Saturday, August 29, 2009

Singing in Swedish

In Swedish, the most common word for “Hello” is “Hej” which is pronounced more or less “Hay”. There is an interesting history to this word posted here about why this greeting was not “the greeting of the masses” until the 1970s. Frequently people repeat it twice: “Hej Hej”. The same website seems to indicate that this implies excitement to see someone (Although the woman at the NORDITA cafeteria seems to use “Hej Hej” for every single customer, and she can’t possibly be excited about every single one).

After a day or two of figuring out that “Hej Hej” is actually a greeting, I started noticing that a lot of people seem to sing it more than say it. Of course not everyone does it the same way, but I’ve heard an awful lot of people who put the second “Hej” almost exactly a major third below the first “Hej”. The Swedish language does have a bit of a sing-song quality to it, but I don’t really detect any other consistent musical intervals in the Language except for when people say hello.

Geek interlude: a major third is a frequency ratio of 5/4=1.25 on a natural scale but is a frequency ratio of the cube root of 2 (1.2599..) on an equally tempered scale.

Speaking of Swedes and music: yes indeed, I have heard a lot of ABBA in Stockholm (Hey, come-on, admit it, you love them too). I was hoping to hear some Ace of Base and the Cardigans too (Yes, they are both Swedish.)
Saturday, August 22, 2009

The International Week of the Gaffnian

Isn’t it great when you manage to get in one room a majority of all the people in the world who are interested in the same esoteric aspect of an already esoteric subfield of theoretical physics?

Keeping in mind that about half of the people who actually care about my particular pet projects within my sub-subfield are actually at NORDITA this week, I took the opportunity to declare last week to be the international week of the Gaffnian, being that most of the speakers at NORDITA were speaking about the Gaffnian in one way or another.

So for the more general physicists in the audience, I’m sure many of you are wondering what on earth a Gaffnian is. The terminology is a mathematical-phonetical-linguistic joke that I managed to get into print. The Gaffnian is a trial quantum Hall wavefunction we introduced in several years ago in this paper (In the paper, the joke is explained… sort of). For the experts it was proposed to describe the nu=2/5 quantum Hall effect, but probably instead describes a nearby critical point. This Gaffnian wavefunction has remained a bit of an enigma. It is nice because it is based on a conformal field theory – but it is not nice because it is based on a nonunitary CFT, which means that it cannot represent a state of matter with a gap (if it is a critical point it makes sense that it has no gap). However, the numerical overlap of this trial wavefunction with other wavefunctions, such as the composite fermion wavefunction, believed to represent the ground state of a gapped state of matter is about 99% even for reasonably large finite sized systems. So apparently the 1% difference between these two wavefunctions makes all the difference in the world to the physics. How this 1% changes the story is not very well understood.

A year or two after we introduced this Gaffnian wavefunction, interest boomed when Bernevig and Haldane pointed out that the Gaffnian is actually just one of the simplest of many possible trial quantum Hall wavefunctions (for general nu=k/r) that can be described as Jack polynomials (these are a family of special functions with lots of nice properties). But after a fair amount of grief it turns out that, except the previously known Read-Rezayi series, all of the Jacks suffer from the same nonunitarity problem that the Gaffnian suffers from – so they probably describe a whole set of quantum Hall critical points. The reason the Gaffnian is so interesting to study is because this is the simplest of the nonunitary Jacks and presumably understanding the physics in that case will tell us a whole lot about the more general cases.

The Gaffnian also has an appealing advantage (not shared by all Jacks) that it is the exact ground state of a particularly simple Hamiltonian. As such, quite a few exact statements about it can be made. This gives one a starting point for analysis that gives a bit of hope that we might actually be able to unravel its mysteries.


When I wake up in the morning in Stockholm, I catch the subway at Thorildsplan, change from the subway to the bus at Odenplan, and then get off the bus at Valhallavägen. I somehow have the feeling I am living in a Marvel Comic. Should I have a constant fear that Ragnarök and Fenris Wolf are just around the corner?

Making the eternal conflict between good and evil all the more cartoonish (or maybe making the eternal conflict between heavy metal and the rest of the world more obvious) it seems that every word in the Swedish language is decorated with an unreasonable number of umläuts.

