

 

This paper, “The Strong Free Will Theorem” by S. Conway and S. Kochen (2008), describes a rigorous theorem of physics called the free will theorem, the meaning of which the authors summarize as follows: “if people have free will then so must elementary particles”. Perhaps a more precise description of the results of this paper is that if people’s actions are not determined solely by their histories than elementary particles are indeterminant as well. Scott Aaronson in “The Ghost in the Turing Machine” (2013) describes the theorem more cynically: “for the indeterminism that’s relevant here is ‘only’ probabilistic: indeed, (people and elementary particles) could be replaced by simple dice-throwing or quantum-state-measuring automata without affecting the theorem at all”. Suffice it to say, we won’t dwell on the arguments of the theorem here (if you’d like to dive deeper check out this video here, or this talk by S. Conway here). Instead, elementary particles, like the electron, seem like as good a starting place as any. After all, they are the most fundamental things in the Universe, and, therefore, feel like a less arbitrary place to start. So, we’ll jump on Freeman Dyson’s bandwagon (opening quotation) and run with it and see where it takes us…

 

In 1922, in the Stern-Gerlach experiment, silver atoms were fired through a magnetic field as depicted in (figure 9). The magnetic field deflected the atoms upward or downward depending on the direction the silver atoms were spinning. Classical physics expected a continuous distribution on a detector on the far side of the apparatus (see -4- in figure 9) because the silver atoms could be spinning in any direction. The distribution that was actually seen was two singular points (see -5- in figure 9) proving that the silver atom’s spin was quantized. We’ve already said that the electron’s spin is quantized and if we perform this experiment on electrons instead of silver atoms we will see the same result – all the electrons will end up at one of two spots depending on whether their spin is pointing up or down. Just like in our toy box metaphor, this Stern-Gerlach apparatus is one way of measuring the spin of the electron. Now, physicists know exactly how to calculate the quantum mechanics of this problem and it says we will see the electron end up at the upper point with fifty percent probability, and at the lower point with fifty percent probability. But, that is all it will say. There is no way to know more about the electron’s spin than this probability – there is no way to predict it. It is truly random.

 

 

Figure 9: Stern–Gerlach experiment: silver atoms travel through an inhomogeneous magnetic field and are deflected up or down depending on their spin. 1: furnace. 2: beam of silver atoms. 3: inhomogeneous magnetic field. 4: expected result. 5: what was actually observed. Image and caption by Tatoute at Wikipedia.

 

So, now, let’s just suppose the 1^st person perspective is present in the electron and it does make a choice. From the electron’s perspective, it feels the quantum probability distribution manifest as a matter of preference. In this case, since the quantum probabilities are equal, it is indifferent, it prefers both choices equally and so it selects one with equal probability. To the physicist, the electron follows the laws of quantum mechanics exactly, the wave function collapses to one of two outcomes that are equally probable as given by the rules of quantum mechanics. Both the 3^rd person and 1^st person descriptions of the event are present, valid, and equivalent – as, obviously, they must be if we are going to claim the electron has a 1^st person perspective dual to quantum mechanics. Now, notice that we forced this choice (measurement) upon the electron. It did not have any say in the matter, we just fired it through the Stern-Gerlach apparatus. Nor does it have any memory of the measurement afterward. After our experiment, the electron goes on its merry way constantly being measured, forced to make choices by the surrounding environment, and having no means to retain a memory of those choices nor any future anticipation of choices to come. No past, no future, just moments of “now”. No understanding, no self-awareness, no consciousness as we know it, just repetitive, uncontrollable, forced choice. Practically speaking, we are talking about roughly nanoseconds before an average electron interacts with something again and is measured – these moments are very brief and fleeting indeed. In a subsequent section, we will explore in more detail what it is like to be an electron.

 

There is nothing magical about a fifty-fifty proposition by the way. The electron could have been prepared, prior to entering the Stern-Gerlach apparatus, in a superposition of 90% up and 10% down, or 70% down and 30% up, etc. Quantum mechanics predicts the outcome for the 3^rd person view precisely, as given by the skewed wave function, and the electron, in the 1^st person, experiences the higher probability choice as feeling more compelling – like ice cream versus spinach – and chooses accordingly. Measurement is still random in 3^rd person and free choice in the 1^st. Measurement and choice are dual to each other. Dual descriptions of the same thing.

Figure 10: A single electron fired through a Stern-Gerlach apparatus. Electrons enter the apparatus moving right to left in a quantum superposition of states – each electron is both spin “up” and spin “down”. The electrons interact with the magnetic field of the Stern-Gerlach apparatus and are projected onto a detector. Those with spin “up” end up at point “A”, with spin “down”, at point “B”. The electron remains in a superposition of “up” and “down” all the way up until measurement at “A” or “B”.

