A little over a year after the highly publicized start-up and break-down of the LHC, the damage has been repaired, new protection systems are in place, and all sectors are cold and ready for beam. Yesterday, the first injection test of 2009 was completed— beams of protons and heavy ions were successfully threaded into the LHC beampipe from its predecessor, the Super Proton Synchrotron (SPS). The beams were allowed to flow as far as the first experiments in both directions, ALICE on the clockwise side, LHCb on the other.
New York City, 1956
Leaning on a Chinese restaurant at a busy street corner in Greenwich Village, I crossed my legs, tipped my hat low, and quietly panicked. This case is turning into a nightmare: dozens of suspects, growing daily, and they all seem to swap places when you’re not looking. A pion couldda done it; pions seem to be some kind of front for the nuclear force that Madame Curie was playing with before she died. But leave a pion to itself and it disintegrates into a muon and a neutrino, neither of which claims to have ever heard of nuclear forces. Radiation in the form of muons and neutrinos has been raining down on us since the beginning of time, and it’s never even hurt. If pions are just glowing with nuclearness, where does the nuclearness go when they die?
For that matter, what is a particle, anyway? I have to admit, I wasn’t suspicious when I first heard the word— I thought they were talking about little rocks or marbles or something. But rocks don’t just change into different kinds of minerals on their own, except for Curie’s rocks, that is. What are these particles? The physicists themselves don’t seem to know: everyone I ask gives a different answer. They seem to be some shadowy energy-clouds, sometimes insubstantial and sometimes infinitely hard. What kind of world are we living in, anyway?
I felt a crumpled slip of paper in my pocket. Pulling it out, I read the well-worn handwriting under my breath, “Seek the Dragon Lady.” I scanned the crowd. I’d bet none of them knew the half of what’s going on, right under their noses! Well, not just their noses, but everywhere in fact. “Any of you folks know a Dragon Lady?”
“Are you looking for Madame Wu?” The young man startled me. From the high-necked sweater and the pipe in the corner of his mouth, I’d reckon he was a student.
Ithaca, NY, 1948
After a wrong turn in Albuquerque, I caught up with Bugs Bunny, alias Richard Feynman, somewhere near the ends of the earth. Up to my elbows in snow-drifts, I spied on the little window to his office, in which he seemed to be doing normal professor-things, plus wild gesticulations. I decided on a particularly frozen morning that I would have to risk visibility if I was to get answers, so I enrolled at Cornell, posing as a G.I. bill student. In Professor Feynman’s introductory physics lectures, I could see that there was something remarkable happening here. People researching physics is about as natural as fish studying water: it’s the very stuff we’re made of. He had a knack of getting down to the ground floor, asking the basic questions, just as much in a block sliding down a plane as in neutrinos.
His teaching assistant, a quirky bow-tied Brit by the name of Freeman Dyson, knew the man personally, so I inquired. “Oh, he’s working on something, yes. The trouble is he just won’t publish, no matter how much I cajole him. He says he’s depressed, but Dick depressed is just a little more cheerful than any other person exuberant. It’s the Bomb, I think, and of course Arlene, his poor wife who died in New Mexico. I probably shouldn’t be telling you this, but Dick and Arlene got married knowing she hadn’t long to live, she having T.B. Bit like Dick to give it a go anyway.”
“What do you suppose he’s up to?”
“Well, he’s got his own private quantum theory for starters. Quantum theory, that’s the theory of the atom and electrons. Until recently, no one’s been able to make it work with Einstein’s relativity; it’s riddled with infinities, you know. Schwinger’s done some remarkable work reconciling the two— all operator theory and renormalization, I’m still trying to understand it. Somehow, there’s a way to replace the infinities with experimental measurements, then the beast is well behaved and gives very nice results. Dick manages calculate the same thing with these funny little pictures, and he puts plus signs between them like they were real mathematical formulae. Quite a ball at conferences: squiggle plus squiggle equals whatever. I mean to pick his brain about it before he flies off to Brazil.”
“Yes, he’s taking a visiting professorship. Says he hates the cold.”
On my way home that evening, I saw a shadow linger on my doorstep, then dart away. I broke into a run to pursue it, but not a trace was left, not even footprints in the snow. With one exception, that is: crinkled under the door and sodden with melt-water was a little envelope. Inside was a note, which read,
“The killer is left-handed.
Los Alamos, 1946
I should have sought Dr. Fermi right away, back when it was easy. When Mademoiselle Curie gave me the lead, Enrico was a quiet university professor in Rome. Since then, he’s got a lot harder to find, and it seems that the professor has government ties— secrets as big as the men who hide them. I chanced upon a tip leading me to a project in Manhattan, and though I found Fermi on the books as a Columbia professor, I had just missed the man himself. Asking an associate about where in the New World this Italian Navigator might be, he turned bright red and insisted that there was absolutely nothing in the basement. Nothing at all.
Sometimes you just get lucky. I asked one of his students to give me a tour of the basement, and was shown a room-sized apparatus for creating artificial radiation. “Artificial radiation?”
“Radiation is just ordinary particles, accelerated to high speeds. Naturally radioactive elements like radium spontaneously break off parts of themselves and shoot them at us, but we can accelerate them on our own with rapidly oscillating electric fields.”
“So this,” I asked, “is a sort of ‘particle-accelerator?’”
“I guess you could call it that.”
I was on the right track! “What do you use it for?”
“Well, Dr. Fermi did a lot of experiments with neutron capture, but by far the most exciting was the splitting of uranium atoms by a neutron beam. He disappeared soon after that.”
So Enrico wasn’t content to let atoms do all of the dirty work— this cat shoots back!
