Covering Lemma 1

Internet was a bit weird lately, and I didn’t want to lose a post half-way through, so I decided to wait in writing this. I seem to have a weakness when it comes to figuring out how metric properties and measure properties interplay. It is almost inevitable that you will need to call upon some “covering lemma” to get the job done. These are used extensively in differentiation, but also less-known in generally defining measures that interact nicely if your space already has a metric.

Lebesgue Number: Given a compact metric space (X, d), then for any open cover, there exists \delta>0 such that if a set has diameter less than \delta it is contained in one of the members of the cover.

Proof: Really you just do all the stadard tricks, and then tie them together. So let \{U_i\}_{i=1}^n be a finite subcover of our cover. (We ignored the whole space being a member for trivial reasons). Now define C_i=X\setminus U_i and a function f: X\to \mathbb{R} that basically averages the distances to each C_i. This gives us f(x)=\frac{1}{n}\sum_{i=1}^nd(x,C_i). Now for any x, we know that x\in U_i for some i, and this set is open so there is an \epsilon>0 small enough so that B_\epsilon (x)\subset U_i, thus d(x, C_i)\geq\epsilon yielding f(x)\geq \frac{\epsilon}{n}.

Since f is continuous and X compact, f achieves its min. We call this \delta. Now if E is a set with diameter less than \delta, then we can pick any x_0\in E and get that B_\delta (x_0)\supset E. Now \delta\leq f(x_0)\leq d(x_0, C_k) where C_k is chosen as the maximal distance to any of the C’s. But then we have it since U_k=X\setminus C_k\supset E. Thus any set of diameter less than delta is contained in a member of the covering.

There are some interesting other ways of proving this. This is I think considered the “standard” since Munkres does it this way.

OK. So I guess I’ll do a series of these. I was going to do two in this post, but I don’t feel like doing the other now that I just typed that out. Also, if I do a series I can do the three that I feel are most important instead of just the two easier ones.


Not Compact Unit Ball

Here is a beautiful little theorem. The unit ball in an infinite dimensional Hilbert space is not compact. The proof is quite simple. So the unit ball B=\{v\in \mathcal{H} : \|v\|\leq 1\}. Recall that since this is a Hilbert space, we have an inner product defining this norm \langle v, v \rangle=\|v\|^2.

Since our space is infinite dimensional, we can choose \{v_1, v_2, \ldots \} to be linearly independent inductively. Basic application of Gram-Schmidt gets us this set to be orthonormal, in particular, each one has norm 1 and so is in the unit ball.

Now look at the distance between any two \|v_i - v_j\|^2=\langle v_i - v_j, v_i - v_j \rangle

= \langle v_i, v_i \rangle -2\langle v_j, v_i \rangle +\langle v_j, v_j\rangle



Thus all are \sqrt{2} apart and hence there is no subsequence that converges. Since sequential compact and compact are the same here, we are done. (Note the inner product is not symmetric, but I knew the middle terms would be zero, so I went ahead and abused that instead of writing two zeros).

I know the result to be true in a Banach space as well, but I don’t see a quick fix without the inner product…

A lot more of these things will start popping up with my analysis prelim two weeks from today.

Rabbit, Run

I promised this a few weeks ago, but just haven’t gotten around to it. For all you literary types out there, don’t look shocked, but this was my first Updike novel. For those of you unfamiliar with Rabbit, Run it is the first book in a series of four. Two of the four won Pulitzer prizes (only one other author has won twice in the category of fiction, and he did it in the same set of novels!). Updike in general is probably more known for novels such as The Witches of Eastwick.

On to the actual book. I was infatuated with the writing for the first 50 pages or so. It was unlike anything I’d ever seen. It was a sort of prose-poetry hybrid. Attention was paid to flow and rhythm and partial rhymes. The metaphorical language was beautifully constructed. Then I started getting into the story and didn’t notice it as much. I guess you could say I got used to it. This only lasted probably another 50 or so pages. Then the language started to really annoy me. What I considered great metaphorical language before was now just plain distracting.

Here is an example. I just opened to a random page completely confident that I would have a whole slew of choices. “Number 61 is a big brick place with white wood trim, a little porch imitating a Greek temple, and a slate roof that shines like the scales of a big fish.” Now that might not seem like much, but when it happens nearly every sentence it is hard to focus. I start thinking about Greek temples and fish. Maybe I just have a short attention span.

The story itself was great, in my opinion. Basically, this guy (people call him Rabbit), feels trapped in his life. He has the job he didn’t think he’d end up in, the wife he didn’t think he’d end up with, one child to feed and another on the way, routine, routine, routine…and he just needs to get out. So he runs. Leaves his pregnant wife one night.

