Rewrite end of introduction to be accurate with the actual plan of chapter 2

[github-actions]

I wrote the intro of chapter III so long ago i don’t even know if it’s accurate anymore

milestones:

https://github.com/tefkah/thesis-writing/blob/feb2b179fdba26c2122ff47145d31634631427f6/Chapters/III. Anyons/1. Why Anyons?.md#L19

---

The main subject of discussion here will be a new class of (quasi-)particles:
anyons. In undergraduate courses, or, if you're lucky, highschool, we learn that two types of particles exist: fermions and bosons. These particles are distinguished by spin, half-integer and integer spin respectively. Anyons, as their name suggests, break this binary and are allowed _any_ type of spin, creating a whole new category of particle.[^1] While anyons are fascinating in their own right, we are interested in them because, according to the canonical explanation, anyons are two-dimensional particles. Phrased more suggestively, the space anyons occupy cannot be _approximately_ $2D$, such as a $3D$ space of $1nm$ height, but _exactly_ two dimensional: a clear case of an infinite(simal) idealization. Of course, we would not be talking about anyons were it not that, at the time of writing, anyons have rather strong empirical backing [@Bartolomei2020].

Another reason why anyons present such an interesting case is that their explanation explicitly requires _topological_ arguments. Topological arguments, as I will show, show up in unexpected places and allow problematic idealizations to sneak in, as they smooth out many of the difficult to solve geometry. By tackling such an explicit use of topology in an infinite idealization, we will be able to use the argumentative structure in our general analysis of infinite idealizations. `The opposite of topology is geometry`

## What are we going to do

The following chapter will consist of four sections. In section 2, we will first attempt to understand what anyons are and why they are commonly thought of as a uniquely 2D phenomenon. The most important feature turns out to be how to define _path-uniqueness_, which is commonly defined either topologically or geometrically, the latter being the focus of section 3. To understand the topological notion of path-uniqueness (homotopy) we will first examine a more simple 3D case before diving into the actual 2D one. Using a toy model we show that anyons are not a uniquely 2D phenomenon by constructing a [[Simply Connected | multiply connected]] 3D space using somewhat plausible physical assumptions.

After working through the problem topologically, we turn our gaze to how path uniqueness is construed using the geometry of the space. One of the downsides of the topological explanation of anyons is that it is just that: an argument for *why* anyons show up in 2D. The topogical account does help us actually calculate the spin of particles in any real physical system, whereas a more specific geometric account could. It turns out that said geometric account (the so-called Berry-Phase) relies heavily on the physical features of the system, making it difficult to construct and evaluate a generic geometric explanation of anyons. We therefore examine a physical system which purportedly produces anyons in order to compare the explanatory power of the topological and geometric explanations: the Fractional Quantum Hall Effect (FQHE).

In section 4 we work through the FQHE, showing why anyons are involved and, importantly, what idealizations play a role in the explanation of the effect. Before doing so, we recap how particles contained in magnetic fields `get` quantized energy levels (Landau levels) and the basics of the simpler variant of the FQHE, the Integer Quantum Hall Effect (IQHE). These can be skipped by the impatient reader, as while some specifics of the IQHE carry over to the FQHE, most of the conceptual basis is retained, which is recapped at the end of section 3.XXX

{/** TODO: Rewrite end of introduction to be accurate with the actual plan of chapter 2 
    * I wrote the intro of chapter III so long ago i don’t even know if it’s accurate anymore
    * labels: III 
    * milestones: 
    */} 


Finally, we put all the pieces together in section 5, and see that while a geometric explanation _could_ be possible, it is not, at the moment, feasible. Later, in chapter `XXX`, I will argue if and why such a potential explanation should still be preferred to the topological account, but such conclusion cannot be drawn until we have gained a more thorough understanding of what a good explanation is in the first place.