But Marvel comic jokes (and jokes about characters in Marvel comics) aside, Stockholm is actually a very nice city. Very much as you might expect of a northern european city, Stockholm has its share of canals, bridges, pedestrian areas, open air markets, gardens, castles, museums, cafe's, and tall blonde people. Overall not much to complain about.

The NORDITA center (Nordic Center for Theoretical Physics) moved to Stockholm from Copenhagen where it had been for half a century (This move struck me as a great loss for Copenhagen). Like the KITP and the Aspen Center, the idea of the center was just to have a place where theoretical physicists could come and hang out for extended periods of time and work together. They've done a great job of making it comfortable and pleasant for all of their guests. I've been here for almost a week already and I'm staying three weeks more --- I think this is going to be a very nice stay.
Saturday, August 15, 2009

The Summer of Being Erdős

Paul Erdős (pronounced Air-Dish) was one of the most prolific mathematicians who ever lived – he was also one of the most eccentric. His life has been immortalized in the documentary film “N is a number” and in the bestselling book “The man who only loved numbers.” (which is a very fun read, even if you don’t like Math).

One of the unusual things about Erdős is that he was a vagabond: Except when he was very young, he never actually had a home - he just spent his life hopping from one mathematics conference to the next. And when there was no conference to visit, he would just pop in on one of his many collaborators for several weeks, working on a mathematical paper or two for a while, before moving on to his next visit. All of his possessions fit into one suitcase. When I read about this lifestyle, it certainly seemed a bit crazy.

I wish I could claim to be as prolific and as important as Erdős was. I’m not. But I am starting to share one of his other eccentricities – being a vagabond. And it isn’t nearly as crazy as it sounds. You see, over the course of this summer, I have been hopping from one locale to the next, living out of a suitcase. Between the day the spring semester (Trinity) ended (June 18) and the day the fall semester (Michaelmas) begins (Oct 4), only a mere handful of days will be spent at my “home” in Oxford. The rest of the time I’ve been all over the map: California, Italy, Colorado, Netherlands, Germany, Sweden, Ireland, …. Right now, I am about two-thirds the way through the summer, and I’m still holding up pretty well (an annoying cold and a damaged rib not withstanding). Living out of a suitcase turns out to not be that difficult.

Wherever I go, there is interesting physics to be done, and interesting people to talk to. And in the modern era it is not even that difficult to keep in touch with friends and family wherever I happen to be. I’m not sure I could reduce all my worldly possessions to one suitcase, but it might not be such a bad idea to try

Oh, and for those who want to know, my Erdős number is 3 (via Mike Freedman and Laszlo Lovasz).

Monday, August 10, 2009

No Satisfaction Redux

A few weeks ago I blogged about hearing the talk by Boris Altshuler purporting to show that the adiabatic quantum computing scheme for solving the 3-sat problem is not going to work.

To review: 3-sat is a particularly simple case of NP complete problem (such as traveling salesman) where it is not known if a solution can be found in polynomial time or not. Showing that NP problems can be solved in polynomial time (or showing the converse) is perhaps the most important outstanding theoretical computer science question. The adiabatic quantum approach purports to have an approach for finding such a solution in polynomial time.

The essence of the scheme is to construct a Hamiltonian H_P whose ground state is the solution to the problem. Initialize the system in a a known ground state of a known simple Hamiltonian H_I . Then adiabatically vary the Hamiltonian

H = t H_P + (1-t) H_I

As t goes from zero to one, the adiabatic theorem keeps the system in the ground state so long as the gap does not close. So the issue boils down to how big is the gap. Most people (including Altshuler) think that the gap gets exponentially small which means that the scheme takes exponentially long to run (which is then uninteresting).

Last week at Aspen I heard an informal talk by Eddie Farhi, one of the proponents of adiabatic quantum computing, on the same subject. Although Altshuler’s talk was pretty compelling, Farhi made a few notable points.

(1) To a computer scientist, the 3-sat problem is “solved” in polynomial time if you have an algorithm that ALWAYS finds a solution to the problem in polynomial time. However, it is still interesting (although not equivalent) if you have an algorithm that typically finds a solution in polynomial time. This is a bit less stringent of a requirement.

