The Free Will Theorem

Today we prove Conway-Kochen free will theorem:

SPIN + TWIN + MIN + (very limited) free choice of the experimenters 

= free choice of the particles

SPIN is just a basic statement about the behavior of a spin-1 particle. It states that angular momentum operators exist and they behave in such a way to make the 101 property be true (see below).

TWIN states that it's possible to entangle two particles, such that their spins, when measured in the same direction, are opposite.

MIN is stated rather obscurely. I believe it means that it is impossible for an event to depend on another event outside of its past light cone. This is basically what special relativity states.

Free choice is defined as "non-functional", that is, not described by a function. In this interpretation, to say I have no free choice in going left or right, is to say that there is a function $f$, such that the direction I am going is $f($everything in my past lightcone$)$.

This proof goes in two steps. The first step is the Kochen-Specker Theorme uses only SPIN. The second step uses TWIN and MIN to construct an entanglement separated by a very long distance (like all those Bell-inequality experiments), and then assume (a very limited amount of) free choice of the experimenters, but not the particles, to get a contradiction.

Step 1: Kochen-Specker Theorem (1966)

This is well-known and I will direct you to plus magazine's proof. First read this, then read this. For those who know a bit more quantum mechanics, here's what 101 property means: Consider a spin-1 particle. Let $S_x$ be the operator of the angular momentum along vector $x$ for the particle, then $S_x$ has three possible eigenvalues: $\hbar, 0, -\hbar$. Normalize by setting $\hbar = 1$, we find that $S_x^2$ has two possible eigenvalues: $0, 1$. Then, it can be shown that for any triple of orthogonal vectors $x, y, z$, we have $S_x^2+S_y^2+S_z^2 = 2$, and so the measurement results must be one of $(1, 1, 0), (1, 0, 1), (0, 1, 1)$.

Notice that since $S_x^2 = S_{-x}^2$, we can safely consider a direction as defined by a line through the origin, rather than a vector.

Another note: sometimes, the configuration of 33 lines is called the Peres configuration. It's easy to verify that, if we represent each line as a vertex, and connect two vertices iff they represent orthogonal lines, then we obtain a graph with 72 edges, making up 16 triangles (corresponding to triple-orthogonal-lines) and 24 edges that do not make up any triangle.

The Stanford Encyclopedia contains more variations and ways to escape the conclusion of the Kochen-Specker theorem.

Tech stack/hierarchy in mathematics

What are layers?

Consider the OSI model. In technology, in building a complex system, one often divides the system into layers with simple exterior appearance, but complex interiors. The layers are basically modules that have a direction: whereas "modules" are egalitarian, and can be assembled in many directions, layers have a fixed pecking order, and can only be assembled in one fixed direction.

Layers = Modules affected by gravity

Causality is not fundamental in the world

Today we tear down the illusion of causality, with quantum mechanics.

This started when I was reading The Order of Time (2018) by the rather poetic physicist Carlo Rovelli, and read that he said that causality is not certain, and two events can have a superposition of causality: a superposition of two possibilities: A causes B, and B causes A.

Quantum corelations with no causal order

The paper that started this seems to be a highly cited (over 200 currently) Quantum correlations with no causal order (2012), Ognyan Oreshkov, Fabio Costa, Časlav Brukner.

The abstract starts with a question: is causality fundamental?

Let's read: $C^*$-algebra, quantum mechanics, Pascual Jordan

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. ---- Hermann Weyl

After 3 courses in analysis I have finally reached Banach algebras, I feel exhausted by the climb and yet there's more theory ahead until the true pinnacle: the theory of $C^*$-algebra, for quantum mechanics!

Let's read, what is being promised?

$C^*$-algebras were first considered in quantum mechanics to model algebras of physical observables... began with Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of $C^*$-algebras which are now known as von Neumann algebras.

So experiments in quantum mechanics → weird behavior of observables in quantum mechanics → Heisenberg matrix mechanics → Jordan purified it into Jordan algebra → von Neumann generalized to rings of operators → von Neumann algebra generalized to $C^*$-algebras.