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Chapters/III. Anyons/1. Why Anyons?.md

@@ -5,7 +5,7 @@ tags:
 
 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`
+Another reason why anyons present such an interesting case-study is that the current prevailing 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
 

Chapters/III. Anyons/2. Topology and Anyons.md

@@ -3,13 +3,17 @@ tags:
   - subchapter
 ---
 
-In a standard physics undergraduate program, you learn that there are two types of particles, fermions and bosons, which are distinguished by their spin, fermions coming in half-integer multiples $(\frac{1}{2}, \frac{3}{2} , ... )$  and bosons in integer multiples $(0, 1, 2, ...)$. A good starting question for understanding anyons would be: why _do_ we think there are only two types of particles? Why not $1$, or $3$, or infintely many? While many similar questions are as of yet unanswerable, such as why there appear to be only three generations of leptons (e.g. electrons, muons, tauons) or three fundamental forces in mhe standard model (electromagnetic, weak, and strong), this one does have an accepted answer. It is, as always, a combination of experimental results and theory. Fermions and bosons' share few features besides the fact that they are particles, but the important commonality is that they are both_indistinguishable_ particles. Every electron is the exact same as every other electron, and every photon is perfectly identical to the next. There is no way to tell whether two fermions or two bosons have switched places when you weren't looking. We will explore the consequences of this in more detail later, but the important consequence of this fact is that it induces an overdescription of the physical situation: we can mathematically describe two situations, one in which we have electron $1$ on the left and electron $2$ on the left. While mathematically distinct, these situations are physically identical, and in order for our physics to make sense we need to account for this distinction. The reasons for assuming indistinguishability could be a thesis topic in its own right, and the above description does not attempt to do it justice. The important point is _how_ we arrive from this indistinguishability to the fact that there are two types of particles: how does indistinguishability help us distinguish particles, and how do we mathematically distinguish these identical states?
+In a standard physics undergraduate program, you learn that there are two types of particles, fermions and bosons, which are distinguished by their spin, fermions coming in half-integer multiples $(\frac{1}{2}, \frac{3}{2} , ... )$  and bosons in integer multiples $(0, 1, 2, ...)$. A good starting question for understanding anyons would be: why _do_ we think there are only two types of particles? Why not $1$, or $3$, or infinitely many? While many similar questions are as of yet unanswerable, such as why there appear to be only three generations of leptons (e.g. electrons, muons, tauons) or three fundamental forces in the standard model (electromagnetic, weak, and strong), the fact that there are only two distinct classes of particles does have an accepted explanation. 
 
-As a warm-up, let us consider a rather handwavey argument. Handwavey, because it relies on the particles still being distinguishable, but it illustrates the point. Say we start with a system $\Psi_0$  of two particles, $x_1$ and $x_2$, each with their own probability distribution $\psi_{i}$ 
+It is, as always, a combination of experimental results and theory. Fermions and bosons share few features besides the fact that they are particles, but the important commonality is that they are both _indistinguishable_ particles. Every electron is the exact same as every other electron, and every photon is perfectly identical to the next. There is no way to tell whether two fermions or two bosons have switched places when you weren't looking. We will explore the consequences of this in more detail later, but the critical consequence of this fact is that it induces an overdescription (a symmetry) of the physical situation: we can mathematically describe two situations, one in which we have electron $1$ on the left and electron $2$ on the left. While mathematically distinct, these situations are physically identical, and in order for our physics to make sense we need to account for this distinction. The reasons for assuming indistinguishability could be a thesis topic in its own right, and the above description does not attempt to do it justice. The important point is _how_ we arrive from this indistinguishability to the fact that there are two types of particles: how does indistinguishability help us distinguish particles, and how do we mathematically distinguish these identical states?
+
+As a warm-up, let us consider a rather handwave-y argument. Handwave-y, because it relies on the particles still being distinguishable, but it illustrates the idea we will be exploring. 
+
+Say we have a system $\Psi_0$ of two particles, $x_1$ and $x_2$, each with their own probability distribution $\psi_{i}$ 
 
 $$\Psi_0=\psi_a(x_1)\psi_b(x_2)$$
 
-Now, for our next trick, we will exchange the two particles, such that particle$x_1$ is in state $\psi_a$ and particle $x_2$ is in state $\psi_b$. If we suggestively take state $\psi_i$ to be mean something like "having probability $1$ of being found around $x=a$", then this exchange can be the physical exchange of the two particles. Now, of course, we do not know which particle is which, so writing down the 1-exchange wave function would be getting ahead of ourselves, but we do know that if we were to exchange the particles once again we should regain our initial wavefunction $\Psi_0$. To denote this, we define an exchange operator $P$ which does just, and say that the wavefunction after exchanging twice (=rotating by 360 degrees) is 
+Now, for our next trick, we will exchange the two particles, such that particle $x_1$ is in state $\psi_a$ and particle $x_2$ is in state $\psi_b$. If we suggestively take state $\psi_\alpha$ to be mean something like "having probability $1$ of being found around $x=a$", then this exchange can be the physical exchange of the two particles. Now, of course, we do not know which particle is which, so writing down the 1-exchange wave function would be getting ahead of ourselves, but we do know that, were we to exchange the particles once again, we ought to regain our initial wavefunction $\Psi_0$, as we would have recovered the initial configuration. To denote this, we define an exchange operator $P$ which does just, and say that the wavefunction after exchanging twice (=rotating by 360 degrees) is 
 $\psi_{2\pi}=P^2\psi_0=1\psi_0$. To find the wavefunction of the 1-exchange system, we find:
 
