…and yet in physics we apply quantum physics to measurement devices all the time. In doing so we include the measurement device as part of a larger quantum system: the ‘measurement’ process then becomes just one more dynamical process, described unitarily in terms of evolution on Hilbert space. Conceptually speaking this should not help at all, since it just pushes the problem back: in order to interpret that larger quantum system we need recourse to a primitive notion of measurement of that system. And if we try to model that process of measurement too, we need yet a third primitive notion of measurement, and so on ad infinitum .

David Wallace

Quantum mechanics studies the underlying structure of our reality. It represents states as normalized vectors in Hilbert space, and observables as operators acting on those states, spitting out the desired quantities. Beyond formalisms, there are as well various interpretations of Quantum mechanics, each attributing significance to the ways our world is configured. This essay will address specifically the Many-World/Everett Interpretation (MWI), and under which how a theory against determinism, as differs from conventional understandings of MWI, might be formulated. We will start by refamiliarizing ourselves with the mathematical formalism and MWI interpretation, and discuss the nature of measurement, collapse, and branching, before eventually arriving at an argument for the existence of free will.

To understand what Quantum mechanics does we must first clarify what physical theories, of which Quantum mechanics is a subset, do or aim to do. A physical theory aims at uncovering the laws under which nature evolves. It commits ontologically to some quantities which we use to parametrize the dynamical laws, and by which we arrive at a predicted state of affairs of the future. For instance, Classical or Analytical Mechanics is a perfect example of physical theory: 1. it posits physical objects (particles) and some properties of physical objects (position, momentum, energy, etc.). 2. These properties, combined with the dynamical laws (Newton's laws), deterministically produce states of affairs of the future. In Quantum Mechanics, too, the two 'axioms' are satisfied, but in a different way. First, the ontology posited is not the physical reality such as particles in a traditional sense. It is instead the wavefunction that propagates, without a definite position and momentum. To find out a particle, the wavefunction must be localized, or observed, so that it emerges more definite properties; we can extract the probability distributions of those properties by acting operators on the wavefunction correspondingly.

Under Everett Interpretation there is no uncertainty. The probability distribution derived from the wave function does not represent an indeterminacy of how the world will evolve, but simply how it branches. In some senses, it represents our ignorance, given the complete information about the universe (its wave function and dynamical laws), we still do not know what branch will we be in after entanglement. The probability, therefore, is not really a probability which denotes true randomness, but our inability to deduction given our embedded, or indexed, position within the world.

This is more or less an agreed understanding of the Everett interpretation. The universe is globally determined, yet we are locally ‘free’. The branching structure of all time is fixed, as the set of our possibilities, but we never know which possibility will we go down eventually. I aim to provide a stronger case against this global determinism.

We are determined, in the sense that we are part of the world. The world is determined in that it can be represented by a wave function, which evolves according to its Schrödinger equation and, even when it branches, the branches are determined. However, the branches are determined only with respect to an external observer. Imagine yourself in the middle of a Stern-Gerlach experiment or outside the box of a Schrödinger’s cat. After the observation, you are entangled with the observed: the electrons are going either up or down, and the cat is either dead or alive. You, together with the world you are observing, branched into one of the two situations which are determined beforehand. However, it is only with you, that is, with entanglement, that branching happens. Without observation, the electrons are not either up or down, and the cat is not either dead or alive. To you, they are still in a state of superposition represented by their wave functions.

What I want to suggest is a fairly simple fact: branching is observer-dependent, therefore the branches that are determined require as well an external observer. However, since the branching happens precisely when the observer becomes gradually entangled with the system, the latter is no longer a completely isolated system, and the wave function describing it changes as the original observer merges into the system. That is to say, there is no completely isolated system with respect to a transtemporal external observer since the observer always becomes part of the system after the observation. Imagine another observer observing this branching process. For her, the wavefunction and thereof the branching will consequently be updated after the first observation. For the second observer, the first observer and the original isolated system become a larger isolated system, and will become no more isolated to her since herself will be merged into it as well after this observation.

If by determinism, as discussed, we mean that even particular entities within the observed system do not know what branches it would go down, all the possible branches are determined and it is not a sensible question to ask as to which branch will it go down, since it goes down all the branch at the same time. However, as the above argument suggested, branching is in fact an observer-dependent process that is changing at all times, for the observer in the process of observing in turn becomes the observed to another observation. The structure of branches is not tree-like as commonly understood, in which one system, after being observed, splits into different worlds as described by its original wavefunction. This structure assumes an atemporal, ‘transcendental’ observer looking at the whole of the world, or worlds, and at precisely which time it splits into what branches.

However, this observer cannot exist, since observation is necessarily an act within the bounds of space and time. Any branching is determined only at, or the moment before, the exact time in which an isolated system is observed or measured. In a sense, branching is a local instead of a global phenomenon, and global determinism as we described relies on the idea of branching being global. Therefore, this determinism does not make sense.

What is determinism, or physical determinism specifically? Physical determinism means that the states of affairs at any temporally later point are determined solely by the initial condition and the laws under which it evolves. There has to be, nevertheless, an agent who is doing this 'determination', that is, who measures the system. If I am within an isolated system with respect to an external observer. When measured, I know my future branch is been determined. Yet, by the same logic, the observer in this process becomes entangled with me and my world. The branching happened, but to determine future branching, another isolated observer is required, and this process of finding external observers recurs ad infinitum , there is no such absolute observer whom we can treat as isolated at all times, which means there is no one capable of doing the determination.

Recall the famous quantum entanglement in which Alice and Bob both take one of the entangled objects and separate by a distance exceeding some unit of lightyear. When Bob observes his objects, the wavefunction of Alice's object suddenly 'collapses' which is a nonlocal action. The MWI explanation of this is that there is no collapse of the wave function, what happened is simply that Bob is entangled with his object as he observes, which is in turn entangled with Alice's object. In a sense, Alice's object's wavefunction 'collapsed,' but only with respect to Bob; There is no universal wavefunction that collapses when it is collapsed with respect to a specific observer, that is, this 'collapse' is local, the wavefunction of that object 'collapses' with respect to Bob does not mean it also 'collapses' with respect to other thing such as Alice. The moral of the story is that we are again secretly falsely imagining an observer external to everything who controls this 'absolute' wavefunction, who measures without being merged or turned into part of the system, and who does the so-called global determination.

To conclude, this short essay has argued for an indeterminism within Everett interpretation of Quantum mechanics. MWI has been seen as a deterministic theory not because every agent indexical to the universe has a determinate worldline, but that its indeterminacy, that is, its structure of branching, is determined. Thereof even every object in itself has no determinate outcome at the end of the day, its possibilities split as its world splits so that it attains all the possibilities at once, and this is determinate. Contrary to the view, this essay argues that the branching structure is not determinate and the world is therefore not determinist. The branching is determined by its wavefunction with respect to an observer to which it is considered isolated. Yet the gist is that, when observation happens, the observer becomes part of the originally isolated system, therefore the world, its wavefunction, and branching all have changed. To determine updated branching recursively requires a new isolated observer, for there is no eternally external observer.

Branching thus is a local phenomenon, and can only be determined at a point in time in which the observer and the system are selected. Its structure, unlike regular trees which branch from all vertices, might be visually a set of connected branching structures in which only the vertices traversed by the agent branch out. The original tree-like structure might emerge out of the union of all these agent-specific branches, there is no, on the contrary, a completed tree-like branching a priori . If you are an agent within the world branching out, your immediate future, that is, the branching, is determined only at the moment where observation happens, no matter whether you are the observer or the observed.