Firefly

I started watching the science fiction TV series "Firefly". It's pretty cool so far -- nice costumes and a wicked sense of humor.

I'll write more as I watch more.

Here are the lyrics to the Firefly theme song:

Take my love, take my land
Take me where I cannot stand
I don't care, I'm still free
You can't take the sky from me
Take me out to the black
Tell them I ain't comin' back
Burn the land and boil the sea
You can't take the sky from me
There's no place I can be
Since I found Serenity
But you can't take the sky from me...

Proposition parties

Here's an interesting idea from a friend of mine:

Some of my friends hold an annual party that's a great idea. They call it a "proposition" or "prop" party. All attendees are invited to study up on a candidate or proposition that will be on the upcoming ballot. At the party, we mix food and alcohol with discussions by the informed among us on each ballot topic. This is not a vitriolic debate, since by and large the group holds very similar views, but rather a chance for all of us to become well-informed about all the ballot options. Thought I'd share the idea since it's a great one. Maybe this kind of event will eventually crop up all over the place.

Dealing with tasks efficiently

Problem
You worry about unfinished tasks, projects, etc.

Solution
If it's something you can do right away (whatever your definition of "right away" is), do it. If you can't, write down what you need to do and schedule it for a later time. I record the item in a calendar (e.g. "Check if department manager has setup voice mail on my phone."). Later, I get an email reminding me to do what needs to be done at the right time.

My idea isn't a new one; it's the foundation principle described in "Getting Things Done," a book by organizational guru David Allen.

Song of the day: "Falling for the First Time" by Barenaked Ladies

A while ago, I heard the song "Falling for the first time" by the Barenaked Ladies (an alternative group). I thought it was a song about falling in love for the first time. But then I looked at the CD liner notes from "Disc One." Steven Page, one of the Barenaked Ladies, says that the song is really about "a perfectionist who discovers the joy in failing -- how he's got to learn to fall before he learns to fly, and how that fall makes him feel a freedom he's never felt before."

I'm so cool, too bad I'm a loser,
I'm so smart, too bad I can't get anything figured out,
I'm so brave, too bad I'm a baby,
I'm so fly, that's probably why it,
Feels just like I'm falling for the first time.

I'm so green, it's really amazing,
I'm so clean, too bad I can't get all the dirt off of me,
I'm so sane, it's driving me crazy,
It's so strange, I can't believe it,
Feels just like I'm falling for the first time.

Anyone perfect must be lying, anything easy has its cost,
Anyone plain can be lovely, anyone loved can be lost,
What if I lost my direction? What if I lost sense of time?
What if I nursed this infection? Maybe the worst is behind.
It feels just like I'm falling for the first time.
It feels just like I'm falling for the first time.

I'm so chill, no wonder it's freezing,
I'm so still, I just can't keep my fingers out of anything,
I'm so thrilled to finally be failing,
I'm so done, turn me over cause it,
Feels just like I'm falling for the first time.

Anything plain can be lovely, anything loved can be lost,
Maybe I lost my direction, what if our love is the cost?
Anyone perfect must be lying, anything easy has its cost,
Anyone plain can be lovely, anyone loved can be lost,
What if I lost my direction? What if I lost sense of time?
What if I nursed this infection? Maybe the worst is behind.

Email management, bookmarks, and listing

More cool links from 43folders:

• How to manage your email according to Mark Hurst
  
• delicious -- a community of users who share interesting bookmarks
  
• list your way out of stuckness

Concentrating, simplified technology

Here are some interesting links I've found on the archives of 43Folders:

• Advice on how to concentrate while studying
  
• Neo -- a dumbed down computer which weighs 2 pounds and only does word processing

Song of the day: "La, La, La"

I think this song is cute; my sister hates it.

From Bert and Ernie's Greatest Hits:

La, la, la, la, lemon,
La, la, la, la, lightbulb,
La, la, la, la, lamppost,
La, la, la, la, lump in my oatmeal!

La, la, la, la, laughter,
La, la, la, la, lullaby,
La, la, la, la, lollipop,
La, la, la, la, lights in the sky!

La, la, la, la, linoleum!

Listen to me,
'Cause "L" is such a
Lovely letter,
For words like
Licorice and lace
The letter "L" lights
Up your face,
So why not, la-la-la-la-la
With me!

Life hacks

There is a community of bloggers devoted to "life hacks," i.e. ways to make your working life more organized. One popular site is "43 Folders," which was mentioned in a recent New York Times article.

Inspired by their example, I am going to post my own life hack ideas or other people's ideas which work for me.

Problem
It's hard to get going in the morning.

Solution
Find a way to wake up first (for me, it's to use a randomly shuffling mp3 player clock on my computer). Then jump in the shower and plan your day while you shower.

Problem
You can't get any work done. Maybe it's because you have too much work, other people are driving you crazy, a hurricane hit your house, or you're plain lazy. Whatever the reason, you now have so much work to do, you feel intimidated and hopeless.

Solution
Find a friend you respect. Make yourself accountable to him or her. Email the person every morning with a list of goals, then email the person at the end of the day with a report on your progress. If you don't accomplish everything, explain why. The idea is to shame yourself into working.

Problem
You're having trouble being organized. You get 100 emails a day, have appointments with people, etc.

Solution
Keep your tasks centralized in one place. I use email. I have a work account and a non-work account. I check my work account constantly during the day since urgent emails will only go there. I also send myself email reminders about tasks I need to do (for instance, see "101 Reminders"). These go to my work account. Less important emails go to my non-work account. I check this account at the end of the day. If irritating, non-essential emails show up in your work account, forward them to your non-work account. The idea is to keep your work email inbox clean. Only important things are in there.