I was called to investigate the recent death of a famous physicist: Marie Curie, born Manya Skłodowska. When I arrived on the scene, she was in her death-bed, her face long and grey, a ghostly shadow in the warm light of the mountain sanatorium. Her daughter Eve was there. “It’s so quiet,” she said, “so fearfully motionless—”
We made our introductions, but she was obviously distracted. “So motionless, those hands. No longer nervously shaking, constantly moving, always working…”
I took a look at the hands, still and limp on the bed. They were hardened, calloused, deeply burned and thick-skinned. “What is this?” I asked myself, but I must have said it out loud because Eve heard me.
“Radium,” she said.
“Those were her last words— ‘Was it done with radium or with mesothorium?’ As she was stirring her tea with a spoon— no, no, not a spoon, but a glass rod or some delicate laboratory instrument… She had drawn away from human beings; she had joined those beloved ‘things’ to which she had devoted her life, and joined them forever.”
A cup of tea and now dead? That didn’t sound good. “Poisoned?” I asked. I never mind stating the obvious.
“Yes, poisoned. By radium. In the laboratory, she always used to say, ‘That polonium has a grudge against me.’”
“Radium— or polonium?”
In my previous post on card shuffling, I established a basic framework in which we will work. We are given a probability distribution on and we wish to determine when first begins to decay exponentially, where is the -fold convolution of One key feature of card shuffling theory, as well as much of finite Markov chains in general, is that the tools available are often very particular to a small class of problems. There just aren’t very many big hammers around. Even though the theorem described in the previous post was quite general, it was non-quantitative, and so not especially useful in practice.
The standard shuffling technique is called the “riffle shuffle.” In this shuffle, the deck is cut in half, and the two halves are zippered together. We need to come up with a mathematical way of describing the riffle shuffle, and I’ll list three different methods (I’m assuming the deck has cards here, but any will do):
First Way. The first thing to think about is how we cut the deck. Mathematically speaking, we will assume the number of cards in the top half of the deck after we cut is binomially distributed. All this means is that to determine the number of cards we cut from the top, flip 52 coins and count the number of heads to figure out how many cards go in the top half. It may seem strange that there is a positive probability of having all 52 cards in the deck sitting in the “top half” but the probability is extremely small and so doesn’t matter so much. For most shufflers, the size of the two “halves” are often quite different. Anyway, suppose that our result is cards in the top half. From here, we think of having 52 boxes lined up and put the cards in them. We pick of the boxes (assuming each box is equally likely) and put the top half of the deck in those boxes, keeping them in the same order. Put the remaining cards in the remaining boxes, keeping them in the same order relative to each other. Stack the cards back up. Note that there are ways to put cards in 52 boxes, so that the probability of any box choice is .
Just about anyone interested in mathematics has studied a little probability and probably done some easy analysis of basic card games and dice games. Completely off topic for a second, am I the only one who has noticed that basic probability homework exercises are the only situation, aside from funerals, that anyone will ever use the word “urn?” For whatever reason, probabilists love that word. Anyway, in any real card game, the computations tend to get complicated rather quickly, and most people get turned off from the discussion. With some ingenuity, however, one can answer some pretty cool (but initially difficult seeming) questions without having to go through a lot of tedious computations.
Take as an example card shuffling. In the face of expert card-counters, the natural question for the dealer is how many times he or she has to shuffle the deck before it’s well-mixed. In the case when the dealer is also playing the game — and is a card-counter at the level of a member of the MIT black jack team, say — he or she could drastically improve their odds by using a shuffling method which seems to shuffle the deck well, but actually is very poor at doing so. Anyway, at this point the question is slightly ill-posed, as we have no obvious way to interpret the word mixed, let alone well. In fact, coming up with a mathematical model of what shuffling means is already fairly difficult. What I’m hoping to do is give a framework which makes the problem more tractable.
A year or two ago, a couple of us were bored and somehow got to thinking about the series
It’s been a while since there has been any math on the blog, so I figured I’d share a recent (trivial) mathematical fact I came upon while passing the time. A less noble goal is that I hope some of you will find it interesting enough to think about it for a while. In other words, I’m too lazy to keep working on it but I hope some others will fall into my trap and let me know the answer.
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In a meeting at Chamonix last week, CERN, the LHC collaboration, and the LHC experiments came up with a 2009 schedule. “Second beams” (as opposed to first beams last year) will start a little later than expected: September 2009 instead of July. Then the goal is to have first collisions at the end of October, making the delay due to The Incident almost exactly one year.
Then after that, the good news starts in earnest. Instead of having a long shutdown over the 2009-2010 winter, as CERN usually does, the shutdown will be short, and we continue running and collecting data until we have something close to 200 pb-1 at 10 TeV, which will probably take about a year, until fall of 2010. That’s great, because it’s just enough for some of the basic discoveries: Z’ and W’ above 1 TeV, Higgs -> WW (if the Tevatron doesn’t see it first), the low-mass region of SUSY/mSUGRA parameter space (the “LM#” points), contact interactions in jets, and maybe a very optimistic extra-dimensions model (see my “Early Discoveries at CMS” talk at Dark Matter and the LHC conference). Running at 6 TeV collision energies was considered, and thankfully rejected, as that would be just below the interesting threshold for a lot of this. (The LHC’s design energy is 14 TeV, which is scheduled for 2011.)
At the risk of sounding naive, I think it’s really going to happen this time. The LHC people must know a lot more about the actual behavior of the beams from their real-data test last year, and given how disappointing last year’s setback was, I’m sure they’ll do everything they can to avoid anything like it. In other words, the argument is based on social reasons, not technical ones, but guessing when we’ll have data is a social science.