(2) Numerics simulating the adiabatic quantum approach to solving the 3-sat problem (for up to 100 bits or so) seem to indicate that it will work after all (that the gap is not exponentially small). These numerics have been done by people like Peter Young, who is certainly no slouch. One can always argue that when you look at more bits the algorithm will suddenly fail, but apparently there is no indication of this as of yet.

(3) Finally, it was pointed out that even if generically you have an exponentially small gap, you still have the freedom to construct an arbitrary

H = t H_P + (1-t) H_I + H_{extra}(t)

And if any H_{extra} can be thought up that generically keeps the gap large, then you have a good solution.

My gut feeling is that despite this evidence to the contrary, in the end, this is not likely to provide a polynomial solution to NP complete problem. I even think it is unlikely to typically find a solution.

Apparently Alexei Kitaev, who is officially a genius by the McArthur foundation, apparently stated that he thought the gap HAD to be exponentially small based on the reasoning “otherwise the algorithm would work”.
The next morning, Carissa and I started out at around 8:30 am to climb Castle Peak. (Lin was having trouble with her feet so she did not join us.) Had we known it was going to be difficult we would have opted for a more alpine start.

Castle Peak, at 14,265 feet, is the tallest mountain in the Elk range (the 12th highest in Colorado) and is also one of the most frequently climbed. From the northeast, there is a very “well worn” route that gets you to the top with little trouble. In fact, you can drive to within about 1500 feet from the summit if you have a sturdy jeep.

The standard routes to ascend Castle Peak is along the northeast ridge (shown in blue on the map). We intended, however, to climb the peak directly from the west from the hotsprings (shown red on the map). The hike from the trailhead to the hot springs is along conundrum creek from the north.

According to our guidebook, the approach from the west is a reasonable climb as well, but it turned out to be much more difficult than we expected. In retrospect, scanning around the web there are several reports of people having trouble on this route exactly the same way we did. The problem, in short, is that the mountain is made of bad rock and is simply crumbling away. There is no clear path up the mountain from the west, just one giant steep slope of scree and talus -–- small rocks that have a tendency to avalanche down the mountain, carrying you with them. (In fact, “scree” is from the norse word for “landslide”. Talus, is from french and means roughly the same thing, but sometimes refers to slightly larger rocks).

Physics 1: Angle of Repose. When rocks are piled on a slope, there is some critical angle of steepness of the slope beyond which the rocks starts falling down the slope. This is known as the angle of repose. If the steepness of your slope is much less than the angle of repose, it is unlikely there will ever be much of an avalanche. If the steepness of the slope is greater than the angle of repose, it is completely unstable and is likely to avalanche at any moment. In the photo here, I am walking over a field of talus that is a bit too close to the angle of repose.

Many physicists have spent many years studying rock-piles and avalanches. What is interesting is that such piles have a tendency to tune themselves precisely to the angle of repose -- an example of what is known as "self-organized criticality". If you assume that the rocks are always being added from above (say, from the mountain itself crumbling higher up), then the angle of the slope continually increases as rocks are added until it hits the angle of repose, then there is an avalanche that reduces the angle a little bit, and the angle starts growing again -- such that the angle of the mountain is always near the critical angle of repose. Another interesting feature is that avalanches occur on all length scales -- sometimes small ones, sometimes huge ones. (For the experts, yes I know that sandpile models do not really behave like real sandpiles and rockpiles, but some of the rough ideas are similar).

At any rate, Carissa is a very experienced climber and is extremely sure-footed and quick over bad surfaces (as well as being very good at trail finding -- to the extent that a trail existed in the first place). At some points I started thinking that she must be half mountain goat. In comparison, I felt very clumsy and slow moving. To make matters worse, there was a great deal of ice, frost, and frozen hail (from the storm the previous night) on many of the surfaces which slowed me down even more.

As shown in red line in the map above, we started by traversing northeast along the side of a smaller mountain (I think called Castleabra) to arrive at a large amphitheater and turn southeast to continue climbing. Half way up the amphitheater, we could finally see the peak, but it was not so obvious how we were supposed to get up from the amphitheater onto the ridge that leads to the summit.

