The beam went three times around the LHC ring: see CMS’s e-commentary and CERN’s twitter.  This is the milestone that was a big media event last year (September 10, 2008).

Update: now it’s 500 times around the ring (about 0.05 seconds).  Last year’s record was about 9 minutes of continuous beam.

Update: up to 9 seconds, 30 seconds (50k events seen by CMS)…

Update: and now a beam in the other direction has made a full orbit.  (All you gotta do is smack ‘em together!)

Update (Nov 21): on Saturday, we got hours of stable beam (single beams, not colliding).  This has never been done before with the LHC: from here on, it’s all new territory.  Now I’ve got to get to work on the offline data, which should be great for detector alignment…

See CMS e-commentary for live updates.

“Beam-splashes” are when a beam is threaded part-way through the LHC ring, then deliberately collided with an absorbing block of tungsten to stop it, upstream of a detector. Many particles are created in this collision, most of them are absorbed, with the exception of the muons and neutrinos. CMS can detect muons, and what it sees is a huge splash of activity, shown in this event display from September, 2008.

The blue bars indicate huge deposits of energy in the calorimeters. They seem to project from the center of the detector, but this is an artifact of the software, which was designed to visualize collisions from the center. The calorimeter cells measure energy, not direction, so when it sees energy coming from a flood of particles arriving from the right, it draws them as though they came from the center.

You can also see little parallel lines surrounding the central burst like a school of fish. These are individual muons seen by the barrel muon detectors, which do measure direction.

Update: here it is, the first CMS beam-splash of 2009 (from the e-commentary page)!

The little red lines are reconstructed muon tracks, blue dots are raw hits, and the yellow/blue starburst in the center is the calorimeter energy.  You can tell that the beam is coming from the right-hand side of the detector (“LHC beam-1″, the clockwise direction around the ring).

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.

The publicity of the turn-on events will be somewhat more subdued this time around. Of the main steps toward high-energy collisions— first circulating beam, 0.9 TeV collisions (accelerated purely by the SPS), 2.2 TeV collisions (breaking the Tevatron’s world record of 1.96 TeV), and 7 TeV collisions— the media will only be invited for the last; the others will be covered by press releases. Circulating beams are planned for the last week of November, with first collisions and a ramp-up to 2.2 TeV in December, followed by a Christmas break. Holding two 1.1 TeV beams in the ring will only require a modest 2,000 Amp currents in the magnets, far from the 8,500 Amp tests that caused the short last year. Early next year, the machine will be checked-out for running at 7 TeV (6,000 Amps in the magnets). After collecting a sizeable quantity of data and experience, the collision energy will be further raised to something between 8 and 10 TeV. While this might be less exciting than the original plan, it’s great for long-term scientific goals because the four experiments will have a year to calibrate on Standard Model processes and obtain a detailed map of the structure of the proton at smaller and smaller lengths scales (the “parton distribution functions”), which is a crucial factor in many calculations. When we do reach beyond the frontier set by the Tevatron’s collision energy and accumulated dataset, we’ll understand our detectors and the shape of the proton well enough to be sure that what we’re seeing is not just a mistake.

A few weeks ago, I watched PBS’s Journey to Palomar about the building of the Palomar telescope and was rather struck by the parallels with the LHC. Huge crowds flocked to Corning, NY in 1934 to watch the pouring of the glass for its enormous 200-inch mirror, and radio commenters were calling it the most significant event of the 20^th century, including the Great War. Contrary to the opinions of astronomers of the day, many expected the great telescope to finally reveal cities on Mars. At the first pouring, spectators witnessed the result of a miscalculation: the high temperature of the glass melted some of the cores in the mold, causing them to float up to the glass surface, ruining the mirror. A second pouring was attempted months later, closed to the public, and this time it was successful.

The telescope took until 1949 to complete, but when observations began, they revolutionized astronomy. The evolution of stars through the main sequence was finally understood. Cepheid variables measured the distances to galaxies 3 million light years away, allowing us to observe the history of the universe through its expansion. Palomar is still a working instrument, making discoveries today: hopefully the LHC will share the same fate.

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.

“You know her?” I asked.

“Not personally, but I have a roommate who calls her that. It’s unkind, but she’s so tough on her students. Keeps them up all night, just so they can get a reading when some who’s-a-ma-whatsit stabilizes.”

“Figures.” I grunted. “Physicists.”

The man grinned in the corner of his mouth, like I was in on some secret. “Something happens to their brains when they go over to physics, you know.” He took the pipe out to gesture, then checked to make sure no one was listening. “I have a friend whose thesis advisor took apart his experiment— the love and pain of years of his life, mind you— and even though all his work’s been ripped up, the fool boy’s been hopping around ecstatic,” he said in an excited whisper. “Telling everyone that the universe is lop-sided. Lop-sided! He falls prey to some silly, jury-rigged, week-end experiment, and he thinks they’ve learned something important about the universe!” He clenched down on the pipe again. “After all, we already know that the universe is absurd.”

“Where can I find this Lady Wu?” The card in my pocket said to seek her in Columbia, so I haunted the streets around Columbia University for weeks.

“Washington, I think. Hmm, yes, that’s right: she’s developed an unhealthy interest in refrigerators.”

Without wasting a moment, I booked a train for the District of Columbia.

Washington, D.C., December 27, 1956

A few thousand inquiries about refrigerators led me directly to the National Bureau of Standards and the coldest cryogenics laboratory on Earth. Madame Wu was a visiting scientist, borrowing the cold to chase down an idea about radiation. She was hard to talk to, rushing about and short on time. “It worked last month,” explained one of her colleagues, “but the result couldn’t be reproduced. The radiation counter was loose… This is too important for shoddy work: too remarkable.”

“What’s the experiment about?”

“Symmetry of the universe!” his eyes opened wide.

“The universe? In there?” They lowered a tiny flask into a hissing cryostat, bellowing cold smoke. There was a click, a clamping into place, more smoke, and then counting machines clattered down to zero. A few seconds passed. One… Two… Three, four… The little numbers flapped down intermittently, counting like an erratic clock.

That seemed to be the moment Madame Wu was waiting for, though she didn’t look relieved. “And now we wait. Thank you, gentlemen.” She was a Chinese immigrant, though she had been in the States for some time. I’m told she was skipping out on an anniversary cruise to do this experiment. Her husband had to go alone.

I coughed to get her attention.

“Yes, that’s right; you want to talk to me.”

“Madame, I just wanted to ask a few questions. About the universe.”

“Are you a reporter?” She seemed sharp, almost scolding.

“What if I were?”

“We know nothing for certain yet— experiment is not conclusive. We need to do more tests.”

“Could you at least tell me what you’re doing?”

She seemed defensive, as though there was some struggle between competitive secrecy and some implicit oath to disseminate all truths. “Yes, of course. Come to my office.”

She didn’t have an office, exactly. The corner of the lab where her desk was looked much like the rest: all plumbing and machine-shop tools. It was only when she sat down that the dignity she projected made it seem like an office. She gestured at a chair and waited. I sat down.

Opening a bottom drawer, she pulled out a large, circular mirror, and set it on edge on the desk. “What do you see?” she asked.

I laughed a little, wondering what this was all about, but she didn’t waver. I stared deep into the mirror. Nothing was coming to me. My mind wandered: the orb in her long-nailed talons could have been the glistening sun— the Dragon poses riddles! “Um, myself?” I stammered.

“You see yourself, and everything in room. But backwards.”

“That’s right.” Oh, that’s all.

“Is there anything unusual about it?” she asked again.

Stupefied, I shook my head.

“If you look at the world through the mirror, would you even know it? Would there be any clues?”

I was good at clues. “I suppose there’d be a lot more southpaws.”

“More left-handed people, yes. And writing would be all backward, of course. Those are human conventions. If there were no humans, or if historical accident had been just a little different, we would all write backwards in real life. We would drive on left side of road.”

“Except in England.”

“Whatever,” she waved her hand. “But the universe: like matter and energy— space-time— if all that were flipped like in a mirror, would anyone ever notice?”

“Like atoms, and radioactivity?”

She slowly set the mirror down— apparently that word meant a lot to her. “Especially radioactivity.”

“No, I don’t suppose we would notice one way or the other. An atom’s round… isn’t it?”

“Not all particles are symmetric. Like pions: there are left-handed pions and right-handed pions. When you have two pions, you can make symmetric state.”

“You’re losing me.”

She held up her two hands. “Left hand,” she clapped with her left hand only, “right hand”, she clapped with her right. “This,” she held her hands together, “is symmetric state. Left-and-right hands clasped together is the same as right-and-left hands together. Reverse both of them and you have the same thing. A symmetric, two-hand particle can decay into a left-handed pion and a right-handed pion, and that looks the same in mirror, but three pions must come from a not-symmetric particle, like two lefts and a right or two rights and a left.”

“Makes sense.”

“So what if a same particle sometimes decays to two pions, and sometimes three?” I could detect a glint in her eye, like she was setting a trap.

“I… I don’t know what that would mean. Maybe a symmetric particle sometimes gets asymmetric?”

“How does it know which asymmetry to choose? Right hand? Left hand? Who’s to say which way it should go?” Now I was sure it was a trap.

“Suppose the universe is not symmetric at all?” She folded her hands. “Suppose radiation favors left side over right side?”

I thought about that for a moment. “You’re saying that radiation is not symmetric, that it looks different in the mirror. What’s that got to do with the universe?”

“Radioactivity is a fundamental law of nature. The weak force is one of four basic laws of the universe. Everything comes from those laws. If radiation is left-handed, the universe is.”

I snuck a glance at the smoking caldron in the middle of the room.

“That’s what we’re testing. Symmetry.” She raised her eyebrows. What is it with physicists and dramatic gestures?

“How exactly are you testing it?”

“Beta decay is the most usual example of weak decay: many elements radiate by weak decay.” She stood and led me to the experiment. “If the mirror symmetry is violated by beta decay, then all weak-force interactions are. In here,” she pointed with a pencil, “we have tiny sample of cobalt-60 and radioactive counter on the bottom.”

“That’s the number here, right?” It was up to 86. 87.

“Yes. Cobalt-60 spins like a top. Like this:” she held out her hand and slowly closed it. “It is spinning the way my fingers curl. Around like that. It has a north pole and a south pole, like the spinning Earth. Now look in mirror.” She held up the circular mirror next to her hand and curled her fingers again. “North pole and south pole are reversed.”

“How do you figure?”

“Look closely: when fingers curl this way on my right hand, in mirror they go the other way.”

“The hand in the mirror is a left hand.”

“We define north pole as the way a right-handed thumb points when curling fingers: do what my mirror-hand is doing with your right hand.”

“I’ll be gum! My thumb is pointing the other way!”

“That’s right: poles are reversed in mirrors. Now suppose that my hand is cobalt-60 and radiates electron upward.”

“In the same direction as your thumb?”

“Yes, thumbward. Now look in mirror.”

“In the mirror, it still comes out the top.”

“But in the mirror, the top is the south pole.”

“I suppose it is.”

“What time is it?” She asked, popping open a watch. “The counter stopped. Did you get the reading?” she was asking one of the Bureau scientists as she pushed the mirror into my hands. The counter read 116. He waved a black lab notebook and reset the counter: the numbers all flapped back down to zero. He pulled a large electrical switch— the kind you see in the pictures— and a humming sound noticeably faded away. “Wait for it,” she cautioned, listening intently. “We have to reverse polarity slowly. Okay, turn it on now. Slowly!” He dialed a knob down to zero, closed the switch in the opposite direction, and turned up the dial— slowly. The humming resumed. When the little needles in a bank of meters stabilized, he clicked on the counter. One… Two…

There was still more work to do, making sure everything was running smoothly, before she had time to talk to me again. I hunkered back, trying not to get in anyone’s way. When she seemed to be just coasting, I reminded her of my presence. “So you were saying… in the mirror-world, an electron jumping off the north pole is really jumping off the south pole?”

“What were we talking about? Oh, yes: that. If the universe is mirror-symmetric, how often do electrons jump off north pole, and how often south pole?”

“I don’t know: I’m not an expert.”

“What if all electrons jumped off of north pole?”

“Then they’d be jumping off the south pole in the mirror-world.”

“That would not be symmetric, would it?”

“I suppose not.”

“If the universe is symmetric, there must be equal north-pole electrons as south pole electrons. You can’t have more, or symmetry would be violate.”

“So you’re counting,” I ventured to guess, “the number of north-pole electrons and the number of south-pole electrons. How do you do that?”

She smiled cleverly. “Inside,” she pointed with a pencil, “we have cobalt-60, and radiation counter on only one side.”

“You told me that.”