Most reviews that I’ve looked up say that they hate the book because they can’t like Rabbit. That it is impossible to identify with a character so morally void. I’m not entirely sure these reviewers have ever truly examined their lives, though. I love what I do and it is exactly what I want to do, yet I can’t help but identify exactly with what Rabbit went through. What about people actually in his situation?

Here is why I condemn those reviewers and am going to set the record straight. I am at a relatively early stage of my life. I haven’t made it quite as far into “the trap” as Rabbit, since I don’t have a long-term job or family. But even my language there indicates how real this trap is. After grad school I will either get a post-doc or take a teaching position. Once I’m in a tenure track position, I’d be crazy to leave. All the while I’m fighting to establish myself as a researcher and teacher to get the tenure. Once you have tenure, you’d be crazy to leave, so that determines where I live for the rest of my life. When you truly examine it, you start to realize, am I deciding my life, or has it been preplanned for me?

There seems to be this socially accepted stages to life. You haven’t lived a successful life unless you go through these. So now you start asking yourself the question, well, I don’t particularly want to change my path, but if I did, could I? Remember, I’m at an early stage, but even this early, how significantly could I really alter my the path I just listed? Luckily, I’m not that far into the trap, so I probably could, but if I had a minimum wage job with a family, I probably couldn’t get up one morning quit and say that I want to become an author or something. Once you realize that you are stuck, it doesn’t seem to matter whether you want to change or not. It is the whole fact that you can’t. It is like someone else is controlling your life and not you anymore.

I should stop harping on this point. To sum up, other great points come up in the novel about religion and its role in situations like these. I don’t want to really give anything else away. I just wanted to emphasize the point that most people hate the book because they can’t identify with Rabbit, and I say how can you not?

New and Improved AMFM

Yes. I just abbreviated A Mind for Madness as AMFM. In an attempt to get more organized I’ve changed the theme to a more organizational friendly one. I’ve added some stuff to get more organized and categorized all the old posts.

I looked at many themes and was dissatisfied with some aspect of almost all of them. This was the closest I think to what I was looking for. I was able to compactly get the categories in a drop down as with archives. I like the idea of a search, so I added that. I also like when I see the latest comments on other people’s blogs, since people could be commenting on something from a long time ago, so I added that.

The thing I don’t like is that this seems a little harsher on the eyes. Oh well. I really wanted a three column instead of a two, but there are very few choices for that.

Blogging Habits

I secretly feel like a really horrible blogger. There are so many useful tools that wordpress has given to me, yet I use almost none of them. Some things I notice and like about other blogs that I don’t do:

1) I believe they are called “trackbacks”? I tend to refer to something from thirty posts ago and don’t give anyone any simple access to that material. I’m sure anyone that doesn’t read this relatively often (most people) aren’t too happy about that and don’t even bother to look up what I was referring to.

2) I tag, but I’m very unorganized. My original intent to not creating categories was that I wanted things to be cross discipline. I felt that if I stuck it in categories I would ruin that effect. Now I just think it is poor blogging technique (and this is my 73rd post so I definitely don’t feel like going back and doing it now).

3) I don’t really use any of the widgets or tabs to make navigation easy. It also creates a formidable effect that it is all one thing and you can’t easily read the parts that are of interest.

Now on to just general blogging techniques I’ve looked up:

1) You should use a narrow focus. Lots of blogs in a special area attracts a lot more readers than lots of random stuff all over the place.

2) I don’t use an RSS feed for reading other blogs. Instead I use the time consuming method of just looking to see if anything new has been posted.

3) I don’t send my stuff out through external RSS or pings that aren’t already built into wordpress.

I’m sure there were others that prompted me to write this. Mainly I think it is that the graph that tells me how many people visit each day stopped working for some reason and I’m taking it as a sign that I’m using poor technique. So I’m thinking about making some major changes, but I’m not sure what yet.

I sort of want to narrow the focus, yet at the same time I like having it as an escape from math on certain days. Get thinking about something else. On other days I like to use it to clarify some things for myself that I found interesting in math. So…

Lateral Thinking Puzzle

When war comes up in conversations, it tends to polarize people. Only two options seem to be available: you’re for it or against it. I don’t fit either one/I fit both.

The problems with war…in every situation. I tend to raise the point that in basically every situation, war has been to fight some sort of violence. The problem with that is that violence feeds violence. Violent people want precisely what war is giving them. More specifcally, you can’t fight a war on terror by causing terror. Comments such as these tend to throw me into the anti-war group.