Even from very far away, we could see that there were some people up on the ridge leading to the summit. These people had obviously arrrived on the ridge from the other side of the mountain. A few of them seemed to be looking down at us and wondering how we were planning to get up there. (More likely they were looking to the west to see if any bad weather was heading this way.)

There was absolutely no one else on the west side of the mountain that day. This was actually a good thing. Many times I would accidentally kick a small rock and, with the mountain being at the angle of repose, it would tumble a long way down --- sometimes starting a rather substantial avalanche. I was very careful never to be either directly above or directly below Carissa (although it seemed that she was starting avalanches far more rarely than I was).

Actually, it was not quite true that no one was on the west side of the mountain. Just about at the place where we turned from northeast to southeast, we ran into a large heard of mountain goats. You have to look hard in this photo to see them --- there are about a dozen of them -- they are pure white in the middle of the frame just below the small cliff band. (Maybe they were fooled by Carissa's sure footing and they thought she must have been a mountain goat too so they came over to say hi.)

Our guidebook simply said something like "Continue up to Gain the Saddle between Castle and Connundrum (to the north), then follow the ridge to the summit". This instruction seemed more and more mysterious as we continued up the amphitheater. Here's a picture of Carissa leading the way up a snow field through the amphitheater.

(Yeah, I know, I'm not a good photographer).

Even though we did not have proper snow equipment, the snow was still easier going than scrambling through the nasty scree.

Physics 2: Static and Sliding Friction. Everyone knows this principle: once you are sliding, your friction on the surface goes down and you slide even more. An obvious point here. For God sake, don't start sliding!

The weather pattern in these mountains is that thunderstorms tend to arrive in the afternoon. To add to this, all week long (and possibly all summer long) storms had been rather frequent and severe (as evidenced by the storm the previous evening). The climb had been very slow going, and at about 12:30 we still had a long way to go to the top. There were starting to be some clouds in the west, and we were starting to get worried about how bad the situation would look if a storm rolled in quickly. If we felt exposed sitting in a thunderstorm at 11,200 feet the night before, it could only be worse up on a peak near 14,000 feet.

"Predictions are hard to make, especially about the future" This quote is sometimes attributed to Yogi Berra, and sometimes to Niels Bohr, and on Wiki-Quote it is attributed to Robert Petersen, and I have no idea who that is. Whoever said this, it seems quite true about the weather. The truth is, I know almost nothing about meteorology except the important principle that today's weather is probably going to be like yesterday's. Unfortunately, yesterday's weather (and the day before and the day before) were not so good.

To add to the meteorological concerns, we still did not see how we were going to "Gain the Saddle". Here is a view towards the summit from where we stood. At any rate, given that the path was not at all clear (and perhaps not even possible without more serious climbing equipment), we made the decision to turn around and start heading down. We still had a very long day ahead of us -- as once we reached the campsite we still intended to walk all the way out to the trailhead that night.

Here is a photo of us where we turned around. (Photo was taken by Mr. Timer, not by a mountain goat). You can see a few clouds behind us, and to the south (off left of the photo) there were a few more. Here's another report on the web of another guy who also turned around at roughly the same place.

You might think that the hardest part was over, having done all the up-climbing at this point. Unfortunately, that was far from true. Down-climbing was just as difficult and slow. With tired legs (and suffering just a bit from the altitude) my legs were feeling pretty shaky, and on a number of occasions I lost my footing. Falling on my ass in the rocks is certainly not pleasant, but what I was really worried about was if I started tumbling. You see, according to the principles of self-organized-criticality described above, some tumbles would likely be short ones, but other ones might be longer ones. And according to the principles of sliding friction, once you get tumbling, it is really hard to stop.

Well, sure enough, at one point, I did take a bit of tumble. Luckily I caught myself in only about 10 feet. Beyond scrapes and so forth, I walked away with a rather bruised rib; and four days two weeks later, I still can't sleep on my left side. Not a serious injury, but a bit annoying. After that fall, I decided that I had better start walking even more slowly and more carefully. As a result of my slowness, we did not make it back to camp until after 3pm.

Here's a picture, just as we are getting close to camp and we are finally off of the nasty scree. Do I look tired yet?