“All inside of strong magnetic field. The field forces the cobalt-60 nuclei to line up and spin in same direction. Before, we had all north poles down; now we have all north poles up.”

“And the big refrigerator?”

“All of that is inside very cold cryostat, cooling cobalt-60 to 0.003 degrees above absolute zero temperature. When the temperature is zero, the nuclei don’t jiggle as much, and they stay where the magnetic field puts them.”

“So that just makes the experiment cleaner?”

“It is very important,” she cautioned me gravely. “This is the universe we are talking about.”

“Madame—” one of her colleagues pointed to the counter. Time was nearly up, and it read only 15. The mirror slipped from my hand and cracked on the floor. Grave faces all around.

“Wow,” I whispered, sagely.

“No story!” she pointed an accusing finger at me. “We do it again, longer this time. And then positrons: cobalt-58! Promise me no story until we do it again!”

“I’ve got a confession to make, Ma’am. I’m not a reporter. I’m a private eye.”

“It’s not conclusive yet. We still have cross-checks to make. No story— promise!”

I held my hands up defensively and backed out of the lab. “I promise,” I said as I opened the door. “You won’t hear a peep out of me.”

She turned to her coworkers. “Take it apart. We do it again.”

Whereabouts Unknown

I must have wandered aimlessly for the next few years. In this suddenly off-kilter universe, I felt as though half the ground had been pulled out from under me. It wasn’t long before I connected it to the other mysterious note: “The killer is left-handed.” The universe is left-handed. Ergo, the universe is the killer: I wanted a simple answer, but this was hardly less unsettling. What was the universe up to, anyway, throwing around a bunch of left-leaning particles that sometimes kill scientists?

I meandered into New Mexico, mostly hitch-hiking. I guess my beard grew out; I saw myself one day in a reflection and it surprised me. Avoiding Los Alamos (they seemed a little tougher there these days), I hooked up with a band of vagabonds, who put up with my sputtering nonsense about parity violation. Under the stars one night, my fellow-traveller said, “You know, it’s all expanding.”

“What’s that?” I asked.

“The universe! Stars, galaxies, everything. Expansion without a center. Or the center is everywhere. Right here is the center of the universe.”

Another of our tribe intoned, “woah.”

“And nobody knows why.”

That was enough for me. “Damnit!” I stood up. “Why’d you have to tell me that?” I left them that night.

Wandering westward, I became a scourge on the fair city of Pasadena, California. Sleeping behind a dumpster one day, a magazine in my blanket caught my eye: “The New Einsteins,” it read. Einstein was a physicist, but I hadn’t thought of talking to him while he was still alive. The article had at least as much to do with the physicists’ personal quirks as their science: who would write such trash? Still, it mentioned Madame Wu’s discovery that the parity symmetry is violated, and somehow it had something to do with Sputnik. But it didn’t stop there— apparently, parity violation was no longer a mystery! It was explained by so-called V-minus-A interactions, something alluded to as horribly complicated, with photographs of well-dressed gentlemen arguing in front of scribbled-over blackboards. “The dapper, cocky Maurry Gell-Mann,” it said, “is many physicists’ choice for the brightest light in his esoteric field. With a Madison Avenue fastidiousness about his clothes, his boyish face and quick tongue, Gell-Mann exhibits a kind of sharp intellectual fussiness that has more than once wounded his colleagues.” Amazingly enough, he also lived in Pasadena, so I resolved to find him and ask him about this V-minus-A.

Strolling onto the Caltech campus, I got a lot of nervous glances. I guess they weren’t used to ragged men such as myself at institutions of higher learning. In the physics building, I found Gell-Mann’s office and pounded on the door until I heard a voice inside: “Come in, come in.” I burst into the room, but couldn’t think of what to say. He gazed at me in horror. “What are particles, anyway?” I asked in a raspy voice. “Are they really shaped like little hands? And why is the universe so lop-sided?”

He pondered in amazement for a moment, then said, “I think you want Mr. Feynman. He’s the expert in these fields.”

“Feynman?” The name was awfully familiar. Wasn’t I supposed to be trailing him?

“He’s the one whose van is covered in Stueckelberg diagrams,” he explained with some contempt. “Go ask him. Shoo-shoo,” he waved me away with his wrist.

Feynman, of course! He’s the guy Fermi told me to follow. He’s the guy who made so much sense teaching introductory physics. And he’s also the one who revolutionized the quantum theory with his little diagrams— if anybody can explain what this is all about, it’s him! And he lives in Pasadena, too!

In the parking lot, I found his van: white with squiggily diagrams drawn all over it. I got into the passenger side seat and waited. I waited long enough to have fallen asleep, but woke up when Dr. Feynman got into the driver’s side. I don’t think he noticed me. Clogged up with sleep, my voice croaked when I asked, “So, what are particles?”

“Woah!” Dr. Feynman jumped out of the van faster than I thought possible. “Woah! Woah!” He was still jerking and jumping about outside.

“If a pion emits a muon when it decays,” I continued, “does that mean that the muon was inside the pion?”

He was speechless for a moment. Then he said, “I’m having that dream again.”

“Dream?”

“You’re— just a dream, right?”

I pondered that for a moment. Given the circumstances, I wasn’t sure. “Are particles even real?”

“Well, sure— well, it depends on what you mean by real. Uh, no, not really.” He thought for a moment, then came to some kind of a decision. “Look, this is gonna take a while. How ’bout I buy you a drink?”

I shrugged.

We went to a strip bar, where the waitress apparently knew him. “The usual, Dick?” She was bright and cheery to Dr. Feynman, but scowled at me. The usual was apparently a soda-pop.

“Drank too much in a bar in Buffalo,” he explained, “and ended up with a lulu of a black eye. Never gonna do that again.”

“And there’s something special about left-versus-right, isn’t there? What is V-minus-A?”

“Jeez, you’re fulla questions, aren’t you? Look, I think I know how to explain this. Just forget about particles: there’s too much mental baggage hanging on that word.”

“Okay.”

“We sometimes say that electrons and whatever are particles, and sometimes that they’re waves, but there’s something wrong with both these words because we already think we know what ‘particles’ and ‘waves’ are. Particles are like little pebbles and waves are like on the ocean, right? Well, if you start thinking like that, you’ll be wrong right away because they’re not either. Electrons, pions, muon: they’re all something entirely different, something new.”

I started to dispair of understanding it at all. “But I like to think in analogies.”

“Analogies are good, yes, but you need to know where the analogies apply, and where they don’t. You can’t say what electrons are in a one-word analogy. It’s gonna take, well, more words.”

I let him go on.

“Let’s start by thinking about a field. A field’s a good word, because you don’t really know what it is from common experience. I mean a field like a magnetic field, not like a field of grass. What do you think of when I talk about a magnetic field? What do you visualize?”

“I think of a kind of invisible glowing: it makes any metal nearby want to fly toward the magnet.”

“Yeah, basically. At every point in space, there’s a number, and that number just says how strong the magnetic forces would be on any metal object that might be there. Near a magnet, the numbers are big, and far away from any magnets, they’re small or zero. It’s three numbers, really, representing a little arrow at every point in space, pointing in different directions with different lengths, swaying this way and that when we move magnets around.”

“Now it is beginning to sound like a grassy field.”

“Yeah,” he seemed amused. “That could be why they called a field in the first place. An analogy, but the arrows are at every infinitesimal point in space, and they’re all invisible. Grass doesn’t do that.”

“So what does this have to do with particles?” I asked, impatiently.

“There you go with the particles again! Forget about particles!” He took a delicate sip of his soda-pop and continued. “So we’re happy with magnetic fields, and then you know there are also electric fields: Maxwell back in the 1800’s figured out how they relate to one another. Magnetic fields and electric fields are actually both just parts of a single electromagnetic field. So there’s just this electromagnetic field, lots of numbers at every point in space, and different components of it were historically named ‘electric’ and ‘magnetic.’ He also found the equation that describes how those numbers relate to their neighbors: if the field has a big value here and a small value right next to it, the small value is gonna want to get bigger and the big value wants to get smaller. Wait a moment in time and they change. Wait a moment longer and they change more. On and on until they equalize, and then even further because the equation says that they tend to overshoot.”

“I’m having trouble visualizing this.”

“Don’t worry— the short story is that large values in the field tend to propagate, so that if I jiggle a magnet or something, I can start these waves undulating in the field; they spread out all over the place and bounce off of mirrors and such.”

“Like water waves,” I added. “Drop a rock in a lake and the waves spread out everywhere.”

“Yeah, the analogy works so far. The difference is that we know what the field is for water waves and we know why its equation works: the field for water waves is the height of the water everywhere on the surface of the lake, one number at every point. When the water is higher at one point than at a neighboring point, it wants to flow downwards, increasing the low field value and decreasing the high field value. Then it overshoots because of inertia: it’s all just gravity and Newton’s laws. But the electromagnetic field is not water, it’s something else. We don’t know what the numbers at each point in space mean or why they have this equation relating them.”

“Ether?” This was sounding familiar, from somewhere.

“Ether was a popular theory, again in the 1800’s, but it was really just a problem of them taking the analogy too far. Ether was this idea that there was a material, uniformly filling all space, but invisible and so evanescent that we don’t notice that we’re walking through it all the time. In fact, planets don’t even slow down when passing through it for millions of years.”

“Are you saying that’s impossible?”

“No, no, anything’s possible, especially when we’re dealing with the unknown. But the problem with the ether theory is that they assumed too much: they thought this ether was made of some kind of weird atoms or something, something with a velocity like you and me, you know? The whole space-filling ether would be a kind of object with a definite position, and we should be able to tell if we’re moving relative to it or not. The electromagnetic wave equations don’t suggest anything like that, they’re just waves in the equation, abstract numbers without any hint of a stationary medium; the ninteenth century physicists just assumed it because of the analogy with water. In fact, a famous experiment showed that the speed of this so-called ether always seems to be zero relative to us, even when we go through it at different speeds, in different directions. An equation can do that, but matter can’t. What made Albert Einstein famous was just looking at the equations and asking what they imply, without ever assuming that they must be waves in some material.”

“So there’s nothing but equations? Nothing’s real?” I felt a “woah” coming on.

“Of course they’re real! When you get to thinking about this stuff, you have to decide what you’re going to count as real. We started out by looking at matter, undeniably real stuff, and asked, ‘what is this stuff, really?’ We find that it behaves differently than we might have thought at first. We find that our traditional descriptions break down, and we have to step back and let Nature tell us what it’s all about. And then we try to describe it in the only precise language we know, a language without preconceptions. That’s mathematics.”

“So let me get the picture, then: the universe is a space-filling field of numbers that obey an equation?”

“In a nutshell.”

“What about electrons and pions and muons and all that? Whenever I talk to physicists, they usually talk about these particles bouncing off of each other and breaking apart.”

“Each of these particles, electrons, muons, and the like, each one of them is a separate field. Or they’re different components of the same field, which is the same thing, just more numbers at each point. Ya know, when we talk about The Electron, we don’t make a distinction between a single electron particle and the whole lot of them. It’s as if we would talk about all the geese in the world by saying, The Goose. And that’s actually the right way to think about it. There’s just one electron field, and it has different intensity values in different places. You’ve heard of photons, right?”

“Photons, they’re bits of light, right?”

“Photons are light— and radio waves, and gamma rays, microwaves, infrared, ultraviolet, X-rays, the whole spectrum. But ‘photons’ are just a modern way of talking about the electromagnetic field that Maxwell discovered. Maxwell discovered that combining the equations for electricity and magnetism, you get waves that propagate all over the place, with any imaginable wavelength. And then it was slowly discovered that all of these different kinds of waves, light included, were really just waves in the electromagnetic field with vastly different wavelengths.”

“But I’ve heard that photons are particles.”

“If you say ‘particle’ one more time, I’m gonna knock your lights out! The whole particle business comes from quantum theory. For some reason— and this gets even more mysterious— energy comes in clumps, ‘quanta,’ a certain fixed quantity, such that you can only have zero, one, or two of them, and so on. For any given wavelength of electromagnetic wave, the intensity of that wave is restricted: it can be enough to give you one quantum of energy, zero, or two, et cetera. Einstein also showed how mass and energy are equivalent, so unsplittable units of energy are unsplittable units of mass, which made people think they were dealing with some kind of indivisible particles.”

“So they were wrong: there are no particles, it just looks like there are.”

From the strained expression on his face, I don’t think he liked that conclusion. “We’re defining words as we discover this stuff, so you could just say that this is what ‘particles’ really are: quantized excitations of fields. A lotta my friends define the word that way just so that they’d be retroactively right. Which is fine, of course.”

I thought for a long drag, trying to take it all in. “That doesn’t explain decays. How does a muon come out of a pion? I mean, they’re both excitations, sure, but how does one excitation come out of another?”

