NCG 4

So this is going slower than I thought it would. I’m on 4 and I’ve basically given a definition that was incomplete. Maybe I’ll quit this endeavor if this one doesn’t go any better than the previous. I’m now going to make the assumption that if you could read NCG 2, then you were bored and so to cover more ground I’m going to make more demands on the reader.

A good example of the overview in NCG 2 is in electromagentism. Let V be a manifold. Let $\mathcal{A}$ be the algebra of functions on V. It turns out the group of unitary elements of $\mathcal{A}$ is the local gauge group of electromagnetism. Let D be the covariant derivative associated to the potential.

We can then express D by $\mathcal{H}\stackrel{D}\to \Omega^1(V)\otimes_\mathcal{A} \mathcal{H}$. Where $\mathcal{H}$ is a $\mathcal{A}$-module and D as a derivation satisfies the Leibniz rule: $D(f\psi)=df\otimes \psi + fD\psi$ where $f\in\mathcal{A}$ and $\psi\in\mathcal{H}$. Now we know that each component of the potential couples equally with the Dirac spinors.

This is tending to be a little physics-like, so I’ll take a little diversion here. If you look at the Dirac equation it requires a collection of four objects satisfying: $\gamma_0^2=I$, $\gamma_1^2=\gamma_2^2=\gamma_3^2=-I$, and $\gamma_i\gamma_j=-\gamma_j\gamma_i$ when $i\neq j$. These $\gamma$ have 4 by 4 matrix representation that I won’t type (too cumbersome, but you can look up anywhere if you are terribly curious). Using Einstein summation convention, we get a nice compact form of the Dirac equation: $i\hbar\gamma^i\partial_i\psi = mc\psi$ where $\partial_i=\frac{\partial}{\partial x_i}$.

So we have an interaction with the EM-field $F^{ij}$ through the potential $(A^0, A^1, A^2, A^3)=(\frac{1}{c}V, A_x, A_y, A_z)$. Just replace $i\hbar \partial^i \to i\hbar\partial^i-eA^i$ to get the new Dirac equation $\gamma_i(i\hbar\partial^i-eA^i)\psi=mc\psi$ where the state function $\psi$ is a column spinor $(\psi_i)_1^4\in\mathbb{C}^4$.

OK. So back to what I was driving at. This means we can identify $\mathcal{H}$ with $\mathcal{A}$. So if we define D by the rule $D1=A\otimes 1=A$ along with the Leibniz rule. This leads to the operator D behaving exactly as I derived (no that wasn’t for nothing) above. After redefining some constants we get $D\psi$ is the left side of the “new” Dirac equation.

And so now we have an electromagnetic theory for noncommutative geometry. Which is actually pretty important since like we said before, if we want the four forces in a unified theory we need them to work in a quantum/noncommutative framework. Hmm…sorry, since I feel like the vote that went for NCG was not a vote for physics examples.

NCG 3

As promised. I’m back to a conceptual description. What I think I’ll do is look to the post before and try to tease apart a single term into its concept.

What is a smooth manifold? Well first of all there are two parts to this “smooth” and “manifold.” We’ll start with manifold, and I’ll quickly say that smooth is a technical term so probably doesn’t mean quite what you think.

A manifold is a topological idea. It is also a “local” idea. This means that different manifolds can look radically different, but when you zoom in at a very small scale they start to look the same. Basically, manifolds are an attempt to generalize are what we generally think of as flat space. The surface of your table or a piece of graph paper gives you 2-dimensional space. It is flat and you can put nice perfect squares on it to give you a location. Likewise you can chunk out cubes of the air around you to give you three dimensional coordinates.

