At least, entropy *usually* increases. If we define entropy by first defining “macrostates” — collections of individual states of the system that are macroscopically indistinguishable from each other — and then taking the logarithm of the number of microstates per macrostate, as portrayed in this blog’s header image, then we don’t expect entropy to *always* increase. According to Boltzmann, the increase of entropy is just really, really probable, since higher-entropy macrostates are much, much bigger than lower-entropy ones. But if we wait long enough — really long, much longer than the age of the universe — a macroscopic system will spontaneously fluctuate into a lower-entropy state. Cream and coffee will unmix, eggs will unbreak, maybe whole universes will come into being. But because the timescales are so long, this is just a matter of intellectual curiosity, not experimental science.

That’s what I was taught, anyway. But since I left grad school, physicists (and chemists, and biologists) have become increasingly interested in ultra-tiny systems, with only a few moving parts. Nanomachines, or the molecular components inside living cells. In systems like that, the occasional downward fluctuation in entropy is not only possible, it’s going to happen relatively frequently — with crucial consequences for how the real world works.

Accordingly, the last fifteen years or so has seen something of a revolution in non-equilibrium statistical mechanics — the study of statistical systems far from their happy resting states. Two of the most important results are the Crooks Fluctuation Theorem (by Gavin Crooks), which relates the probability of a process forward in time to the probability of its time-reverse, and the Jarzynski Equality (by Christopher Jarzynski), which relates the change in free energy between two states to the average amount of work done on a journey between them. (Professional statistical mechanics are so used to dealing with **in**equalities that when they finally do have an honest equation, they call it an “equality.”) There is a sense in which these relations underlie the good old Second Law; the Jarzynski equality can be derived from the Crooks Fluctuation Theorem, and the Second Law can be derived from the Jarzynski Equality. (Though the three relations were discovered in reverse chronological order from how they are used to derive each other.)

Still, there is a mystery lurking in how we think about entropy and the Second Law — a puzzle that, like many such puzzles, I never really thought about until we came up with a solution. Boltzmann’s definition of entropy (logarithm of number of microstates in a macrostate) is very conceptually clear, and good enough to be engraved on his tombstone. But it’s not the only definition of entropy, and it’s not even the one that people use most often.

Rather than referring to macrostates, we can think of entropy as characterizing something more subjective: our knowledge of the state of the system. That is, we might not know the exact position ** x** and momentum

That is, we take the probability distribution ρ, multiply it by its own logarithm, and integrate the result over all the possible states of the system, to get (minus) the entropy. A formula like this was introduced by Boltzmann himself, but these days is often associated with Josiah Willard Gibbs, unless you are into information theory, where it’s credited to Claude Shannon. Don’t worry if the symbols are totally opaque; the point is that low entropy means we know a lot about the specific state a system is in, and high entropy means we don’t know much at all.

In appropriate circumstances, the Boltzmann and Gibbs formulations of entropy and the Second Law are closely related to each other. But there’s a crucial difference: in a perfectly isolated system, the Boltzmann entropy tends to increase, but the Gibbs entropy stays exactly constant. In an open system — allowed to interact with the environment — the Gibbs entropy will go up, but it will *only* go up. It will never fluctuate down. (Entropy can decrease through heat loss, if you put your system in a refrigerator or something, but you know what I mean.) The Gibbs entropy is about our knowledge of the system, and as the system is randomly buffeted by its environment we know less and less about its specific state. So what, from the Gibbs point of view, can we possibly mean by “entropy rarely, but occasionally, will fluctuate downward”?

I won’t hold you in suspense. Since the Gibbs/Shannon entropy is a feature of our knowledge of the system, the way it can fluctuate downward is for us to *look* at the system and notice that it is in a relatively unlikely state — thereby gaining knowledge.

But this operation of “looking at the system” doesn’t have a ready implementation in how we usually formulate statistical mechanics. Until now! My collaborators Tony Bartolotta, Stefan Leichenauer, Jason Pollack, and I have written a paper formulating statistical mechanics with explicit knowledge updating via measurement outcomes. (Some extra figures, animations, and codes are available at this web page.)

The Bayesian Second Law of Thermodynamics

Anthony Bartolotta, Sean M. Carroll, Stefan Leichenauer, and Jason PollackWe derive a generalization of the Second Law of Thermodynamics that uses Bayesian updates to explicitly incorporate the effects of a measurement of a system at some point in its evolution. By allowing an experimenter’s knowledge to be updated by the measurement process, this formulation resolves a tension between the fact that the entropy of a statistical system can sometimes fluctuate downward and the information-theoretic idea that knowledge of a stochastically-evolving system degrades over time. The Bayesian Second Law can be written as ΔH(ρ

_{m},ρ)+⟨Q⟩_{F|m}≥0, where ΔH(ρ_{m},ρ) is the change in the cross entropy between the original phase-space probability distribution ρ and the measurement-updated distribution ρ_{m}, and ⟨Q⟩_{F|m}is the expectation value of a generalized heat flow out of the system. We also derive refined versions of the Second Law that bound the entropy increase from below by a non-negative number, as well as Bayesian versions of the Jarzynski equality. We demonstrate the formalism using simple analytical and numerical examples.