“Sit tight: I haven’t given you the whole story, yet.” He thought for a moment. “I said that the field obeys an equation; the equation tells it how to propagate. Maxwell’s equations for electromagnetism are only the simplest example. You can re-write the equations in a form called the Lagrange Equation, which makes explicit how energy wants to flow from one point in space to another, and from one field to another. The Lagrangian is an expression that Nature wants to minimize, so you can think of it almost like economics: it’s a cost function. In classical physics, Nature exactly minimizes it— a strict book-keeper— but in the quantum theory, Nature allows a little loss here and there if it will get a bigger return on its investment later.”

“Should I be taking this analogy literally?”

“No. The form of the Lagrangian is a series of terms added together, and each one has a physical meaning. The electromagnetic Lagrangian, the one that comes strictly from Maxwell’s equations, has only one term: changes in field values multiplied by changes in field values.” He wrote something down on a napkin, something with lots of subscripts and superscripts. “By ‘changes,’ I mean derivatives in calculus, a calculation of how much the field values vary from one place to the next and how much the field values vary from one time-step to the next. By multiplying them together and minimizing that, the way economists do with cost functions, you get waves. Another way to look at this term is as a kind of connector: it connects field values at one point in space and time with field values at the next point over; hence, energy flows from place to place.

“When we look at a different kind of field, like an electron field, we need to add another term to the Lagrangian. Electrons propagate all over the place, just like photons, so that first term is similar, but unlike photons, electrons also have mass. This allows them to do a curious thing: sit still. As a consequence of the wave equation, photons have no choice but to zip around at the speed of light. Photon energy can’t sit still unless you put it between two mirrors or something. For photons, energy absolutely must flow from place to place, because only neighboring points are connected by a term in the Lagrangian, but electron energy can sit still, or move at any speed less than the speed of light. The second term in the electron’s Lagrangian is just the field multiplied by itself, effectively connecting the field value at one point with the field value at that same point. Therefore, energy doesn’t need to flow from one point to the next, it can flow from the point to the same point again, going nowhere.”

“That’s another reason that electrons look like particles.”

“If you like your particles to be stationary, then this is a good feature for you. Thinking of this term as a cost function, what do you suppose it looks like?”

Reading his napkin closely, I said, “it looks like two tridents next to each other, like Neptune had twins.”

“What? Naw, I don’t mean the letter.” He scribbled out what he had been writing, probably only writing by instinct. “The letter psi is just how I’m representing the numerical value of the field as an unknown in the equation. I mean what does the graph look like, the graph of psi times psi, or if you like, x times x— x squared?”

“I wasn’t expecting an algebra exam!”

“Why shouldn’t you be? We’ve been talking about numbers and equations— didn’t you think that math would come in at some level? I can tell you how the equations generally go, but to learn this in any detail, you’ve gotta do some math. Besides, this is easy— high school stuff.”

I thought really hard, remembering only Sister Drummy’s hard-edged ruler. Then it came to me in a flash: I drew a large cross for the x and y axis, then x squared was a U-shaped cup whose lowest point passed through the x-y intersection. “It looks like a trident!” I exclaimed, with some glee.

“Well, I’ll be durned,” he chuckled, “it does look like a trident, after all. Okay, where were we? Oh yeah: this U-shaped cup, that’s the value of x squared for each value of x. When x is 1, x squared is 1, when x is 2, x squared is 4, when x is minus-2, x squared is 4 again, et cetera. This ‘x,’ or my ‘psi,’ is the value of the field at some point in space, like here.” He picked a point in space from the air in front of him, somewhere over his soda-pop. “If the field is fluctuating around small values like 1 or minus-2, this term in the Lagrangian is small but always positive. If the field is fluctuating around large values like a thousand, the term in the Lagrangian is large, like a million.”

“Does that mean it has more energy?”

“EXACTLY!” He seemed very happy that I got something right. “The Lagrangian describes energy flow: thinking economically, this mass term is like a bank. You can deposit energy into a point in space and it stays there: a phenomenon we’ve traditionally called mass. And since you can only do it in fixed amounts, one dollar increments, let’s say, you can only create an integer number of particles. Then you can just as easily take the energy out of the particles and use it to make motion if you want.”

“This is why particles decay, isn’t it?”

“YES!” He was still very excited. “And it’s how you can make them in collisions, too. It’s just another term in the Lagrangian; there’s nothing more mysterious about making and destroying particles than there is in things moving from place to place. Heck, ‘moving’ is essentially the same thing as destroying a particle in one place and making another one right next to where the first used to be. Take the energy out of one point in space and put it in the next— like switching banks. And that gets to your question about pion decay: when a pion decays into a muon and a neutrino, energy comes out of the pion field, part of it goes into the muon field, and part of it goes into the neutrino field. There’s a bit left over because the mass of one pion is more than the mass of the muon and the neutrino, so that’s put into the motion of the two final particles, er, waves.”

“How does the pion know it should decay into a muon and neutrino?”

“Oh, I forgot to tell you: there’s another term in the Lagrangian, expressing the connection between the different types of fields. The Lagrangian is a wonderful little package: it explains the existence of wave motion, stationary mass, and transitions between all the types of particles in one stroke. They’re all just connections between field values, a subway map saying where energy can flow to. You can write Lagrangians down to describe all sorts of situations. You know, I was listening to a concert cellist the other day, thinking about the way that sonorous old cello reverberated around the room, and I came up with this cartoon Lagrangian,

L = bow*string + string*string + string*cello + 100 cello*cello + cello*air + d(air)*d(air) + air*ear

She pulled the bow across the string, transferring energy to the string, the string reverberated with itself like a little particle, then the string shook the cello body and the cello body reverberated with itself a lot, a really heavy, massive particle— that’s why there’s a factor of 100 there— and then that flowed into the air and propagated through the air as waves. That’s why it’s got the derivatives term. Then that is connected to my ears and I heard it. Like some kind of complicated decay chain, a Xi particle to a Lambda to a pion to a muon to an electron.”

“Didn’t the music reverberate in your head?”

“Oh yeah, let’s add that! + ear*head + 10 head*head. It didn’t reverberate as loudly in my head as it did in the cello, but it had a longer lifetime before it decayed. Look, we can also draw it like this.” He drew a chain of lines, each labeled “bow,” “string,” “cello” (with a loop), “air” (as a wavy line), “ear,” “head,” loop. “This diagram encapsulates some of the information in the equation, which can be really helpful when things get complicated. Like the cascading decay from the Xi to electron. That’s a big, hairy tree of a diagram: Xi to Lambda pion, pion to muon neutrino, muon to electron neutrino, Lambda to proton pion, pion to muon neutrino, and finally, muon to electron neutrino. All told, 13 particles! All just coming from one Xi: it has quite a lot of mass energy to divvy up. Each one of these points, where the energy flows from one field into two others, is related to a term in the Lagrangian. There’s a prescription for turning diagrams into equations, and now these funny little pictures are all over the literature.”

It looked like the drawings on his van. “Are these Stueckelberg diagrams?”

“These,” he roused in contempt, “are Feynman diagrams! Hey! Have you been talking to Maurry?”

“Gell-Mann? He sent me to you.”

“That rat! Oh, no offense— it’s been a pleasure talking with you. When I see him tomorrow, though, I’m gonna pop him!” He doodled for a while.

“So what about forces?” I asked. “If everything is a field with a Lagrangian Equation, how come there are forces? Don’t these waves just flow through each other, being different fields?”

“Forces are the same thing,” he seemed a little surprised, “interactions in the Lagrangian. Energy of motion flowing from one particle to another. Here, look, I’ll give you an example.” He started drawing again: two kinked lines with a wavey line connecting them. “These are two electrons, and the wavey line is a photon. The line represents the trail that the particle follows in space and time, occupying one point in space which changes as a function of time.”

“I thought that electrons are waves of energy, kinda spread out.”

“Yeah, yeah, you got me there,” he bobbed his head, chuckling, “but that would be hard to draw. The idea behind these diagrams is that they just describe the connections and approximate the space-time description by showing one possible path. When we convert this into equations to try to calculate something, we have to add together all the possible paths that this represents.”

He waited for me to assent. “Okay, fine. Go on.”

“Okay! So remember that photons don’t have a mass term. There’s no minimum balance in the photon account. You can make photons with as little energy as you want. Well, this little photon wave propagates from this electron here to that electron there, taking not just energy but momentum: that’s why the top electron pulls away, to balance the momentum given to the photon. And then when the photon wave gets to this other electron, it gets absorbed, and pushes the other electron away. Thus, two electrons repel each other when they get close. Neat, huh?”

“That’s why all electrons repel each other?”

“That’s why any charged particles attract or repel. They transform part of their momentum-energy into an intermediary, which then gives it to another particle nearby.”

“What if you replace the photon with a pion? The pion’s the nuclear photon, I’ve heard.”

“For a bum off the street, you know a lot about some things! Yes, we could put a pion in there in place of the photon, but then the electrons wouldn’t notice, because electrons don’t connect to pions. There’s no such term in the Lagrangian. But also replace the electrons with, say, protons, or neutrons, and then the pion is very happy to connect them with a force. That’s the nuclear force, holding the nucleus together!”

“The Lagrangian describes everything, doesn’t it?”

“You’re tellin’ me! We’ve got motion, mass, particle decays— which is radiation— and now forces, all in one neat package. All that’s left is to find out what exactly is in the universe’s Lagrangian, the weltformel, or ‘world-formula,’ as Heisenberg liked to call it.”

I blinked. “Wow. And I thought physics was a mess. All these perplexing, contradictory discoveries, the lop-sided universe…”

“These are the best of times!” He seemed stunned. “Perplexing discoveries are a physicist’s dream come true! It’s like being lost in a candy shop: all these mysteries to unravel. Did you ever experience the joy of figuring something out?”

“I’m a private detective…”

“It’s wonderful! Absolute pleasure. So much fun when you’ve got it that you’ll willingly spend months or years struggling with another problem to get it again. Did I ever tell you about the time I figured out V-minus-A?”

“You know, I’d love to know.”

“I had this idea in Brazil, when I was coming home from vacation. I was thinking about these parity violation experiments, the ones that were showing that the universe is mirror-unsymmetric.”

“Like Madame Wu’s experiment?”

“Madame Wu’s was the best, but there were others. Leon Lederman— now there’s a character— he took apart his poor student’s thesis project to get a quick result and scooped her. But I think we wouldn’t have totally believed it without Wu’s result. There were a few others, and some of them contradicted each other in the exact way that the symmetry is violated. You could put some terms into the Lagrangian to explain some of the experiments, but not other experiments.”

“Sounds like a mess to me.”

“A mess, sure, but exciting! So I tried this one particular combination: a vector current minus an axial one— that’s the ‘V’ minus the ‘A,’ they’re mathematical forms which have the properties of arrows, that’s the vector, and rotational poles, like the north pole and the south pole. That’s the axial form.”

I could see he wasn’t sure if that was a good explanation. I held out my hand and curled my fingers to show that I was at least partially on board. It seemed to be a universal physicist’s gesture, like a secret hand-shake.

“Yeah, that’s right. One flips direction when you look at it in the mirror and the other doesn’t. If you subtract the two, you get only the flipping. On a lark, I tried assuming that was the right form and just seeing what happened, if you push it all the way through the equations. And wouldn’t you know it— I got the right answer, nearly, for all of the experiments! All except one, and that one turned out to be wrong. It was… sparkling!”

“Sparkling?”

“Yeah, sparkling! As I thought about it that night, as I beheld it in my mind’s eye, the goddamn thing just shone brilliantly! You know, it was the first time, and probably the only time in my scientific career, that I knew a fundamental law of nature that no one else knew. Now, it wasn’t as beautiful as Maxwell’s, but it was a bit like that. It was the first time that I discovered a new theory, rather than just a more efficient method of calculating someone else’s theory, or a little solution to a problem. It was the most amazing night of my life.”

“Until you got to work the next morning,” said a woman with a Yorkshire accent, “and found out that Maurry had thought of it first.” Who was she? We weren’t in the bar anymore, but in Feynman’s backyard in Altadena, under the stars. When did we relocate?

“Now Gwen,” (that must have been his wife,) “Maurry han’t entirely worked it out.”

“And he got it from that poor student you scooped.” She was having fun with this, apparently not for the first time.

“I couldn’t possibly have known about that. I was in Brazil at the time!”

“Still, you weren’t the only one in the world knowing it. Including the student’s advisor, I’d count four.”

“Okay, so I knew something only four people in the universe knew about. Still, it was beautiful!”

She smiled. “I’ve heard all this before,” she said as she got up. “I’m off to bed. You two have fun.”

“Good night, Gwen.”