We finally made it back to camp, and Lin had very kindly prepared us dinner with lots of water. We then packed up our stuff and started the 8.5 mile hike back to the trailhead that night. We had headlamps just in case, but we were hoping to make it back before dark nonetheless, so we tried to hustle.

From the hike out, we had one last really good look at Castle peak.

To quote Carissa in a classic use of double negative, "Well, it is certainly not unsteep".

We made it to the trailhead by dark.
Tuesday, August 4, 2009

Hiking and Climbing and Science: Day 1.

Last weekend while hiking with Carissa and Lin, I realized that the great outdoors is actually a great place to learn about science. This blog posting is going to be a combined report of our little weekend adventure, and some fun science along the way.

Conundrum Creek Trail:
The 8.5 mile hike from the trailhead is not a particularly hard one, even with full packs. It rises only a few thousand feet, and the trail is a smooth and beautiful walk. At the end of the hike, at 11,200 feet above sea level, is a campsite with some delightful hot springs which are almost famous according to this article. (No, that is not us in the photo).

The fact that such geothermal hot springs exist at all simply amazes me, but indeed they do exist, and are not even so uncommon in this part of the country where the ground was once volcanically active. In short, the water flows deep enough underground to get near a geothermal hot spot, and it comes up plenty warm. Sometimes it even comes up steaming as a geyser, or simply as water too hot to enjoy sitting in. The conundrum creek hot springs are particularly nice because they don’t smell like sulfur and they are just about bath temperature. After our hike, we had a nice soak.

Biophysics at Altitude 1:As you go to higher altitude the air pressure drops. At 11,200 feet, the pressure is only about 2/3 what it is at sea level. It is kind of surprising that the human body can adapt to such changes so readily. (However at only another few thousand more feet higher, extremely serious altitude sickness becomes common). A frequent word of advice when you travel to altitude is to drink a lot of fluids. This is not just because it tends to be dry up at altitude, but rather has something to do with helping your body develop a new chemical balance with the lower ambient oxygen levels. I scanned the web for a more detailed description of exactly why fluids help you adjust to altitude, but I did not find any good answers. If any biologists or MDs want to leave a comment I’d be much appreciative.

Since I only arrived at altitude in Aspen a few days earlier, I should have been a bit more careful about getting enough fluids. I had some pretty nasty leg cramps that evening (in places I have never had them before) and I think I can place the blame squarely on the altitude and my not taking enough water.

Biophysics at altitude 2: Maybe mosquitoes in Colorado are just plain stupid, or maybe they are genetically different from other mosquitoes – but in short, these critters are about as fast as Slowpoke Rodriguez, the slowest mouse in all of Mexico (i.e., very slow – and I apologize for the politically incorrect reference to one of my favorite bizarre loony tune characters). On the east coast (where people even talk quickly) you have to really have fast reactions to successfully swat a skeeter. However, out in Colorado up at 11,200 feet, swatting skeeters is like racing a running snail. Maybe this is because the air is so thin that they can’t get enough “traction” with their wings to fly away quickly. Or maybe they are just lethargic from lack of oxygen in their blood. Any entymologists want to ring in on this one?

Added: I asked around, and most people seem to think that the average mosquito is this slow and it is the east coast variety that is Speedy Gonzales.

As air rises and expands, it gets colder. A rough rule of thumb is that air drops about 5.5 degrees Farenheit per 1000 feet (roughly 1 degree Celcius drop per 100 meters). If you want a nice physics explanation of this phenomenon, see this page. At any rate, since Aspen is at 8000 feet, and at night it gets to be about 50 degrees, I should have expected it to reach about freezing up at 11,200 feet – and indeed, this was just about the temperature at night. Brrrr…

Since it was quickly getting cold out once the sun started to set, we had a quick dinner then jumped into our tents and into sleeping bags to keep warm. Unfortunately, at about 11pm, the weather turned very suddenly nasty and we were hit with a massive thunderstorm complete with violent hail. The thunderstorm hovered overhead for only a few minutes – but being rather exposed up at 11,200 feet – those few minutes passed extremely slowly.