Clearly, the nonviolent side isn’t correct either. A million nonviolent people will be killed by a single violent one. Of course, nonviolence has worked in the past, and in fact it wasn’t a form of non-action as many people try to say. The problem is that first times have changed. The nonviolence of Ghandi and King Jr. clearly won’t work against things like nuclear or biological weapons. The other part that made those nonviolent actions effective, in my opinion, is that it relied on the fact that the people hitting the nonviolent protestors did not actually want to hurt them. They were under orders and doing their job. The nonviolence brought out a feeling of immorality in the violent side. I don’t think that this is the case in our current situation. Suicide bombers kill innocent people, and this is the point of the action. A huge group of nonviolent protestors will just make their job easier.

So the two sides of the argument don’t work. Both have had some form of success in the past, and people tend to polarize toward one or the other, but I think our times have changed enough that pursuing either one is completely hopeless.

I now propose a lateral thinking puzzle: What type of solution at least has some sort of potential to work?

A Refutation of a Refutation of Moral Relativism

First off, I am not a relativist, but there is one refutation of relativism that has always seemed a little fishy to me. Lots of famous people have used this argument including Sam Harris featured several posts ago.

Relativism says that there is no universal moral code, and that all moral standards need to be created within a cultural/social/individual situation. So there are no absolutes in right or wrong, it all needs to be considered in context.

So lets look at the argument against it. It comes from logic. There are no universals. But that statement seems to indicate a universal moral principle: we must always make moral judments based on context. So it is a condtradictory theory in that it proposes a statement that can’t be true or false. “There are no universals” is sort of Godel-esque. If it is true, then it itself cannot exist, and if it is false, then the theory is wrong.

There are a few issues I have with this analysis. First off, I think that we change levels when we talk about that statement. The statement’s truth or false refers to the content of the theory, whereas to talk about the truth or false of the statement is a meta-level higher. For example, you can globally define a variable in programming (not recommended, by the way). You have a universal for your program, but each time it is instantiated it could take on a different or varying meaning. This is like saying relativism can have the universal principle to not harm others, but in different contexts this could be displayed rather differently. Maybe a few people have to be harmed to prevent a lot of others from being harmed. Maybe two people get pleasure out of harming eachother that outweighs the harm. So there is a level distinction, where even if relativists don’t make any of these universal claims, we can see how a universal claim can be implemented relativistically.

Well, I used the term “first” but accidentally did both of my refutations in one. I was going to separate how logically statements involving universals can be done at different levels, and then the second point as how there could be implicit universals in relativism without changing the essence of the theory.

As with any post on ethics, I’m not sure if this has been done before or if this is a good argument, but it has always been something that bothered me since I didn’t think it really had any merit in attacking relativism. (Remember, it is sort of weird that I’m defending it since I don’t subscribe to it for reasons not mentioned here).

QFT take 3

I’m going to try to clean up some stuff from last time. I’ve realized that I was pretty sloppy. Given a classical field theory we can get to a QFT. To do this we just take the algebra \mathcal{A} to be the universal *-algebra generated by the classical fields. Now to get the inner product to behave the way we want it, take any state \omega: \mathcal{A}\to R such that \omega^*=\omega. Now we just define the inner product by (a,b)=\omega(ab^*) and the module D=\mathcal{A}\diagup\ker(\cdot, \cdot). If you want more details on why this works, it is known as the Gelfand-Naimark-Segal Construction.

Now we need to somehow get \omega on the algebra \mathcal{A}. In physics people call this the Feynman integral. So we get something that is probably familiar to people that have taken a quantum mechanics class, \displaystyle \int (\int\phi(x)^*f(x)dx) e^{i\int L(\phi)d^4x}d\phi. Here we run into another of those math vs physics problems. Since our space could be infinite dimensional, we don’t have a well-defined notion of what a measure is there. We can’t even define a Radon measure and do our trick of using bump functions.

Let’s beat the problem this time by ignoring the how to define on things we don’t care about. Look at that Feynman integral. We need to define the integral of things that look like \{Jet space forms\times e^{quadratics}\}. We create what is called the Feynman measure, which is just a measure on this particular space. Now although this measure is not unique, it equates to picking a renormalization scheme, so we have a group acting on the space of F. measures and Lagrangians that preserve the QFT. This is essentially what gives rise to what are known as “anomalies.”

Well, I don’t think I want to go into the nitty-gritty of the specifics of everything. This is a pretty rough idea that I just threw out there in case people were wondering. If you’ve followed this and would like some more details, just comment. Otherwise, my guess is that for the most part people don’t really care and so I’ll move on to something else.