“Good night,” I added, still a little shaken by the change in locale. How did I not notice driving here?

“Listen—” he turned to me, conspiratorially. “I want you to help me play a trick on my old pal Maurry. It’s only fair, after all.”

“Okay,” I said, hesitantly.

“Tomorrow we’ll be working in his office on some Buddhist thing of his. I want you to barge in, claiming to be Chairman Mao’s secret police, come to put a stop to all this illegal religion.”

“Buddhist thing?”

“He’s got a theory, but for fun he’s phrasing it all in Buddhist terms. I think it’s time someone zings him for that, clouding up the literature with weird words.”

“So you want me to barge in… again?”

“‘Strangeness!’ Can you believe it? He got everybody saying things like ’strangeness.’ What’ll be next, ‘quirkiness?’”

“Sure! Why not?” I said. “Do you think I could pass for Chinese?”

“Listen, I’ll drop you off at your place. Where is that?”

“Oh.” And then I proceeded to explain where I lived.

Pasadena, January, 1961

I had forgotten the simple luxury of taking a bath in a hotel. To make the most of the experience, I shaved and cut my hair, too. What a mess! The whole sink was full of hair, and much of the floor, too. While I was cleaning it up, getting on my hands and knees to sweep it up in my fingers, you wouldn’t believe what I found. Stuck in there, behind the fixture, another note! My fingers trembled as I shook it out, and read,

“You struggle in vain to make out what you see,
But I run in your loops, and you don’t know me.

– signed, 0⁺⁺“

“Zero-Plus-Plus? What in Hades… Feynman! All this time, it was him! All those notes, stringing me along, and I thought I was following him. That fink!”

In a moment, I was dressed and out the door. I returned to the sunny campus a changed man, literally, as I was now dressed in the grey trenchcoat and fedora that let me blend into a crowd. I burst into Dr. Gell-Mann’s door, and pointed an accusing finger at Feynman, sitting smug while Gell-Mann caressed the blackboard with Kabbalistic diagrams. He was already laughing.

“You!” I shouted.

“Me?” he seemed a little surprised, the cunning fox.

“You! You thought you were so clever, but I’m on to your ruse.”

“Me?” he asked again, still laughing, probably more now than ever, “Don’t you mean him?” Gell-Mann was equally surprised.

“You’ve been sending me all these notes!” I held out the latest one.

Until this point, Gell-Mann’s mouth had been hanging open like a trout. He composed himself and asked, “Do you two need privacy?”

“No,” we both said, at the same time.

Feynman, through fits and chortles, managed to get out, “Weren’t you supposed to be Chinese?”

“Let me see this,” Gell-Man took the note from my hand. “‘I run in your loops?’ What kind of love-letter is it, hmm?”

“It’s not a love-letter— it’s a red herring!” I exclaimed, “He’s been steering me off the path all this time!”

“I should say that’s not unlikely.” He pushed up his thick-framed glasses with his middle finger. “‘Signed Zero-Plus-Plus,’ well, that is interesting.”

“Lemme see it,” said Feynman, grabbing for the note. Gell-Mann didn’t let him have it.

“‘You struggle in vain to make out what you see,’ he read, and actually started writing it on the chalkboard, amid all of his mathematics. ‘But I run in your loops, and you don’t know me.’Repeated stress on the last three syllables: how appropriate.”

“But I do know him— now,” I explained.

“‘Signed, Zero-Plus-Plus.’ You suppose that Dick is the mysterious Zero-Plus-Plus, do you?”

“Lemme see that,” Feynman grabbed for it again, this time successfully. “I didn’t write this.”

“Oh, come now— who else could have?” I said triumphantly. “Ithaca, Rio de Janerio, Ithaca again, and now my hotel room in Pasadena. Who else would have known my hotel room?”

“That does sound like Dick,” Gell-Mann wrote the cities in a column on the board.

“I didn’t write this,” he protested, but not too hard.

“No?”

“No, this isn’t even my handwriting.”

“Who else could it have been?” I demanded. “And who’s Zero-Plus-Plus?”

“That’s the question,” Gell-Mann continued, “the heart of the riddle. Do you suppose it might be the J^PC?”

“Does that stand for something I don’t know about?” I asked.

“J^PC, like quantum numbers?”

“It would be a scalar,” he continued, “invariant under parity and charge conjugation…”

“Are you saying that a particle wrote the note?” Feynman was obviously following this.

“Could you two please explain what you’re talking about?”

“Sure,” Feynman turned to me and said, “each type of particle, each field like we were talking about last night, has certain properties that can be neatly written up in in a few numbers, its so-called quantum numbers, and we call that a signature.”

“Purely a book-keeping device,” Gell-Mann chimed in.

“So this ’signed’ business followed by ‘0⁺⁺‘ looks suspiciously like a particle’s signature. ‘J’ is the internal angular momentum, a scalar without any spin at all, just a single number at each point in space.”

“The minimum possible structure.” Gell-Mann added. “Do we know of any 0⁺⁺ particles?”

“The first plus is the parity, what the field does when you replace it with its mirror-image. Plus means nothing happens at all. Same thing for the second plus, that’s what the particle does when you reverse the charge, replacing all particles by anti-particles. This 0⁺⁺ is unaffected. In a sense, it is its own anti-particle.”

“I don’t think there are any 0⁺⁺ particles,” Gell-Mann had written all the particles he could think of on the board, with their signatures. He was running out of room.

“Maybe we just haven’t thought of it yet,” Feynman answered. “After all, the note does say, ‘you don’t know me.’”

“Are you two saying that a particle wrote these notes?” I asked, finally catching on.

“Stranger things have happened,” Gell-Mann answered dryly.

“Could it be a bound state?” Feynman asked, now fully engaged in this puzzle.

“I don’t know everything there is to know about particles,” I chipped in, “but I’m fairly sure they don’t write letters.”

“What about that Omega-minus in your theory?” he offered.

Gell-Mann seemed a little indigant. “That’s three-halves-plus, you remember?”

“Oh, yeah, of course. Yeah, it’s a baryon, that’s right.”

“‘I run in your loops’ suggests a virtual particle,” Gell-Mann went on, “That was the first thing I noticed. What else runs in loops?”

“Dogs?” I offered.

“If it’s a virtual particle, it could be anything.” Feynman was scratching his head.

“What’s a virtual particle?”

Feynman graciously paused to explain again, “Do you remember those diagrams I drew last night?” He grabbed the chalk from Gell-Mann and drew a Feynman diagram with a circle in the middle of it.

“Stueckelberg diagrams.”

“Shuddup, Maurry. These diagrams are schematic; they just describe connections. When you calculate something from them, you have to add up all the possible paths with the same topology. These lines on the outside, they’re external legs, corresponding to particles we measure in the laboratory. Their momentum is fixed. We measure it, so there’s only one possibility for what it can be. With me so far?”

“Go on.”

“This particle in the middle, the loop, it has to transfer the right momentum to and from each of these legs, but its own momentum can be anything.”

“ANY-thing.” Gell-Mann stressed.

“It could be something close to zero, something on the same order as the external legs, or an enormous value. It could be thousands, millions, billions of times the momentum of the external legs, there’s no limit.”

“You have to integrate to infinity,” Gell-Mann added, nodding helpfully.

“The trouble is,” Feynman shook the chalk in the air while he talked, “the trouble is that there could be dragons up there.”

He let the import of that settle on the room, but I was perplexed. “Dragon particles?”

Gell-Mann rolled his eyes. “Metaphorical dragons.”

“New physics, things we haven’t discovered. We only know how the universe works at low energies, low energies and whatever we can cook up in colliders, which isn’t much.”

“Much less than infinity.”

“So if the universe has new laws, new interactions, well, new to us at least, it would have an effect on any diagrams involving a loop, and we wouldn’t know what it is.”

“Oh.” So any diagrams that have internal loops are essentially uncalculable. “How many physics processes involve diagrams with loops?” I asked.

Gell-Mann smirked. “All of them.”

“All of them? Then you’re telling me you can’t calculate anything?”

Feynman seemed a little defensive. “We can and we do.”

“But they’ve all got this uncertainty! You just told me that you can’t calculate that, that— dragon diagram, there.”

“It could have contributions from as-yet unknown particles,” he admitted.

“That seems to be what the couplet is plainly saying,” said Gell-Mann.

Feynman chuckled at my indignation. “God himself could be running in the loops.”

“I prefer to think it’s a yin-yang.”

“Right, Maurry, I forgot you were on your Buddhist kick.”

“Please, Dick, the yin-yang is Taoist. And I rather like the way we can’t know anything until we know everything, don’t you?”

I stamped my foot. “But that’s not science!”

Silence. I struck a chord.

“If we don’t proceed from the known to the unknown,” I lectured, “we can’t be confident in our conclusions. It’s all just guessing games.”

Gell-Mann smiled from ear to ear. “You’re quite right. Sorry, Dick, we don’t know what we’re doing.”

“Actually, I think his point needs a serious response—”

“I am serious. I mean what I say. Methodical discovery is impossible. It’s as impossible as learning to talk.”

Feynman gave up his offensive, sat down and crossed his legs. For my benefit, he said, “Keep in mind that this is the guy who know some— how many is it? Twenty languages or so.”

“It’s impossible,” he raised both eyebrows, “to learn even one.”

“I seem to have managed it,” I said.

“How do you learn new words?” Gell-Mann challenged me.

“I look them up in the dictionary, I reckon.”

“And what do you find in the dictionary?”

“The definition?”

“Words! More words! How did you learn them?”

“People told me what they meant?”

“In words! Tell me, how far back does this chain of words go?”

“To my mother’s knee?”

“Where you learned simple words, ‘Mama, Dada,’ and the like, but did anyone ever tell you what they mean?”

“They were evident from context.”

“But that context had to be communicated somehow, didn’t it?”

“With gestures.”

“How did you learn what the gestures mean?”

“They were obvious. Look, I don’t know where you’re going with this.”

“‘Obvious’ doesn’t cut it, does it? Aren’t we scientists?”

“Is this an analogy?”

“Science is communicated in words— wouldn’t it be infected with the same uncertainty?”

“Well, there’s no way around that, is there?”

“How ’bout math?” Feynman chimed in.

Gell-Mann turned to him. “How did you learn math?”

“From books. But it’s not like I was just told that it works, and had to trust it. I fiddled around with it. After I read about something, I fiddled around to see how it works. Then I knew first-hand.”

“Are you two saying that’s what science is?” I asked.

“I think that’s where Maurry’s driving.”

“That’s EXACTLY where I’m driving. Look, as far as I’m concerned, learning to read Nature’s book is like learning any other language.”

“Except that it wasn’t written by people,” I protested.

“That’s irrelevant. Look, suppose you’re dropped in France and you know no French. You listen. If you listen for a while and you’re clever enough, patterns emerge. At least you think they’re patterns, so you try them out, you ‘fiddle around’ as Dick was saying. If you’re right, people understand what you’re saying and they sell you cheese. If you’re not, they shrug and you try again.”

“Sounds like the theory-experiment cycle to me,” said Feynman.

“It goes deeper than that. Look,” he erased a space on the blackboard, “what patterns do you hear when you’re learning, French, for instance? Conjugation, primarily. Je sais, tu sais, il sait, nous savons, vous savez, ils savent.” He drew them in a table on the board. “Singular on the left, plural on the right, first person, second person, third person. Every one of these must be filled, because any one of them might come up in conversation. Most verbs follow the same patterns.”

“There are irregular verbs.”

“Yes, Dick, and those are the interesting ones, aren’t they?” Feynman shrugged. “The pattern must be generated somehow. It just wouldn’t do to have ways of saying plural, ways of saying first person, second person, and third person, without, say, a plural third person, would it?”

“But you could,” I complained, “in principle.”

“Maybe. But probably not. And you start by assuming not. Only when some strange feature is forced upon you, like parity violation, do you assume the non-obvious. And so it is with science. What Dick and I were discussing before we were interrupted is the perfect example. Tell me what you think of this,” he pointed to his Kabbalistic diagram, “This is the Eightfold Way—”

“The Buddhist thing!” I interrupted.

“— indeed. It is nice how its eight group generators can be mapped onto Buddhist precepts, but given the variety of religions in the world, something could probably have been found for any small number. This diagram is generated by a simple but abstract mathematical object called a group, and the wonderful thing about this particular group is that its operations can be associated with particle properties, quantum numbers like strangeness, isospin, and the like— their meaning isn’t important, just the fact that they have labels that are semi-preserved— and each element, each spot on the diagram, can be associated with a known particle.”

“Except one.”

“Yes, Dick, except one. But I think that’s the beauty of it, don’t you? That particle hasn’t been discovered yet, and by predicting its properties from this model, we’ll pose a little challenge to see if we were right. It would lend credibility to the model.”

