Lifetimes

Dear Diary,


You see, Diary, we work in one dimension where some normal descriptions of matter break down.  We like one dimension because things are often easier to solve, but we need to be careful to avoid nasty physics (or hard math) that differs from problems we might want to solve in two or three dimensions.

One such instance is the description, known as a Fermi-liquid, that describes excitations that are close to the Fermi-surface very well in three dimensions (and two).  But if you try to calculate the quantum field properties in one dimension, you get a bad answer.  Specifically, the self-energy has a big imaginary part to it.  That's bad because it means the quasi-particles (or hypothetical weakly or non-interacting particles that would act like the many interacting particles) in our system have a short lifetime.  That means it's a bad description!  Ideally, we'd like the true quasi-particle that only weakly interacts in our system.  This means the math we wrote down to describe the system can be solved (perhaps more simply than the original problem if we find the right quasi-particle!).

For a real world example, let's describe how SCARY would tell the story of the camping trip last week.  A lot of that story would contain an untruthful description of the real events of that system that have a large imaginary part, but that means a made up story wouldn't have a very long lie-fetime!  One would say the made up quasi-particles in that system are a bad description of the actual events that happened.

You see, physics is useful for describing things.

The radius of convergence is zero

Dear Diary,

Let's talk about a Taylor series' radius of convergence for a minute.  When you're writing down a mathematical structure for a particular object, sometimes the best you can do is generate the Taylor series in a small area of the problem.

I'll liken this to a color gradient.  If you have a panel that slowly varies from red to white like this:

If you start on the white side of the figure and move towards the red, you need to slowly add red to the spot you're standing on.  This is what a Taylor series is doing.  As you move around on the color gradient from right to left, you need to add more color...but in the physics or math problem, you add more mathematical terms to the Taylor series!

You can also a define a radius of convergence of the Taylor series.  This would mean, how far can you want with your Taylor series.  Say our gradient abruptly turned blue to the left of the above gradient.  You can't add more red to get to blue, primarily.  So, the radius of convergence of that color set up would be the width of the rectangle.

There are some things in physics that have a radius of convergence of zero.  As soon as you take one step away from where you started, the Taylor series is already bad!  Most famously, this happens in quantum field theory when you try to see how far away the electron's mass is from the bare mass in the Lagrangian.  You get an infinite distance; in other words, your Taylor series is a miserable description everywhere!  Instead, you can only use the experimental mass in your computation and then everything works!

But there are also real life examples.  For starters, consider the distribution of camping trips I've been on in my entire life.  The one I went on last week had a radius of convergence of zero about it.  You can't Taylor expand to compare it to any other camping trip!  It was that bad!  I have a bunch of trips that were good, bad, and otherwise interesting, but this one was SCARY!  I find it hard to compare to any other experience.  A description from any other experience would not be adequate.  Just like a Taylor series with a radius of convergence of zero.

Back in the Saddle again

Dear Diary,

It's great to be back!  I really mean it this time!  Physicists have a long tradition of getting outdoors and coming back for great physics.  Robert Oppenheimer, leader of the team that made the nuclear bomb, loved horseback riding and took physicists up to his cabin in the woods for some rest and relaxation.  He also made sure there were many (epic? lame?) parties while everyone was working on the Manhattan project.  What a guy!

This last week, I went on a camping trip with a physicist whom I actually did a lot of productive thinking with in terms of physics (and ideas!), but we also had one other person who completely ruined the whole thing.  I may feel the need to refer to this person again in the future as SCARY.

I suppose the less on here is to only take people who have a vested interest in getting out of the woods in one piece on a camping trip.  Further, we came up with an idea that I could implement as soon as I know DMRG!  How exciting!  I better get to it!

It does look like I was more productive while I was away, though.  The paper I'm on with Lucas, Miles, Steve, and Kieron got published an editor's choice (see the publications page on dft.uci.edu!).  Wow!  I must say the review process on this paper was leaps and bounds easier than any paper I've ever submitted!  We didn't even need to appeal any decisions!  I also resubmitted another paper, got some data for Attila, and now have to figure out a new Kuali purchase order form (whatever the hell that is).  So, while I should have spent extra time here, I guess I didn't get too far behind.

But time to get moving!