Edited: Well, darn. I’m sure I’m going to keep remembering things I left out since I took such a scatter brained approach to this. Well, at the absolute very least I should include the Wightman axioms. We want to extrapolate from what has been presented so far to get what the axioms of a QFT are.

1) We’ve seen this explicitly along with the rationale for it. We want \mathcal{A} to be generated by \phi(f) where f is a classical field with compact support.

2) The inner product is positive definite, for pretty intuitive reasons.

3) We have Lorentz and translation invariance. This is also intuitive since we want QFT to work with relativistically.

4)  An operator that pushes things forward in time is positive. This amounts to an operator that increases energy is positive ((Ex, x)\geq 0 for any x\in H).

5) We have a “locality” condition. This amounts to \phi(f)\phi(g)-\phi(g)\phi(f)=0 if the supports of f and g are spacelike separated (the commutator is 0).

6) There is a vacuum vector. This amounts to something being fixed by the Lorentz group.

There are some other minor axioms as well. Remember these things kind of vary depending on the situation. We are still not sure of the correct formalism. I should also emphasize that everything I’ve done so far is the free Hermition scalar QFT. OK. Hopefully that will cover me for now.

QFT Take 2

Let’s actually try to make some progress on QFT today. There are three parts to make a minimal definition. First, you need a module D over a *-commutative ring. So to get a few definitions on the table. A *-ring, R, is pretty easy. You just have a ring with an antiautomorphism and involutive mapping *: R\to R. This means that (i) (a+b)^*=a^*+b^*, (ii) (ab)^*=b^*a^*, (iii) 1^*=1, and (iv) (x^*)^*=x. So if you’ve seen rings, this shouldn’t be out of grasp. An example would be complex numbers with complex conjugation. A <a href=””>module</a&gt; is basically a generalization of a vector space.

The second part is a Hermitian inner product (\cdot, \cdot): D\times D\to \mathbb{R}. So recall that Hermitian just means that it is self-adjoint. You could think of this as when you express the operator as a matrix the conjugate transpose is itself again. Lots of operators satisfy this, like the differential operator. Essentially the property Hermitian is in place, because if something is obsevable then it is Hermitian.

The last part is that we need a *-algebra, \mathcal{A}, of operators acting on D. Let’s jump out to a bigger picture for a second. The details here are sort of the details of getting around a problem. What we really want is basic. We want a Hilbert space H and an operator satisfying the axioms we want. So our field \phi: \mathbb{R}\times M\to M, and our operator defined at each x\in M as \phi(x) (an operator on H). The problem we are skirting is one of how to get around \phi(x)\phi(y) when x and y get arbitrarily close (an uncertainty problem as you might guess).

So we do the standard trick of “smoothing out the singularities.” Instead of points we will use bump functions. A bump function on M is just a smooth function with compact support. We redefine the operator then to be \phi(f)=\int \phi(x)f(x)d^nx. Here is why I jumped out to the big picture we are skirting around. \mathcal{A} is generated by \phi(f).

Some examples will be instructive. Let G be a group and D an orthogonal representation. Then \mathcal{A} is the group-ring of G, with “*” as g^*=g^{-1}. Or we could let L be a Lie algebra acting on a vector space D with an invariant symmetric inner product. The algebra can be \mathcal{A}=U(L) with a^*=a. Or we could take \mathcal{A} as any C^*-algebra or von Neumann algebra and D any Hilbert space that is a *-representation.

These three examples should make us notice something. These are not things physicists typically work with (unless they are doing mathematical foundations of QFT or something). So despite having a definition in place, we might need to make some restrictions or correlations to what computations are being made down the road. These three examples are QFT’s, but that is sort of weird, since we usually speak of “QFT” and not “a QFT” or “this QFT” as if there is only one.

New OS

Well, technically I don’t have a new OS, but I did spend today putting a new disrtibution on my computer. I was going to post the next QFT, but I just spent too long trying to get this thing running the way I want it to. I switched from Ubuntu Linux to LinuxMint (the slogan is: From freedom comes elegance).

So far, I’m not overly thrilled. It is lighter weight, so I thought it would run faster, but the visuals are better which seems to make it run slower. There is an especially slow factor in internet. I’m not sure if my wireless connection is causing it, or firefox, or this new distro. I will experiment more tomorrow to try to figure that out.

The nice thing I noticed immediately, though, is that the weird timing issue with the old Ubuntu kernel that made composing using a sequencer impossible is not a problem here. Yay for being able to compose like a normal human, now.

Darn! I was planning on finishing Rabbit, Run by Updike today. I have lots to say about it, so maybe once this QFT stuff is done I’ll do a post on that.