“We would get the cheese.”

“Hopefully.” Then he turned to me. “But do you see how this is like guessing at a language? These properties, strangeness, parity, spin, these are properties like singular versus plural. Knowing about pluralness tells us to build a table and be on the look-out for words to fill the empty slots. Then when we realize there’s a past tense, we have to expand the table, and we have a lot more slots to fill. But it’s always provisional, because at heart it’s a guess.”

“The experiments, too, have their share of uncertainty.”

“There’s that, too.”

I gazed at the diagram; each corner was labeled with a Greek letter. “Are those all of the particles?”

“It’s most of them,” he said, proudly.

“You have to remember,” added Feynman, “that this is no small trick. We’d been bewildered by all these different particles for a while. I keep saying there ought to be a Periodic Table of them, just like the elements in chemistry, and it looks like this could finally be it.”

“It’s not rectangular like the Periodic Table I remember from school.”

“No,” said Gell-Mann proudly, “it has an SU(3) structure.”

“All the suspects,” I said to myself, “in an SU(3) line-up…”

“The thing I’m looking forward to,” said Feynman, “is learning what it means. Figuring out how to line them up is just the first step.”

I nodded.

“Say,” Feynman pondered, “where would 0⁺⁺ go in here?”

“It wouldn’t,” Gell-Mann answered him flatly.

“It’s another mystery, then,” I sighed, taking back the card. “I’d best be going.”

As I put my hat on and opened the door, Gell-Mann smiled and said, “Let us know if you find that left-handed killer, gumshoe.” Feynman seemed a little puzzled, but I gave them both a thumbs-up and disappeared into the sunset.

I couldn’t help but feel that there was something not quite right.

That night

I was pacing in my hotel room all night. Something had been left unsettled, and I wouldn’t sleep until I knew what it was. Frustrated, I threw myself into an armchair and let my eyes rest for a moment. “That’s it!” I shouted. “Of course!” I threw on my coat. “Gell-Mann!”

I raced to the Caltech physics department in the moonless night. Only one light on under the door— he was still there! I burst through and pointed, “You!”

He was seated behind a typewriter with drafts of his Eightfold Way paper scattered about in heaps. He pulled off his glasses and said angrily, “Oh, what now?”

“You!” I pointed again. “You never explained how to calculate loop diagrams!”

“I never what?” He put his glasses back on.

“There’s a serious problem with the theory— loop diagrams depend on unknown physics— and you gave me a lecture on the methodology of science! You evaded the question! How do you use quantum field theory at all if the loop diagrams can’t be calculated?”

He thought hard to remember the conversation, then laughed out loud. With the low light casting horn-like shadows across his face from his glasses, it was kinda scary. “You want to be let in on a little secret?”

I nodded.

“Close the door.”

I closed the door.

“Technically, they can be calculated. It’s hard, but Dick and a few others found a way to do it. But it’s completely unsatisfactory.”

“How so?”

“We can’t use the theory as it ought to be used, calculating physical quantities such as the mass of a particle from first principles. As you know, gremlins run in the loops and change the answer.”

“Dragons.”

“Dragons, too. But things like the masses of particles are experimentally known. In dry tables of particle masses are clues to the Omnitheory, Naure’s fundamental rule book. Physics at all possible scales has been averaged over and is sitting in the mass of an electron.”

“Golly, I’ll never look at an electron the same way again.”

“I never do.” He licked his lips. “Now, with the answer sitting in front of us, Dick does the obvious thing— he takes that answer and runs the calculation backward.”

“What does he learn by that?”

“Nothing, at least nothing about the problem he cheated on. But it gives us a handle on certain theoretical quantities that can be plugged into other problems, and now we’re able to solve them because of the experimental input.”

“It doesn’t sound kosher.”

“Many of us think it isn’t. But don’t dispair— there’s another theory, something a few of us have been working on to go deeper than quantum field theory, something more fundamental than fields.”

“What is it?” I was whispering.

“You know what?” he said, a bit too briskly and loud. “That would take too long, and I have work to do. I really need to get this finished.”

“What? You can’t leave me hanging!”

“No, I really can’t be bothered. I can send you to someone who would only be too happy to tell you all the details.”

“Who?” I asked, clutching his desk.

“A brilliant mind named Lev Landau. Yes, he’s probably thought more about this than the rest of us put together. Yes, you should see Lev.”

“Where can I find him?”

“Siberia.”

Moscow, February, 1961

The train chugged slowly to a stop, hissing steam and casting shafts of light into the darkness. Snow and smoke whirrled together in the spotlights as metal ground against metal, then relaxed. I lowered myself onto the platform and into a black and white film— the grainy kind, where blacks are blacker and whites are whiter than they should be. I lit a cigarette: the only spot of color in the dark night.

A lone figure approached along the quai, lit from behind, arms folded behind his back. I tipped my hat, but he knew who I was already. As he came closer, I saw that his hair didn’t lie about his head, but shot up in the middle like a saluting guard. Somewhere in the distance, dogs barked.

“Are you Dr. Landau?”

I could barely tell he nodded.

“Good to meet you,” I held out my hand. “I’m a private eye.”

He didn’t take it. After a steely silence, he boomed, “Aren’t you aSHAMED of yourself?”

I was taken aback; I didn’t know how to answer.

“Shame on you! They tell me you actually beLIEVE in quantum field theory.”

“Can’t say that I’ve committed to the question either way,” I fumbled.

“Very well. Follow me.” He turned around. I followed. I never even saw his face.

To be continued!

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.”

“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.

—an Insider”

Somewhere in South America, 1950

In the months that followed, I learned nothing more about the note and my mysterious informer, and soon it was time to pack my bags for Brazil.  In the meantime, Dyson and Professor Feynman ventured on a road-trip together across-country, and when they came back, Dyson was drawing the little diagrams, too.  I could sense we were on the verge of an outbreak, so I kept my distance.

Appearing as a student in Rio de Janeiro would strain credibility, so I took an apartment on the other end of town.  It was in one of those old lacey buildings, with a balcony made of metal flowers overlooking the street.  There always seemed to be parades going on, with lots of dancing, and often enough the distinguished professor could be seen revelling with the rest of them, banging away on his bongo drums.

Down in the crowd, I caught sight of a black derby hat, swimming toward my door.  When I noticed the man, wearing a black trenchcoat and derby, stuffing an envelope through my letterbox, I realized it was my old friend.  “Hey!” I shouted.  It looked up.  The phantom had no face, as far as I could see, just a grin in the corner of its mouth.  Into the hallway and down the stairs I raced, hoping to catch him up in the crowd.  On my way into the street, I scooped up the envelope, just in time to see a black coat-tail vanish around the block.  I chased him into the alleyway, which led out into another street, amid sounds of livestock and festival-goers.  Always he was just ahead of me, disappearing just around the corner.  I came at last to a dead-end, no way out, and he was gone.

I sat against the wall, catching my breath and fanning my face with my hat.  A gaggle of teenagers hung around— I had interrupted them, and they thought I was the funniest thing they had ever seen.  I winked.  Then I remembered the note.  It read,

“What is the mesotron?

—the Informant”

The mesotron, I pondered.  Sounds like some kind of robot.  “Hey, you kids ever hear of a mesotron?”

There was a lot more giggling and chattering in Portuguese, and every now and then I heard emphatic references to “mésons” and “Yukawa.”  Before long they had circled me, and gestured that I follow them.

They took me to a loading dock where a young Brazilian was piling shipments in wooden boxes onto his covered pick-up truck.  The boxes were stamped, “Ilford, Essex, U.K.”  My new friends greeted him and told him all about me in the most rapid outpouring of syllables.  He turned to me and asked, “You want to see a mesotron?”

I replied that I did.

“Well, come on!  Hop in back!”  He patted the bumper of his truck.  A few of the kids were already climbing on board, so I did too.  Among the Ilford boxes, I made a seat just as our driver swung shut the curtains at the end of the covered canopy.  Soon we were off.

Through the boards lining the sides of the bed, I could see that we were driving out of town.  After a few hours, I decided it was going to be a long trek, so I covered my face with my hat and fell asleep.  I woke when we drove over a large bump, and it was already dark.  A few hours later, we pulled over to the side of the road and camped out.  I couldn’t see where we were, but it seemed to be no more than rank jungle, anyway.  We ate the hottest soup of my life, and my new friends found my reaction to that hilarious as well.

The next day was the same.  A long, unbroken stretch of bumpy driving through the jungle, camping by the side of the road at night.  I asked, “Where are we going?”

“Chacaltaya,” César, our driver, informed me.

“Where’s that?”

“Bolivia.”

More roads, more jungle, and now I could sense a general upward trend, as we must have been approaching the Andes.  The going got harder; we made more frequent stops to let the engine cool off.  We seemed to be going up and down for ages…

I woke at night, with one of the youths shaking me from sleep.  The truck was stopped.  “Where are we?” I asked.

“Halfway to outer space!” he answered in an excited whisper.

The curtain opened, letting in a brilliant light.  I covered my eyes.  When they adjusted, I saw that there were only stars, blinding stars, stars so bright I still had to squint.  White-hot singular stars, peppered by freckles of half-light, flowing in milky clouds of haze.  I couldn’t see the ground.

“Outer space?” I asked, then wheezed.  The air wasn’t good.

They took me by the elbows, helping me stand up and stumble toward the end of the truck.  Over the tailgate, I could see soil and rock, barren and strange, everything in long, flat slabs.  They helped me down: I hopped off the edge and fell in slow-motion, gradually tumbling to my knees, watching little puffs of otherworldly dust float up around me.  Despite the slowness of everything, my knees and right ankle smarted something fierce.  They helped me to my feet as I struggled to breathe.

They hoisted me up by both shoulders, and led me along the sloped plain.  It was cold— colder than Ithaca, but a dry cold.  In the night, I could see the lights of a house, and that’s where they brought me.  Inside was pretty barren: a sofa, some chairs, chalkboard, and some kind of projector.  I could smell chemicals.  One guy got up off the end of the sofa, everybody moved over to fill his spot, and another sat down on the other end.  They were all wearing knit caps that covered their ears, like aviator’s hats.  They sat me down in a desk-chair.

There was a lot of going to and fro, stacking the Ilford boxes in a corner, as I breathed deeply and came to a greater understanding of just how much my knees and ankle were in pain.  There was César, our driver, but the students called him Professor Lattes.  If I thought Dick Feynman was young to be a professor, this guy beat him by at least a decade.  When all was settled, he seemed to notice me, and pulled up a chair.  Sitting backwards, he faced me and said, “You wanted to see the meson?”

“Mesotron,” I corrected, hoping this wasn’t all a case of mistaken identity.

“Meson, mesotron, mesoton, whatever people call it.  It’s the Yukawa force you want to see, isn’t it?”

“Yukawa?”

“The force of the nucleus!  The energy of the atomic bomb!  We’ve found it— coming from outer space.”

“Are we in space?” I asked weakly.

“We are on top of Chacaltaya, about as close as anyone can get to space.  It’s to see the cosmic rays.  Here, there are many, many more of the cosmic rays than at sea level.  Let me show you.”

He stood up and unlatched something in the projector in the middle of the room.  Then he retrieved a thick stack of something from a black paper bag.  “This,” he said, “is a photographic emulsion.  It’s like the film for a camera, but it is many, many layers thick.”  He hefted it onto the projector tray, then slid it inside.  “We don’t expose it to light.  We put it outside, for many months and let the cosmic rays go through it.”

“Um, excuse me,” I raised a finger, “cosmic rays?”

“They are radiation falling from space.”  Something long dormant perked up in my brain.  There it was again, radiation— the cause of Marie Curie’s death!  That something told the rest of my brain it had better pay attention.  “Radiation is falling on us all the time, coming from somewhere in outer space.  Nobody knows where exactly or how.  But the cosmic ray particles have more energy than any radioactive elements or cyclotrons on earth.”

“More energy than radium?”

“Far, far more energy than radium.”

“Then why aren’t we killed by it?”

“Because they are low intensity.  Ah—” he smiled, “you are wondering how they can be low intensity and high energy.  I mean that there are not so many individual particles, but the ones that do are travelling very, very fast.”

“So what is this Yukawa force?”

“That is the nuclear force.  Do you know anything about the forces?”

It jogged my memory of a chalkboard conversation long ago.  “Four forces,” I said, “gravity, electromagnetism, the nuclear force, and…” and what?

“There are two nuclear forces: the weak force and the strong force.”

“That’s right.”

“To be honest, we say that there are four forces, but really there are just four categories of force.  We see all kinds of nuclear transformations, and notice that one group of them happens rapidly, while the other happens slowly.  Fermi described the weak force, the one that happens slowly, and it has long been clear that the strong force holds the nuclei in atoms together.  It’s only when something overcomes the strong force that nuclei break apart, and that releases much energy because the force was very strong.”