 $$\psi_{\pi}=P\psi_0=\sqrt{1}\psi_0=\pm 1\psi_0$$
@@ -18,8 +22,7 @@ This yields two possibilities for particles: those for with $P=-1$ and those for
 
 The above argument is too handwavey, so we would like to make it more concrete.
 
-The way to define what we mean by "exchange*ability*". To start, the suggestively italicized _ability_ part of exchangeability hints at the fact that the _possible_ ways a particle _is able to be_ exchanged is of central importance.
-One natural way of defining this is by looking at all the possible paths the particle can take. We would need to look at the _configuration space_ of the particles: the possible configurations of them and connections between these configurations.[^2] Now we need to supplant this with a notion of what it means for two paths to be _the same_, or, equivalently, what it means for two paths to be different. This idea of path-similarity depends on which factors we judge relevant when considering manipulating quantum particles, which we need to choose if we wish to obtain the most general description possible. Like most of physics, we would not want our description to depend on some sort of absolute position or orientation in space, so our description better ignore those. Beyond that, however, identifying the relevant factors of the space becomes more tricky. The two most promising candidates are a _geometric_ or _topological_ notion of path similarity. As it turns out, the latter is used in the most commonly accepted explanation of anyons, the main reason being the mathematical and conceptual simplicity it brings.
+To start, the _ability_ part of exchange*ability* hints at the fact that the _possible ways_ a particle _is able to be_ exchanged is of central importance. How can we make this more concrete? One natural way of defining this is by looking at all the possible paths the particle can take. We would need to look at the _configuration space_ of the particles: the possible configurations of them and connections between these configurations.[^2] Now we need to supplant this with a notion of what it means for two paths to be _the same_, or, equivalently, what it means for two paths to be different. This idea of path-similarity depends on which factors we judge relevant when considering manipulating quantum particles, which we need to choose if we wish to obtain the most general description possible. Like most of physics, we would not want our description to depend on some sort of absolute position or orientation in space, so our description better ignore those. Beyond that, however, identifying the relevant factors of the space becomes more tricky. The two most promising candidates are a _geometric_ or _topological_ notion of path similarity. As it turns out, the latter is used in the most commonly accepted explanation of anyons, the main reason being the mathematical and conceptual simplicity it brings.
 
 The main notion of similarity used to explain anyons, namely the topological notion of _homotopy equivalence_, roughly means that two paths are the same if they can be continuously deformed into one another. This differs wildly from_geometric_ path equivalence: there paths are only "the same" if they traverse the same path in the same space, they need to be `isomorphic`. Consider the difference between a mountain ridge versus a nice meadow. Geometrically, the two are very different, and if we were to care about how exhausting a hike over that distance would be we would certainly not neglect to take those differences into consideration. On the other hand, if we were an amateur nautical cartographer interested in plotting out all bodies of water in the area, we would consider both spaces identically topologically: neither of them has any lakes. It is clear why a topological approach to particle paths would be attractive, as it massively simplifies the set of possibilities down to the bare essentials. It is clear that sometimes we do need to consider the relevant geometry however: our hiker will no doubt complain if their map-making friend confuses meadows for mountains, or vice versa depending on their constitution.
 
@@ -120,6 +123,8 @@ Returning back our example of two indistinguishable particles in three dimension
 3) A hemisphere which behaves strangely at the boundary
 4) A mobius band which also has its "sides" glued together in the same way a regular mobius band is constructed, namely by twisting it once.
 
+![[Pasted image 20221011123855.png]]
+
 ![](../../media/20210610_174954screenshot.png)
 
 ![](../media/20210610_175017screenshot.png)

Definitions/Group.md

@@ -12,5 +12,5 @@ A group $G$ is any set of functions $f_1 ...$ and a multiplication operator $\ci
 
 1) [[Closure (group)|Closure]] $\forall i,j \quad f_i\circ f_j \in G$
 2) [[Associativity]]  $\forall i,j,k \quad (f_i\circ f_j)\circ f_k=f_i \circ (f_j \circ f_j)$
-3) Identity element: $\exists e : \forall i \quad e \circ f_i=f_i$
+3) Identity element: $\exists e \in G : \forall i \quad e \circ f_i=f_i$
 4) Inverse element: $\forall i \exists f_i^-1\quad f_i\circ f_i^-1=e$