When we generalize we can have curved weird looking things, as long as at any given place we can zoom in really close and get something that looks like a piece of our flat nice space. This is the property of being “locally Euclidean.” Try it with your coffee cup. Zoom in really close at any chosen place. Pretend you’re an ant crawling around on the cup. If you were a small enough ant, even the curved parts are going to seem flat to you. So examples of a 1-dimensional would be a circle or curve. Examples of 2-dimensional manifolds would be the surface of your coffee cup or the surface of a sphere. Examples of 3-dimensional manifolds would be … heh. Notice how 1-dimensional manifolds have been living in 2 dimensional space and 2 dimensional manifolds have been living in 3-dimensional space. (There is a theorem about this I won’t get to that any n-dimensional manifold can live in 2n-dimensional space.)

There are technically two other properties that we’ve ignored. A manifold has to be second countable and Hausdorff. This was a triviality since the examples I gave were living in (were a subspace of) a higher dimensional space that we knew had these properties. So second countable is a pretty abstract idea. Technically it means that there is a countable basis. Or if you take the numbers 1,2,3,… you can for each number assign an open set. Taking this collection we can get all the other open sets. This is sort of meaningless since I didn’t describe a topological space… Hausdorff is easier. It means that given any two points that aren’t the same, we can zoom in close enough on each one, so that the zoom in of the one doesn’t have any of the same points as the zoom in of the other.

So you’re thinking, all those properties seem to be everywhere, how could something not satisfy those? That’s because a manifold is a generalization of our intuition of space. All the concepts of it are meant to be intuitive. It actually takes a little bit of work to find things that don’t satisfy those properties. I think those examples are for another time (along with what it means to be a smooth manifold).

NCG 2

I’ve decided that I must alternate between technical jargon and conceptual posts. If all my posts were like my last one, I wouldn’t make it to the first page of the article. So I think you know which category you belong to. NCG “odd number” will be the conceptual posts, and NCG “even number” will be more technical discussions.

Although Part 1 contained lots of interesting motivation to study NCG, I don’t really want to get into that. So we’ll just say, it occurs in physics a lot, and leave it at that. It turns out that when we say NCG we really mean noncommutative differential geometry as developed by A. Connes. This makes me happy, since I am a fan of differential geometry, and not so much of algebraic geometry (sorry to those aspiring algebraic geometers), and my big fear was that NCG was algebraic geometry but harder since things don’t commute.

I also need to apologize, because I assumed this was heading towards rings of functions which is why I brought up rings, but apparently we are headed down the noncommutative algebras path. So I’ll get on to exploring my first area of interest that has popped up.

In smooth manifolds we can give differential structure through the algebra of differential forms. In other words, given a smooth manifold M, and the exterior differential d, we have $(\Omega^*(M), d)$ that we call “the differential calculus over the algebra M.” Now I must admit that this language doesn’t really fit (since M is not an algebra), and I think what is meant is that where we have M we should really have $C^\infty (M)$ or the algebra of smooth functions on M. This is actually an algebra (the algebra of scalars), and we have the added bonus of being able to reconstruct M from its algebra of scalars.

So the moral of the story is, why do we need so much? Given an arbitrary algebra $\mathcal{A}$ and a differential (which we now need to define) satisfying $d^2=0$, we have $(\Omega^*(\mathcal{A}), d)$ or the differential calculus over the algebra $\mathcal{A}$.

So the generalization for the differential is not immediately obvious. Let $f\in\mathcal{A}$ such that the algebra has a representation on a Hilbert space $\mathcal{H}$. If we let F be an operator on the Hilbert space with spectral properties that make it look like the Dirac operator, then we define $df=i[F,f]$. This definition is really the product of an analogy with the Dirac operator and its commutation relations.

So this probably left everyone feeling extremely unsatisfied, but next time (4) I’ll build some examples as well as providing more solid grounds for interpreting spaces as their algebra of scalars.

NCG 1

From here on out Noncommutative Geometry will be denoted NCG. I will admit upfront that this is based on the article NCG for Pedestrians. I may or may not have found this article from the n-Category Cafe. Just trying to cover my bases. I am far from an expert (in fact this is my first encounter with it), so any person with some experience that finds incorrect information, please point that out.