“Like potential energy— springs,” I remembered Professor Feynman’s class.

“Yes, very much like a spring.  A very, very stiff spring.  This is the energy of the atomic bomb.  Well, the curious thing about the nuclear forces is that they act over a very short range.  Only when protons and neutrons are very close together do they feel any force at all.”

“When electromagnetic charges are far apart, they feel less force, too,” I said, proud of myself.

“But much, much more so.  The force between electromagnetic charges is inversely proportional to the square of the distance between them, but with nuclear forces, it’s exponential.”

“Oh.”  I had learned about exponentials.  Every step decreased the force by a factor of 10, like the Richter scale for earthquakes.  That’s a lot more dramatic than an inverse square law.

“How come they are different?  Well, you know that in the quantum theory, the force between electromagnetic charges happens because a photon goes between the two charges, carrying momentum away from one and giving it to the other.  That way, they both feel the force, and both change their motion as a result.”

“I didn’t know that, but go on.”

“This can happen no matter how far away the charges are, and in fact the intensity is inversely proportional to the square of the distance between them, just like the light of a light-bulb.  If you shine a light-bulb on a wall, the intensity of the light on the wall is inversely proportional to the square of the distance from the bulb, because as the light travels out, it expands as the surface of a sphere around the bulb.  The area of a sphere is , so the intensity decreases as , with being the distance to the bulb.

Suppose that nuclear force works like electromagnetism.  We don’t know that it does, but it’s a good assumption to start with.  In between the nuclear particles, protons and neutrons, there must be some particle like the photon, but for nuclear forces.  A nuclear photon!  Dr. Hideki Yukawa was thinking about that, and he knew that in the quantum theory, particles are never quite stationary, but are always buzzing around within a wavefunction, a region of space where their position might be.”

“They don’t teach this in school, do they doc?”

“Well,” he tipped his head back and forth, “not at the beginning level.  The positions of particles is never quite definite, especially when they’re travelling slowly.  The wavefunction is a little cloud of space describing how likely the particle will be at each point, and this blob is usually exponential, with an exponent proportional to the particle’s mass.”

“So a particle with a large mass,” I gestured with my hands, getting it wrong the first time and correcting myself, “is usually found in a very tight region of space, and a particle with a very small mass could be very spread out.”

“Exactly.  And a particle with zero mass—”

“— can be anywhere!” I interrupted.

“It has to travel at the speed of light, like the photon.  The photon is massless.  It gets anywhere it has to go, and so the electromagnetic force can be felt between charged particles that are light-years apart; it’s just the intensity goes down by .”

“But not so for the nuclear photon, right?”

“Right.  Or that is what Yukawa was thinking.  Yukawa said, what if there is a nuclear photon, and its mass is such-and-such: it would cause attraction between the proton and the neutron, but only when the proton and the neutron are so close to each other as to be within the wavefunction of a stationary nuclear photon.  As soon as a proton or a neutron goes out of the wavefunction, zip—” his hand flew up into the air, “no more nuclear force, and all the energy that held it in is released.”

“Like in an atomic bomb?”

“Yes.”  He nodded gravely.  “In the atomic nucleus, all the protons are electrically charged and nuclearly charged, with the electromagnetic force pushing out and the nuclear force pulling in.  All we have to do is push it over the edge… potential energy released, becomes kinetic, light, and heat… and a very big bomb.”

We sat in silence for a moment.

“Well!”  He slapped his knees.  “Let me show you the meson!”

“You know, professor, you never did tell me what is this meson, or mesotron, or whatever.”

“It is the nuclear photon!” he seemed a little aghast.

“Oh, right.”

Back at the projector, he clamped a few levers into place, and turned on the bulb.  A fan churned and a dull light ignited, shining on the wall.  He tuned the focus, and a spattering of dark spots appeared.  It was like a shadow-play.  “What you are seeing is the developed film, and these are the cosmic rays.”

“I thought you said you never exposed the film.”

“We didn’t—” a little smile came to his face, “the cosmic rays did.  Radiation exposes film, in just the same way that light exposes film, but radiation can pass through dark paper, especially these cosmic rays which are so very, very energetic.”  He tuned a dial, and the little spots moved.  “Now I am changing the focus, so that you see different layers.  When the cosmic rays went through the film stack, they exposed little lines, leaving little trails, and those trails look like spots if you focus on a single depth with a microscope like this one.  But now watch this!”  He was getting giddy— the high altitude air must have been getting to him.  “Watch this one spot: see, it is moving along with a certain speed, and then there!  It changes direction entirely!”

“Did it hit something?”

“No, too much energy to change that dramatically from a collision.  Look at it again.  See?  It just turns.  Immediately.  At one spot.”

He seemed to be waiting for a suggestion, but nothing was coming to me.

“It has decayed!”  He shouted.  “It decays into another particle; in this case, an electron.  The top part of the path is meson, and the bottom part is electron.  Now— question for you is, where did the momentum go?”

The momentum— that word recalled table-top demos and homework problems with cars crashing into each other.  The meson’s momentum before decay was clearly different from the electron’s momentum after decay, simply by the fact that they went in different directions.  “I don’t see where it went,” I had to confess.

“Exactly!  You don’t see it at all!”  He was on the verge of a cackling laughter, and I wondered for my safety.  “The rest of the momentum went with the neutrino, and the neutrino is invisible.”

“No such thing!”  One of his fellow researchers stood up, pointing an accusing finger.  “You can’t just invent ghost particles when you don’t see the momentum.”

I clearly remembered Fermi talking about neutrinos with no suggestion that they might not be real.

“Then how else do you explain the loss of momentum?  Look!”  César Lattes tuned the dial again.

“I’ll believe it when I see it, but I haven’t seen it.”  He sat down again on the couch, arms folded.

“A true experimentalist,” César chuckled, “that’s good.  That’s good.  Well, someday, we will see it.  Right now, no.”

“So this is the meson that causes neutrons and protons to stay together in the nucleus?” I asked, trying to get back to the subject.

“No, it is not.”

“What?!” I felt I had been lied to.

“By measuring the energy of the electron, we know that this meson’s mass is consistent with what Yukawa needs for the nuclear photon, but this particle has no strong nuclear interactions, as it would need.”

I had been lied to.

“That’s why, what I really want to show you is this—” he tuned the focus quickly, past lots of spots, some of them with kinks, until he found what he was looking for.  “This one here.”  It was a spot like all the rest.  “This meson decays, not unusual, but then watch the decay product.”

A second kink.  “It decays again?”

“It decays again.”  He spoke as though his words were full of import.

“Why is that important?”

“Electrons don’t decay.”  He seemed puzzled that I didn’t get it.  “If after the second kink, the particle is an electron, what was it before the first kink?”

It slowly dawned on me.  “There are two mesons?”

“Two mesons, yes!”  He raised his eyebrows.  “A new meson decays into the standard meson, and then that standard meson decays into an electron, throwing off neutrinos each time.  We’ve labeled the top one , and the bottom one , Greek letters, just to keep them straight.  And then we see this over and over again, but only at very high altitudes.  So I went to Berkeley where they have a cyclotron.  Even with the cyclotron, we were able to see this two-decay pattern, and isolated the pi mesons and studied its properties.  It indeed feels nuclear forces— we have found the nuclear photon!”

“It came from outer space,” I mused.  “Imagine that.”

The next morning, César took me up a little hill to see an emulsion stack, a wooden box holding a few feet of film, all wrapped in black.  I couldn’t get over my dizziness— the view was spectacular!

As I wasn’t getting any better by the mid-afternoon, César asked one of his students to take me down the mountain, to which he responded with a bit of annoyance.  If he was supposed to take me further than the nearest village, I’ll never know, but that’s how far he took me.  It was then up to me to get myself back to Brazil.  I limped eastward.

Rio de Janeiro, 1956

When I finally made it back to my apartment in Rio, I found myself face-to-face with an irate landlord.  Promising 6 years of back-rent, I went back to work on the case.  With this new pi meson, I decided to round up a list of all the suspects.  With a little help of the university library, I compiled the following:

• electron: orbits atom, flows through wires as electricity, electrically charged (–1), unaffected by nuclear force
• proton: constituent of atomic nucleus, electrically charged (+1), affected by nuclear force
• neutron: constituent of nucleus, uncharged electrically, but affected by nuclear force
• alpha particle: alias for a Helium nucleus, consists of two protons and two neutrons
• beta particle: alias for electron
• X-ray, gamma particle, photon: light of various energies, called by different names.  Communicates electrical forces between charged particles but uncharged itself.  Doesn’t like to get involved.
• neutrino: electrically neutral, unaffected by nuclear strong force, only nuclear weak force.  Proposed to explain momentum imbalance.  Hypothetical.
• mu meson: all properties identical to electron except that it is 200 times heavier and decays to electron and neutrino.  Weird.
• pi meson: communicates nuclear forces, like the photon for electromagnetism.  Comes in three forms: electrically charged (+1 and –1) and neutral, unlike the photon.

After this point, it started to get a bit more complicated.  After a few conversations in the physics department, and I had to add:

• positron: “antimatter” twin of electron, all properties the same but reversed charge (+1).  Required by equations linking quantum theory with Einstein’s relativity.  Discovered in 1932, but rare.
• antiproton: antimatter twin of proton, same deal: negatively charged (–1).  Discovered last year.
• V-particle, or K meson: comes in three forms: electrically charged and neutral; decays into pi mesons, mu mesons, electrons, and a smattering of neutrinos.  The chain just gets longer and longer!
• Lambda baryon: a heavy “V-particle” that decays into neucleons (protons or neutrons) and mesons, neutral
• Xi baryon: decays into lambdas and pi mesons! leading to a whole chain of decays on both sides.  Negatively charged (–1).
• Sigma baryon: charged forms (–1 and +1) also decay into nucleons and mesons, neutral form decays into lambdas and mesons.

More and more suspects, with no end in sight!  It seems if you just look hard enough, you’ll find something new that decays twenty times and ends up a puddle of electrons, photons, protons, and neutrinos.  After a little talking to, I could understand the antimatter versions of everything: for each +1 charged particle, there must be a –1 particle, and I could understand why we need the pi meson to glue nucleons together, but all those extra mesons and baryons smell distinctly of red herring.

It was at this time that I also realized Dr. Feynman was gone, back to Cornell.  So I’m headed north.

Ithaca, NY, 1956

No sign of him here, either.  Disappear for a few years, and everything changes.  When I left, the basement of Rockefeller Hall was being emptied out for one of those cyclotron-accelerators, and now it had already been dismantled and pieces of it were part of a new synchrotron in a new building.

“Is it going to discover more particles?” I asked, a little warily.

“Could be,” said the graduate student, who had abruptly introduced himself as Karl Berkelman.  “But it would be better if it explained all of those new particles.”

“And the Mystery of the Missing Neutrino,” I added.

“Missing Neutrino?  The neutrino’s been discovered.  In Georgia, in a bubble chamber next to a nuclear reactor.”

“Really?  I thought it was too weakly interacting.”

“Not if you have enough of them and wait long enough,” he answered ardently.  “Sitting next to a nuclear reactor is like having a sun next door.  Then all you have to do is wait for beta decay to happen in reverse: neutrino plus proton goes to neutron plus positron.  The neutron gets captured and the positron finds an electron to annihilate with, so it looks like these two rare events happening at the same time.”

“I bet Fermi was pleased.”

“Fermi’s dead.  Stomach cancer— it got two of his students, too.  Probably all of that radiation from the reactor they built.”

I was about to ask if he wasn’t a little worried himself, when he introduced me to the director of the laboratory, Bob Wilson.  Bob was a genial fella, somehow part-cowboy and part-nerd without contradiction.

“They let me carry a gun at Los Alamos,” he said with some glee.

“What?” I asked.

“At Los Alamos, I packed a six-shooter.”

I was a little mystified.  “Why are you telling me this?”

“Like you said, the nerd-cowboy thing.  Born in Wyoming, I learned to shoot before I could stand up straight.  So when there was a sample of enriched Uranium that needed protection, I asked Oppie if I could be registered to carry a gun.”

I need to be careful what I narrate aloud.

“Anyway,” he continued, “that’s the right metaphor for what we’re doing.  This is about as near to the wild frontier as it gets!  The subatomic world: it’s not just a nucleus and some mesons holding it together, it’s a whole landscape of weird new particles and even weirder interactions.  Did you know that empty space is just teeming with electron-positron pairs, creating themselves out of nothing and annihilating spontaneously?”

“I assure you, I had no idea.”