Direct product vs. tensor product

Suppose you have two vector spaces. How can you combine them into another vector space?

The simplest way is to use a direct product. For example, suppose I have vector space A with Cartesian coordinates a1 and a2 and vector space B with Cartesian coordinates b1, b2, and b3. The direct product of A and B is a new vector space with coordinates a1, a2, b1, b2, and b3. Basically, the idea is to add the dimensions of A and B together. Notice that A has 2 dimensions, B has 3 dimensions, and their direct product has 5 dimensions. Pictorially we might imagine a direct product as taking the union of vector spaces A and B. Then we get a new vector space where you label the elements of the set by a set of coordinates in A (a1, a2) and a set of coordinates in B (b1, b2, b3).

A slightly more interesting thing we can do is to take a tensor product. Let's call the basis vectors of A, ^a1 and ^a2, and similarly the basis vectors of B, ^b1, ^b2, and ^b3. The tensor product of A and B is a new vector space with basis vector space with basis vectors ^a1 x ^b1, ^a1 x ^b2, ^a1 x ^b3, ^a2 x ^b1, ^a2 x ^b2, and ^a2 x ^b3, where "x" connotes tensor product. So the tensor product is kind of like multiplying two vector spaces. The tensor product space of A and B has 6 = 2 x 3 dimensions.

In physics, a simple example of direct and tensor products is spin-1/2 particles. A single spin-1/2 particle is a direct product of |&uarr> and |&darr> -- dimension 2. However, the Hilbert space of 2 spin-1/2 particles is a tensor product of two single spin-1/2 Hilbert spaces -- dimension 4 = 2 x 2. We see that in general that a tensor product space is larger than its corresponding direct product space. For example, the direct product of 3 spin-1/2 particles has dimension 6 = 2 + 2 + 2, but the tensor product of 3 spin-1/2 particles has dimension 8 = 2 x 2 x 2. That means that there are vectors in the tensor product space that can't be mapped to the corresponding direct product space. We know that spins combine in tensor products because we observe "entanglement." A direct product of 2 spin-1/2 particles would have states like |&uarr&uarr> but not Bell states like |&uarr&darr> - |&darr&uarr>.

Chaotic perspective on solving quantum mechanics problems

I heard something very interesting in class recently.

The lecturer nicely summarized the main methods of solving quantum mechanics problems.

1. Separable solution: gives you special functions (Hamiltonian has some symmetry)
   
2. Perturbation theory: sometimes works well, but not always
   
3. Semiclassical method: works for large wavelengths
   
4. Numerical analysis: broadly applicable but doesn't give much physical intuition

The first method isn't realistic; we don't have perfect rectangular and spherical potentials. Perturbation theory doesn't work very well in highly degenerate systems (e.g. the high energy levels of a stadium billiard). So for analytic solutions to general problems, we can only use the semiclassical method.

The lecturer's insight was that the generic Hamiltonian in quantum mechanics is partly chaotic and partly quasi-periodic. We can deform a symmetric Hamiltonian or quantum system a small amount and if the system is in certain areas of phase space, the system still has locally conserved quantities (KAM theorem from chaos theory).

This sounds a little bit like Noether's theorem where you make an infinitesimal transformation which leaves the Hamiltonian invariant. Noether's theorem says that for a continuous symmetry like this, there is a corresponding conserved quantity. The KAM theory adds to this by saying that symmetric Hamiltonians can be robust to small perturbations, which break their symmetry.

A friend of mine raised the following objection:

Probably true, but don't forget scattering theory for the continuum, which has lots of powerful methods related to perturbation theory and semiclassics. Also the partial wave expansion of scattering theory is powerful even when the system is not spherically symmetric in the same way that the multipole expansion is useful for arbitrary current and charge distributions when considering radiation and other distant fields.

For fun, here's a Poincare section I made in a graduate level class on mechanics and chaos:

The history of gauge theories in physics

The most basic gauge theory in physics is Maxwellian electrodynamics. While Maxwell developed his theory in the late 19th century, it was not realized until the 1950s and 1960s that the concept of gauge invariance was crucial to developing theories that explain the fundamental forces between elementary particles.

But what does gauge mean? Here's a hint from the Webster dictionary:

A measurement (as of linear dimension) according to some standard or system: as (1) : the distance between the rails of a railroad (2) : the size of a shotgun barrel's inner diameter nominally expressed as the number of lead balls each just fitting that diameter required to make a pound [a 12-gauge shotgun] (3) : the thickness of a thin material (as sheet metal or plastic film) (4) : the diameter of a slender object (as wire or a hypodermic needle) (5) : the fineness of a knitted fabric expressed by the number of loops per unit width

Hermann Weyl is the scientist who first introduced the idea of gauge invariance, but in a different context. He was trying to come up with a theory to unify electromagnetism and gravitation. For him, gauge meant "scale". He thought that physics might be invariant under a change of scale at the local level.

Weyl's ideas were a remarkable insight at the time, but unfortunately he was wrong -- his theory does not describe nature. In modern physics, gauge theories refer to physical theories that are preserved under certain local symmetry transformation. For instance, in quantum electrodynamics, the Lagrangian is preserved under multiplication by a complex phase (technically known as a U(1) symmetry).

But whenever you encounter "gauge" in physics literature, it might amuse you to think of railroad tracks!

A public declaration

All right, this is the last straw. You know that the world is messed up when even your mom has heard of string theory.

The public seems unaware that there are other equally interesting areas of physics that have nothing to do with strings, extra dimensions, subatomic particles, or the cosmos.

When I get tenure, I'm going to write a popular science book on condensed matter physics. You just wait!