Speed of Sound: When lightning strikes, you see the flash essentially immediately, since the speed of light is extremely high. But the speed of sound is slow, so you don’t hear the thunder until a few moments later. With sound moving at about 300 meters per second it is easy to figure out how close the lightning is striking. Ever since I was a kid, I’ve been counting the time between flash and bang to see how far the lightning really is. Every five seconds is a mile. When the time between flash and bang got to be about a second, (and the thunder was sounding awfully loud) I started worrying that we might be in a bit of trouble. Then it got down to half a second. --- then a bit less. Lightning striking within 100 meters from my tent. Yikes!

Electricity: It is hard to overstate the strength of lightning. The temperature of a lightning bolt can reach tens of thousands of degrees Celcius, and lightning hitting a tree can easily cause the tree to simply explode. The electrical power of a lightning bolt can reach a billion watts - on the order of the power output of a large nuclear power plant. And while the power of a single bolt remains “on” for only about a second, it can do an awful lot of damage in that time.

So, given the danger of lightning, what can you do to avoid getting zapped? The principle to keep in mind is to try to avoid having the lightning go through you. The best thing to do is to get inside a modern building or a metal vehicle. The metal in these objects is a great conductor (a so-called Faraday cage) and even if it does get hit, the electricity gets diverted around you. Unfortunately, when camping, this is not really an option. Here are some tips from the physicist for what one can do

(1) Don’t be the tallest thing around. (This should be obvious. You don’t want to be on the very top of a mountain).

(2) Don’t stand near the tallest thing around, like a tree, or anything sharp or metallic that might attract the lightning. (Lightning can strike the top of the tree, run down the tree and then jump to you)

(3) Don’t touch anything conductive like a metal fence, a long wet rope, or a large pool of water. The lightning can strike the object far away and then run along the conductive object to hit you.

(4) Touch the ground at only one point – i.e., keep your feet close together and do not lie down. The point of that when electricity is running through the ground away from a strike, you do not want the electricity to find you to be a more conductive path between two points than the ground is – thus going up from the ground, through you, and back down into the ground. Cows are often victims of lightning because their feet are so far apart.

(5) The thicker the insulation between you and the ground, the better. Wear your thick boots, stand on your backpack, or on a plastic ridge-rest or similar.

So if you are out camping, the best thing you can do in a lightning storm is to get away from tall trees in some low area (but not into the center of a flat field where you are the tallest thing around), stand on something insulating, put your feet together and crouch down.

Anyway, as the lightning storm quickly approached, I started putting on my hiking boots –preparing to follow my own advice. But in the dark, finding the headlamp, to then find my shoes was not so easy, and by the time I had them on, the storm was already receding, and I had already avoided getting zapped. Also, by that time, my tent was almost collapsing from the weight of the hail that had fallen on it. Once the hail stopped falling, I got out of my tent, looked around to see if anything was burning (nothing was), brushed off all of the hail from my tent, and climbed back into my sleeping bag to go to sleep.

Carissa reported that Lin’s first comment in the morning was “I hope Steve is still alive”
Sunday, August 2, 2009


The Aspen Center for Physics is summer camp for nerds. Supposedly the rarified atmosphere and the natural beauty of the environment inspires the thinking of very deep thoughts. Whether or not this is true, the Center is still a great place to do science. Just on the border of the town of Aspen Colorado, and within earshot of the Aspen music festival tent and the Aspen Institute, the Center is about as pleasant as any one could imagine for just hanging out and thinking. Theoretical physicists (along with a few mathematicians and a few experimentalists) come from all over the world every summer to spend a few weeks here, to talk to other experts, and just to decompress and clear their minds. At any given time during the summer, something like 50 scientists are here.

The scientists are housed in ski-lodges that are vacant for the summer (sometimes these can be pretty ritzy) and are given shared offices at the center. We are given bikes for transport (unless you prefer walking, like I do – since it is harder to stop to pet the dogs when on a bike). Coffee is always plentiful, cookies are at 4pm, and barbeques are every Tuesday. Hiking, Climbing, and Mountain Biking are always nearby and are very popular among us nerds.

Like the KITP in Santa Barbara, the success of this place is, to a large extent, due to the fact that everyone wants to come here, so the Aspen committee is always able to choose visitors who are smart and interesting and fun to talk to.

A short video about the Aspen Center for Physics is spliced at the beginning of my public lecture from my previous post.