“You can scatter high-energy photons off of them.  We did it just a few years ago with the old machine.  Ah, here it is,” he opened the door to a basement filled with a giant mechanical doughnut.  “Ain’t she a beauty?”

“This is the new sychrotron?”

“Yup.  Come and take a look!”  He showed me all the ins and outs of the machine: how it accelerated electrons pretty close to the speed of light, how the gradient was alternated for strong focusing of the beam, the beamlines that led to different experiments; everything seemed to be in a perpetual state of dismantling and reconstruction.

“Does it work?” I asked, perhaps naïvely.

“What’s that?  Oh, not right now, but we’ll get it to work again.”

“You mean you got all these people together, all this electronics built, and it doesn’t work?”

He waved me down.  “Don’t worry about it.  Something that works right away is over-designed, and will have taken too long to build and cost too much.  When you see all of these precision instruments, you shouldn’t think that if we put it all together and it doesn’t work, we’ll go home sad.  No, when it doesn’t work, it doesn’t work because of a few fixable problems, and there’s loads of diagnostics to figure out which those are.  Besides, flying loose like this means that we have the flexibility to try new ideas first, before the big labs finish their paperwork.  See this strong focusing I told you about?  There’s a story about that.

“So about 10 years ago, a bunch of universities got together and founded an accelerator complex on an old army base near Brookhaven, NY with Manhattan Project money.  There they built the world’s largest synchrotron, called the Cosmotron—”

“Cosmotron?”

“Yeah!  Cosmotron!  It’s a great name!  It delivers energies on par with cosmic rays, yet it’s a synchrotron, so why not?  Well, the Europeans got jealous and decided they wanted a big accelerator lab, too, and after a lot of negotiations, sent some scientists to Brookhaven to learn how it’s done.  Seeing that the magnets could only focus horizontally, but not vertically, one of them asks, ‘Why not alternate horizontal and vertical focusing?’  Well, they had never thought of that.  With more focusing, you need less magnet, which is easily the most expensive part, so in the long run you can get more energy for the buck.  Based on this idea, Brookhaven is building an even bigger synchrotron, and the Europeans are building their own, but neither of those are finished yet.  We heard about it while we were still building this guy here, and just changed the design mid-stream to take advantage of it.  And it’s payed off big-time.  A number of other accelerator projects around the world weren’t quite as flexible, and they’re still building living dinosaurs.  That’s the trick: keep it flexible and keep it cheap, and reap the rewards.”

“How cheap was that wrench you melted?”  One of the other physicists challenged, eyes gleaming.

“My old boss was even more spend-thrift than I am,” Bob explained.  “He had the unfavorable experience of inventing cyclotrons in the Depression.  He fired me once for melting a wrench.”

“A whole wrench!” the other physicist emphasized.  “Melted into a pool of metal!”

“That was only the second time he fired me.”

“So I take it you’re not a fan of big laboratories,” I asked.

“Oh, no, they’re fine.  It’s the only way you can make the really big machines and reach the really high energies.  Some things you just can’t find out without going up there and looking.  But the sad thing about them is that they’re so far from the classroom: they’re breeding a generation of researchers who’ve never taught.  Ideally, a physicist ought to be able to teach a roomful of students at 2 o’clock, calibrate an instrument at 3, then be back in his office for homework questions at 4.  I think we’re starting to lose that connection between teaching physics and doing it.”

I thanked him for showing me around, but I had to catch up with Feynman.  “Oh, he’s at Caltech now.  Likes the warmth.”  Caltech.  Another cross-country trip.  So be it.

Before packing my bags, I took a leisurely stroll around Bebe lake.  It was early autumn, and I was beginning to realize how nice the north-east could be from time to time.  I was standing on the top of an arched stone bridge when I heard a rustling in the woods.  I could have sworn I saw something dark and sinister between the trees, up the steep hill, but there was nothing now but swinging leaves.  Leaves, and a little envelope.  Another envelope!  I snatched it up and, looking over my shoulder twice, read it.

“Seek the Dragon Lady in Columbia.

—an Old Friend”

“I’ve had enough of this!”  I shouted at the undergrowth.  “Dragon Lady?”  I thumbed the card, then pulled the two old cards from my pocket, rough and weathered next to this one.  “There is no way I’m going back to South America!”  I shouted again.

I waited, but no answer was forthcoming.  Clearly, I was going to need some help with this, so I took all three notes back to the nuclear studies lab.  Sure enough, Bob wasn’t there; he was in his office to answer homework problems.  I brought him mine.

“Dragon Lady??” he asked, obviously highly amused.  “No, I don’t know any Dragon Lady, and I’ve never even been to Columbia.  What are these other ones?  You say it’s a guy in a trenchcoat?”

“He followed me to Brazil.”

“Hard to imagine anybody with that much time on their hands.”  I ignored the inferable insult.  “What’s this?  The ‘mesotron’?  God, I haven’t head anybody use that word in ages.”

“I found out about that one right away— and it was a good lead.  I was just in time to catch a pickup to Chacaltaya, where I saw one of the first photos of a pi meson.”

“Wow, that sounds like fun.”

I put the long walk home out of my mind.  “Why’s it called a mesotron, or meson, anyway?”

“It’s Greek.  Greek for ‘medium.’  Electrons and neutrinos, they’re leptons, ‘leptos’ for ‘light’, all below an MeV or so in mass.  Mesons are medium at a few hundred MeV, and baryons are heavy— ‘barys’ is like thick or stout.  Baryons are usually a thousand MeV or more.  That’s the advantage of a big accelerator: you can’t get heavy particles without putting in enough energy to create their mass.  A consequence of relativity, this conversion of energy into mass.”

“So these particles, they’re named by weight?”

“Sure, that’s almost all we know about them.  Like I told you about this being a frontier— we don’t know much of anything yet.  We’ve just dumped the puzzle on the table and all we can do so far is group the edge pieces with the edge pieces, the middle pieces with the middle pieces, and maybe decide where the corner pieces go.  There’s something special that makes baryons a thousand times heavier than leptons, and considering that none of the leptons feel the strong nuclear force, while all of the baryons and some of the mesons do, that’s gotta be a clue!”

“Like the four forces— they’re just groups,” I added.

“Yeah, like the four forces.  We think that all the weak decays are related by some Universal Fermi Mechanism, and we think that all the strong forces are due to Yukawa’s meson, but nobody’s really solved that.  Nobody’s written down an equation that works for all of them.”

He flipped to the next card, and lingered.  Something must have struck a chord.  “The killer is left-handed,” he muttered.

“Do you know who it is?”

“No, no, it’s just—” back to the new card— “the Dragon Lady…”

“Is she left-handed?”

“I have a hunch,” he said finally, handing me back the cards.  “Go to Columbia University in New York City, the physics department, and ask for Madame Wu.  Ask her about left-handedness.  But don’t call her ‘Dragon Lady.’”

“Do you think this will solve the mystery?”

“I’m sure it’s a clue,” he smiled confidently.  “The game is surely afoot.”

July, 1934

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.

“Radium?”

“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?”

“Both.”

“A conspiracy?”

I could see the news was shocking to her— she didn’t know what to say.  A doctor entered the room to pick up a chart.  I followed him; maybe he’d have some answers.  “The disease was an aplastic pernicious anaemia of rapid, feverish development,” he said, and I scribbled in my notebook as fast as I could to get it all down.  “The bone marrow didn’t react, probably because it had been injured by a long accumulation of radiations.”

“Radiations, eh?”

“That’s her killer.”  My ears perked.  He went on to tell me that the most distinguished doctors in France had attended her case, to no avail.  Sometimes they said it was grippe and sometimes bronchitis; no one had ever seen the like.  But one thing was sure: those radium and polonium figures were behind it all.  The doctor gave me a lead, one Professor Regaud, a colleague of the late Madame.

“Marie Curie can be counted among the eventual victims of the radioactive bodies which she and her husband discovered.”

“She knew her killer?”

“Better than anyone in the world, and yet not well enough, I’m afraid.”

“Tell me more about this discovery of hers— or theirs?  Where is her husband?”

“Oh, dear, you don’t know?  He died more than twenty years ago.  Run over by a car.”

“Sounds unrelated.  What about this radium?”

“Yes, it was the first element discovered by radiation.  It makes the air conductive to electricity— Pierre and his brother Jacques invented an instrument sensitive to its subtle rays, an electrometer.  Well, the electrometer actually measures the small electric currents that are then induced through the conductive air, but the energy creating those currents came from radiation.  Marie used the electrometer to identify all of the radioactive elements, and then to discover a radioactive source more powerful than any known element, buried in pitchblende.  Since it could not have been one of the known elements, she postulated that a new element was at work: radium!”

Scribbling all of this down, I double-underlined pitchblende.  “What’s pitchblende?”

“Oh, nasty stuff.  It’s black and sooty— mostly uranium oxide, but the Curies used a depleted ore in their studies.  You see, she wasn’t interested in the uranium, that was known; it was the idea that there are new elements, producing the same sort of rays but far more active, that excited her.  When the carriage arrived from Bohemia with the stuff, she dug her hands into it— she just couldn’t wait to extract its secrets!”

“Her hands!  The marking on her hands!” I exclaimed.  “It’s because she went digging around in radioactive soot!”

“I should say it was not merely that incident,” the Professor continued, somewhat annoyed, “it could be attributed to a lifetime of radiochemical research, handling highly concentrated radioelements with only a glass vial or small metal box to protect her hands, stirring literally tonnes of bubbling pitchblende in a cauldron, draining off the inert metals, in a stinking shed with no proper ventilation: that’s your culprit, gumshoe!  Dedicated service to Humanity!”

So early in the case, and already solved: radioactive mud and service to humanity— it seemed so simple.  I caught up with Eve at her home, where she was writing her mother’s biography.  Papers were stacked on a concert piano, and old scientific notebooks were open and strewn about the house.

“Look at this—” she said, holding up what must have been an electrometer, “they’re still active.  Even the notebooks are radioactive after all of these years.  Their ‘living activity’ has outlived their authors.”

“That’s the key to the mystery,” I informed her, “radioactivity killed your mother.  That, and humanity.”

She seemed underwhelmed.  “Well you’re the last to be surprised,” she said finally.  “It’s hardly mysterious that Mé’s intimacy with her work was killing her.  Her blood tests were always abnormal.  She made her pupils handle tubes with pincers and protect themselves with lead bucklers, but it was too late for her.  Here,” she said, pulling out a transcript from an old journal, “this is Papa’s description of the lesion he raised on his arm.  As soon as he heard that the radium rays had physiological effects, he exposed his arm, indifferent to the danger, and overjoyed to see the bruise develop.

I scanned the scientific-looking description, and could detect the glee hidden behind the report: “After the action of the rays,” it read, “the skin became red over a surface of six square centimeters; the appearance was that of a burn, but the skin was not painful, or barely so.  At the end of several days the redness, without growing larger, began to increase in intensity; on the twentieth day it formed scabs, and then a wound which was dressed with bandages…”  It went on and on, including similar lesions unintentionally inflicted upon Marie’s hands.

“But for all of that, we still don’t know what it is,” she mused.

“Sure we do— it’s radiation.  It comes from chemicals and kills people.  Mysterious rays…”

She snorted.  “Is that the depth of your investigations, detective?  Don’t you have the least curiosity of what really killed her?  Whence cometh the radio-atom? Listen to this: this was Mé’s favorite story about those days.  She and Papa had just spent four years extracting the millionth part from pitchblende, and only a tenth of a gram of radium salts to show for it.  At home, after putting the baby to bed (that’s my older sister, Irène), Marie sat down and made some stitches on the hem of Irène’s new apron.”  (She was at this point reading from her manuscript.)  “One of her principles was never to buy ready-made clothes for the child: she thought them too fancy and impractical.  In the days when Bronya was in Paris, the two sisters cut out their children’s dresses together, according to patterns of their own invention.  These patterns still served for Marie.

“But this evening, she could not fix her attention.  Nervous, she got up; then, suddenly: ‘Suppose we go down there for a moment?’ There was a note of supplication in her voice— altogether superfluous, for Pierre, like herself, longed to go back to the shed they had left two hours before.  Radium, fanciful as a living creature, endearing as a love, called them back to its dwelling, to the wretched laboratory.

“The day’s work had been hard, and it would have been more reasonable for the couple to rest.  But Pierre and Marie were not always reasonable.  As soon as they had put on their coats and told Dr. Curie of their flight (they lived with Pierre’s father), they were in the street.  They went on foot, arm in arm, exchanging few words.  After the crowded streets of this queer district, with its factory buildings, wastelands and poor tenements, they arrived in the Rue Lhomond and crossed the little courtyard.  Pierre put the key in the lock.  The door squeaked, as it had squeaked thousands of times, and admitted them to their realm, to their dream.