Now, what happens if we run this experiment with a system of 3 entangled electrons connected in some way (a 3 electron chain) as shown in (figure 11)? Several things change in very interesting ways (we ignore the technical details and practical challenges in doing this for now). First, there are four possible outcomes instead of just two: (1) we can have all 3 spins pointing up, (2) two spins up and one spin down, (3) two spins down and one spin up, or (4) all three spins pointing down. Incidentally, states (1) and (4) form something known in quantum computing as a GHZ state (Greenberger-Horne-Zeilinger state). Also, note there are three different combinations of forming state (2) and state (3), but just one way to form states (1) and (4). Second, the system may exist in a superposition of up to eight different states at once, , so our little electron system is beginning to hint of a basic quantum computer. Third, even after measurement, unless all the spins are pointing up (point A in figure 11) or all the spins are down (point D in figure 11), the system is still left in a superposition of states. There are three different states it could be in assuming it ended up at point B (figure 11), and three different states it could be in if it ended up at point C (figure 11). Measurement does not destroy the system. It reduces the superposition, certainly, but the entanglement persists. Fourth, if the electrons end up at B or C (in figure 11), the superposition retains something that could serve as elementary memory: all the remaining three states have a symmetry to them – the sum of their spins is +1/2 (at B), or -1/2 (at C). If the system is subjected to the same measurement again, it will end up at the same spot – a memory of the prior measurement results is stored in the superposition. The system is in an eigenstate of the measurement with eigenvalue +1/2 (at B), or -1/2 (at C). For example, looking at (figure 11), all three states of the electron system, if it ends up at B, have two spins up and one spin down. Each spin up is +1/2, each down spin is -1/2, so the sum for each state is +1/2 (e.g.+ ½ + ½ – ½). Last, because we measured an aggregate property (its total spin) of the system, the entanglement is not broken – it remains one system and can be measured again. We will come back to how this quantum memory can then be converted into a more permanent memory in subsequent sections. Other properties of the system can, in some cases, be measured too, without disrupting the system. This can happen if, in quantum-speak, the measurement operators commute (the interested reader can dive deeper into quantum operators here). For example, we can measure the momentum of an electron along the x-axis without disrupting the momentum along the y-axis. Different directions of spin can’t be measured simultaneously, however, but, some quasiparticles, which we will come to later, have infinitely many conserved quantities and can therefore be measured in infinitely many ways.

 

In the 1^st person the entangled system chose from four different subjective preferences, say ice cream, pretzels, brussels sprouts, or spinach. Some choices may be more likely than others with the electron’s subjective preference for each equal to the quantum probability of that outcome. A rudimentary memory persists of its choices. If subjected to the same choices again, the answer will be the same. And, the system’s identity as one thing persists beyond making a choice – it lives to choose again! J

 

 

Figure 11: Chain of three entangled electrons represented as
are fired through the Stern-Gerlach apparatus. (3^rd person description): The 3-electron-system going into the apparatus may be in a superposition of eight states simultaneously. The magnetic field will deflect the electrons where they will be detected (measured) at one of four locations, labeled A, B, C, or D, depending on the spin state. The probability the electron-system is in each state is given by a coefficient , where sum = 1. States with greater net spins are deflected more severely. If deflected to locations C or B, the electrons still persist in a superposition, albeit a reduced one. In any case the system remains entangled. (1^st person description): The 3-electron-system must make a choice upon traveling through the apparatus. The appeal of each choice is equivalent to the quantum probability of finding the system in that state. The electron system after its choice retains a memory of that choice and its identity as “one thing” persists.

 

As the number of entangled electrons grows larger, say to 100, or 1000, suddenly our system takes on the appearance of a powerful quantum computer – it could be in or different states simultaneously. The system of electrons still does not get to decide when to make a choice – we continue to force choice upon it by firing it through the apparatus, but the number of choices available to it, and the vastness of the superposition that can survive a choice increases substantially. For example, say a system of 100 electrons was found to have total spin of +1/2 up. There are ~ ways that 100 electrons, each having spin +1/2 or -1/2 can sum to a net spin of +1/2, so the system could still be in a superposition of this many states even after measurement. Also, the system could potentially choose from an array of 200 different possibilities (there are 200 different possible total spins). The degree of freedom of choice has increased substantially, although nothing resembling our free will yet, and a complex memory reflecting the results of those choices has emerged.