For all you non-mathematicians, I thank you, and will attempt to keep this as nontechnical as possible, but I know your thoughts at this point. Uh, commutative, wasn’t that thing we learned that said $a+b=b+a$? How could that ever not happen? And geometry was that stuff with triangles and circles we did in 10th grade. So NCG is like saying when you add a triangle to a circle it isn’t the same as adding a circle to a triangle.

Let’s start there then. What is NCG? Like all fields of math, it is usually extremely difficult to give a well-defined definition. Yet at the same it is often the case that you can point to an explicit paper on ___ and go, “Oh yeah, that’s ___.” I don’t remember who said this, but someone said that mathematics has two pillars: algebra and analysis (of course topology is the heavens and not of the lowly corporal world). Essentially NCG falls on the side of algebra. When doing NCG, we are trying to interpret (noncommutative) algebraic structures in a geometric way.

So I guess I should try to convince those out there with no experience in this that there is such a thing as noncommutative algebra. We are actually most concerned with rings. So a ring is a set S, with two operations, call them + and * (this is rather abstract, + and * DO NOT have to be the standard addition and multiplication, heck S doesn’t have to contain numbers so addition and multiplication might not even make sense) such that the following properties are satisfied:

+ is associative: $a+(b+c)=(a+b)+c$; + is commutative: $a+b=b+a$; there is an + identity: $0+a=a+0=a$; + has an inverses: $a+(-a)=(-a)+a=0$;

* is associative: $a*(b*c)=(a*b)*c$;

There is a distributive law: $a*(b+c)=a*b+a*c$ and $(a+b)*c=a*c+b*c$.

Things to note: + is a lot more restricted than *. We don’t need an identity or inverses in *. Most importantly, when we say noncommutative we mean in the * operation, since + is required to be commutative.

The example that should convince you that these things exist. At my high school we were taught about matrices, but I’m not sure about the general public on that. Anyway, check for yourself that if you take, say 2 by 2 matrices, that with matrix addition you satisfy the first set of properties, and with matrix multiplication you satisfy associativity, but you do not satisfy commutativity. This seems to be an extremely important example considering you have something most people have heard of not satisfying commutativity, so it is more common than you might think.

So I may have gone a little off topic, and I didn’t even get to the article. I might as well jump into the article next considering there are probably two types of people reading this: those that are like come on this is basic why aren’t you introducing terms like fiber bundle, affine schemes, and Weyl quantization? and the other type is going I have no idea what just happened, but I think it hurt…

Vote for a Direction

There are many ways I could go for the next couple of weeks. All are fascinating to me, so I’ll let you decide. Vote now!

1. I could lay out in simple terms my favorite Millennium Problem: The Hodge Conjecture. I wrote this up last winter, and it is for all levels. It is conceptual and basically no details rear their ugly head. So if you’re interested in what these million dollar prize problems are like vote 1.

2. Several art related things that are well worth analyzing/discussing have come up.
2a. Literature: My Gravity’s Rainbow Challenge is well under way or I never really discussed my thoughts on my first Haruki Murakami experience.
2b. Film: I saw my first Pedro Almodovar film. Other directors worth discussing that have come up recently are Harmony Korine, Werner Herzog, Shinya Tsukamoto, and Shyamalan’s newest The Happening.
2c. Music: Who has popped up this year as exceptional (Bon Iver, Son Lux, Extra Life, etc) and who has let me down (Death Cab). I have a harsh opinion people don’t want to hear.

3. Philosophy: The standard philosophy of mind and language that I’ve been reading, or some ethical debates (more on Sam Harris maybe?).

4. Choose your own adventure: Anything you’ve seen or heard of lately pertaining to math, physics, philosophy, or art that you think I may be able to shed some light on. I have an article entitled “Noncommutative Geometry for Pedestrians” that I’ve been looking for an excuse to read. Also, I have a library system and netflix, so basically any book or film you bring up I should be able to get my hands on.