“‘Don’t light the lamps!’ Marie said in the darkness.  Then she added with a little laugh: ‘Do you remember the day when you said to me, “I should like radium to have a beautiful color”?’

“The reality was more entrancing than the simple wish of long ago.  Radium had something better than ‘a beautiful color’: it was spontaneously luminous.  And in the somber shed where, in the absence of cupboards, the precious particles in their tiny glass receivers were placed on tables or on shelves nailed to the wall, their phosphorescent bluish outlines gleamed, suspended in the night.

“‘Look… Look!’ the young woman murmured.

“She went forward cautiously, looked for and found a straw-bottomed chair.  She sat down in the darkness and silence.  Their two faces turned toward the pale glimmering, the mysterious sources of radiation, toward radium— their radium.  Her body leaning forward, her head eager, Marie took up again the attitude which had been hers an hour earlier at the bedside of her sleeping child.

“Her companion’s hand lightly touched her hair.

“She was to remember forever this evening of glowworms, this magic.”

I let her revel in the magic for a moment before asking the hard questions.  “Where can I learn more about this radioactivity?”  The killer was evidently more intimate with the Curies— both of them— than I had thought.  I hate messy cases.

“Well now that the world’s foremost expert is no longer with us,” Eve began slowly, “I suppose you could ask an Italian by the name of Fermi, Enrico Fermi.  I hear from Irène that he has just proposed an exciting new explanation for radioactive beta decay— it has all our colleagues buzzing.”

“Italian, eh?”  I could only think of one thing: mob connections.  This case was getting messier by the minute.  Still, there’s a killer out there, and I swear I’m going to get to the bottom of it, no matter how long it takes.

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 .

Second Way. Cut the deck into two halves just like before (according to a binomial distribution). Suppose that cards are in the left hand and cards in the right hand. We decide to drop a card from the the left hand with probability and from the right hand with probability If a card from the left hand drops, then we do the same thing: drop a card from the left hand now with probability and from the right hand with probability In other words, the probability at each step that we drop from the left hand or the right hand will be proportional to the number of cards in the left hand or right hand. Continue this process until all the cards have been dropped.

Third Way. This is the inverse shuffle. For each card in the deck, flip a coin and label the back of the card H or T depending on whether the coin landed heads or tails. Take all the cards labeled H out of the deck, maintaining them in relative order to one another and put them on top. This is another way to think of the riffle shuffle (even though it seems strange).

Each of these ways to view the riffle shuffle is actually the same in the sense that if we start at a particular order, the probability of getting the deck into any particular new ordering is the same in all three shuffles. To see that #1 and #2 are the same, observe that one flips coins in the same way. Assuming that heads come up, exactly cards end up in the left hand and cards in the right hand. In #1, one ends up with a sequence of 52 L’s and R’s: the first is an L if the top card after the shuffle came from the left hand and R if it came from the right. Likewise with the second entry, and so on. Each of these orderings is equally likely. In #2, one ends up with a sequence of L’s and R’s depending on the order in which the cards dropped. The probability of any particular ordering is always , which is the same as the probability we found in #1. To see that #1 and #3 are the same, notice that we end up with a sequence of L’s and R’s and H’s and T’s. For the moment, assume that L=H and R=T. Then we end up with the same possible subset of sequences. The probability of each sequence is the same since in each case each choice has the same probability and there are the same number of sequences possible.

This model of the riffle shuffle is rather good for amateur shufflers. Professionals (casino dealers, magicians, and so forth) are not modeled quite as well by this technique since they tend to have less variation in the size of each half of the deck and also tend to be able to have one card drop from one hand and then one card drop from the other. But for the casual card player, this is just about enough.

It is worth noting at this point, that if one were to view shuffles as a Markov chain, the matrix produced by the riffle shuffle is, first, gigantic (), and, second, hard to write down regardless.  So, it seems like estimating the decay from the point-of-view of analyzing the behavior of the second-largest eigenvalue would be difficult in this case.  Let’s try a different means of attacking the problem.

To derive the an upper bound on the number of times one has to shuffle to start decaying exponentially to the uniform distribution we will make use of a technical lemma. But first, one needs a couple of definitions, the first of which being one that probably any mathematician should know. I’m writing the definition with respect to this problem, but the basic idea is the same, for the most part, even in more general situations.

Definition. Stopping Time
A stopping time is an integer-valued function defined on the sequence space of shuffles. One gives the stopping time a sequence of shuffles as input and the stopping time looks for the first shuffle in the sequence for which a particular condition is satisfied. If the shuffle is where this occurs, then the stopping time outputs . The key technical property of is that it only requires finite information. In particular, if for a sequence , , then for any sequence which agrees with in the first entries. As an example, we might say, “Stop when the deck has an ace on top.” Then we keep shuffling the deck, keep track of how many times we’ve shuffled, and then write down how many times we had to wait until an ace was on top.

Definition. Strong Uniform Time
Suppose we have a stopping time . Let denote the order of our deck after shuffles. is a strong uniform time if the probability that and , simultaneously, is constant in . In other words, the outcomes of shuffles under the requirement that all are equally likely. It can be slightly confusing to think about what the probability that means. One has to define a probability on the sequences of shuffles (state space). I won’t do that here, but there are plenty of references out there that can if you’re really curious.

Here is a lemma from Persi Diaconis’s book, Group Representations in Probability and Statistics, 1988:

Lemma. Let be any probability distribution defined a finite group, Let be a strong uniform time for . Then for all ,

.

The proof of the lemma is clever but not very hard — and not especially enlightening either (see page 70 of Diaconis’s book if interested). What it does give us is a way to use a strong stopping time to produce an upper bound estimate on the number of shuffles required. So, we have two goals now: construct a strong uniform time for the riffle shuffle and then compute the probability that it is bigger than .

We construct the uniform stopping time as follows. List the cards of the deck as the rows of a matrix. We perform repeated inverse shuffles. At each shuffle, add an additional column to the matrix. Put an H (or T) in the row for each card if in that shuffle, that card was associated with a heads (or tails). The rows of the columns produce a way to order the cards. For instance — looking at the first two columns of our matrix — cards which have HH are above cards which are TH. TH cards are above HT cards, and TT cards are all on the bottom. For the first three columns, the ordering would be HHH, THH, HTH, TTH, HHT, THT, HTT, TTT.

For a sequence of shuffles, we construct a (very large) matrix . Let be defined on sequences of shuffles so that when is the minimum number of columns necessary to make the rows of distinct (that is, no two rows are exactly the same). For 52 cards, . is a stopping time because if , we are able to determine this based on just the first 10 shuffles (we just need the first 10 columns of , after all).

is a strong uniform time because, assuming that , one could easily interchange any rows in the matrix to produce an equally valid sequence of (inverse) riffle shuffles which still has . Since the sequence in each row determines the position of each card in the deck, this gives us a way to produce any ordering of the deck, and thus there are ways of reordering the rows. But the sequence of shuffles which have the same rows as our matrix are all equally likely, and so the probability of getting a particular outcome , assuming , has probability . Hence is a strong uniform time.

The situation we are in with is similar to the Birthday Problem. Here we think of the cards as people and think of their associated rows as their birthdays. is the probability that at least two cards have the same rows of length : there are rows of length (since there are 2 possibilities for each entry, H or T), and thus birthdays. There are cards. Hence

By choosing , and applying some simple manipulations, one can see that for large ,

So, we need at most shuffles, provided is large enough (note that is “large enough”). In 1983, David Aldous made a much more careful analysis to determine that is actually sufficient for large . For , this number is approximately 8.5, so somewhere around 8 shuffles should suffice.

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.

One can take the following abstract point of view. The fundamental object we are studying is the symmetric group where, more often than not, . Each element of corresponds to a particular way to shuffle the deck. Alternatively, one can think of every element of as a particular ordering of the deck, starting from some prescribed order (i.e. however the deck was ordered when we took it out of the box). The identity element corresponds to the “no-shuffle” shuffle (alternatively, the original order). Transpositions correspond to interchanging the cards at position i and j, respectively, and so on. To model a collection of shuffles, one defines a probability measure on For example, one could put the cards in the deck side-by-side and shuffle as follows: with your left hand pick a card uniformly at random, and with your right hand pick a card uniformly at random, then interchange the two cards. The probability measure defined by this rule is as follows:

Once one has a probability measure , one can define the transition matrix so that Heuristically, the entry of corresponds to the probability of starting at ordering and ending at ordering . As has all non-negative entries and the rows sum to 1, this matrix corresponds to a Markov Chain (which one might call a random walk on ). It is worth noting here that the matrix has entries, which is something to the tune of when Not much help from a computer to be found with numbers like that kicking around. The power of , then correspond to the transition probabilities of going from state to (this, for example, proves the Chapman-Kolmogorov equations for finite state spaces). Working backwards, one can use to produce a probability distribution by This really corresponds to the -fold convolution of with itself, where convolution means the usual thing, i.e.

Heuristically, this new measure represents the probability of starting from a standard order getting to any other order by way of shuffles.

Okay, with this framework in mind, one can now define the difference between two probability distributions, and :

This is a pretty intuitive idea for distance; aside from the factor of , the formula is basically the norm. We will mostly be interested in this quantity when and where is the uniform distribution, i.e. . One key feature of this difference, is that it is (in a sense) submultiplicative with respect to convolution:

This is not very difficult to check. This property is important since:

where we have made use of the fact that the uniform distribution convolved with any other distribution is the uniform distribution (this fact characterizes the uniform distribution, actually) as well as the fact that . In particular, this means that

Hence as soon as gets smaller than , we have rapid (exponential) decay to the uniform distribution. There is a theorem due to Koss which holds in a more general setting where one asks a similar type of question on any compact group:

Theorem (Koss, 1959). Let be a compact group. Let be a probability on such that for some and with and for all ,

for all open sets

Then, for all ,

The additional hypothesis of the theorem says that the shuffling eventually doesn’t avoid a particular subgroup. So, in general, the plot looks something like

Taken from Persi Diaconis's book, Group Representations in Probability and Statistics, 1988

For a long time, this theorem was the end of the road. In our setting of , it is extremely relieving to know that any reasonable shuffling method will eventually converge very rapidly to the uniform distribution — no reasonable shuffling method would leave particular subgroups of out. However, in no way is the theorem useful from a practical point of view: we still have no idea how many times we need to shuffle the deck!

The goal, then, is to compute the best in the statement of Koss’s theorem, so that we know precisely how many times we have to shuffle until we converge exponentially to the uniform distribution. This turns out to be a difficult problem in general, for which, unsurprisingly, no general principle seems to work. That is to say that every type of shuffling technique seems to require its own special treatment. In my next post (which should be in a couple of days), I’ll describe how to model the standard shuffle, known as the riffle shuffle, and derive some estimates on how depends on .

A year or two ago, a couple of us were bored and somehow got to thinking about the series

It’s pretty easy to compute the series for For , one simply has to reorder the sum:

In the original sum, one adds vertically and then horizontally. Adding horizontally and then vertically — and tacitly making use of Fubini’s Theorem — one obtains

In general, one can iterate this idea to determine the sum in terms of the previous sums. For ,

The first row sums to one. The next three rows each sum to . The next five rows sum to . So in general, one obtains

Performing this same idea for general , one ends up with a sequence of rows. The first row always sums to 1. Then a number of rows follows each of which sums to . Then some rows summing to , and so on. To determine the number of rows summing to , one sees that there are . Hence we can write that

Note that the numerator of the right side of the equality is a polynomial of degree , and so by breaking the sum apart into each constituent power of , we have an iterative method of computing the sum for general . In particular,

and if

then

As a relevant aside, while teaching calculus I went through an example in class of how to use power series to compute sums; At the board I realized I had just computed a power series which could be used to compute the sum for any denominator. If

then one can easily see that

All these series have radius of convergence 1, so in particular makes sense when .

Anyway, the original series seems to grow extremely quickly in , much faster than . The approximate growth rate with respect to is pretty obvious by considering the integral,

Since must be smaller than 1, we changed to so that it’s more clear the integral actually converges.Making the change of variables , one gets to something close to the function:

For large enough , depending on , one can replace the lower-limit by 0 without making too great an error. This is because the maximum of the integrand gets larger and moves further and further to the right as gets bigger. In fact the approximation is very good as long as isn’t too small, in particular if isn’t bigger than . Anyway, the whole point of this is that the final integral is almost exactly equal to . For , the integral is 119.99 rather than 5!=120.

Hence,

For , this formula is good to 2 decimal places.

We worked for a while to figure out if there was a way to get a closed-form formula (i.e. a formula for finding the th sum without having to know all the sums smaller than ), but didn’t get too far. If anyone happens to know it, feel obliged to provide a reference!