The crucial word “Bayesian” here refers to Bayes’s Theorem, a central result in probability theory. Bayes’s theorem tells us how to update the probability we assign to any given idea, after we’ve received relevant new information. In the case of statistical mechanics, we start with some probability distribution for the system, then let it evolve (by being influenced by the outside world, or simply by interacting with a heat bath). Then we make some measurement — but a realistic measurement, which tells us something about the system but not everything. So we can use Bayes’s Theorem to update our knowledge and get a new probability distribution.

So far, all perfectly standard. We go a bit farther by also updating the *initial* distribution that we started with — our knowledge of the measurement outcome influences what we think we know about the system at the beginning of the experiment. Then we derive the Bayesian Second Law of Thermodynamics, which relates the original (un-updated) distribution at initial and final times to the updated distribution at initial and final times.

That relationship makes use of the cross entropy between two distributions, which you actually don’t see that often in information theory. Think of how much you would expect to learn by being told the specific state of a system, when all you originally knew was some probability distribution. If that distribution were sharply peaked around some value, you don’t expect to learn very much — you basically already know what state the system is in. But if it’s spread out, you expect to learn a bit more. Indeed, we can think of the Gibbs/Shannon entropy *S*(ρ) as “the average amount we expect to learn by being told the exact state of the system, given that it is described by a probability distribution ρ.”

By contrast, the cross-entropy *H*(ρ, ω) is a function of two distributions: the “assumed” distribution ω, and the “true” distribution ρ. Now we’re imagining that there are two sources of uncertainty: one because the actual distribution has a nonzero entropy, and another because we’re not even using the right distribution! The cross entropy between those two distributions is “the average amount we expect to learn by being told the exact state of the system, given that we think it is described by a probability distribution ω but it is actually described by a probability distribution ρ.” And the Bayesian Second Law (BSL) tells us that this lack of knowledge — the amount we would learn on average by being told the exact state of the system, given that we were using the un-updated distribution — is always larger at the end of the experiment than at the beginning (up to corrections because the system may be emitting heat). So the BSL gives us a nice information-theoretic way of incorporating the act of “looking at the system” into the formalism of statistical mechanics.

I’m very happy with how this paper turned out, and as usual my hard-working collaborators deserve most of the credit. Of course, none of us actually does statistical mechanics for a living — we’re all particle/field theorists who have wandered off the reservation. What inspired our wandering was actually this article by Natalie Wolchover in *Quanta* magazine, about work by Jeremy England at MIT. I had read the *Quanta* article, and Stefan had seen a discussion of it on Reddit, so we got to talking about it at lunch. We thought there was more we could do along these lines, and here we are.

It will be interesting to see what we can do with the BSL, now that we have it. As mentioned, occasional fluctuations downward in entropy happen all the time in small systems, and are especially important in biophysics, perhaps even for the origin of life. While we have phrased the BSL in terms of a measurement carried out by an observer, it’s certainly not necessary to have an actual person there doing the observing. All of our equations hold perfectly well if we simply ask “what happens, given that the system ends up in a certain kind of probability distribution.” That final conditioning might be “a bacteria has replicated,” or “an RNA molecule has assembled itself.” It’s an exciting connection between fundamental principles of physics and the messy reality of our fluctuating world.

]]>So let me point you to this guest post by Anton Garrett that Peter Coles just posted at his blog:

It’s quite a nice explanation of how the state of play looks to someone who is sympathetic to a hidden-variables view. (Fans of Bell’s Theorem should remember that what Bell did was to show that such variables must be nonlocal, not that they are totally ruled out.)

As a dialogue, it shares a feature that has been common to that format since the days of Plato: there are two characters, and the character that sympathizes with the author is the one who gets all the good lines. In this case the interlocutors are a modern physicist Neo, and a smart recently-resurrected nineteenth-century physicist Nino. Trained in the miraculous successes of the Newtonian paradigm, Nino is very disappointed that physicists of the present era are so willing to simply accept a theory that can’t do better than predicting probabilistic outcomes for experiments. More in sorrow than in anger, he urges us to do better!

My own takeaway from this is that it’s not a good idea to take advice from nineteenth-century physicists. Of course we should try to do better, since we should alway try that. But we should also feel free to abandon features of our best previous theories when new data and ideas come along.

A nice feature of the dialogue between Nino and Neo is the way in which it illuminates the fact that much of one’s attitude toward formulations of quantum mechanics is driven by which basic assumptions about the world we are most happy to abandon, and which we prefer to cling to at any cost. That’s true for any of us — such is the case when there is legitimate ambiguity about the best way to move forward in science. It’s a feature, not a bug. The hope is that eventually we will be driven, by better data and theories, toward a common conclusion.

What I like about Many-Worlds is that it is perfectly realistic, deterministic, and ontologically minimal, and of course it fits the data perfectly. Equally importantly, it is a robust and flexible framework: you give me your favorite Hamiltonian, and we instantly know what the many-worlds formulation of the theory looks like. You don’t have to think anew and invent new variables for each physical situation, whether it’s a harmonic oscillator or quantum gravity.

Of course, one gives something up: in Many-Worlds, while the underlying theory is deterministic, the experiences of individual observers are not predictable. (In that sense, I would say, it’s a nice compromise between our preferences and our experience.) It’s neither manifestly local nor Lorentz-invariant; those properties should emerge in appropriate situations, as often happens in physics. Of course there are all those worlds, but that doesn’t bother me in the slightest. For Many-Worlds, it’s the *technical* problems that bother me, not the philosophical ones — deriving classicality, recovering the Born Rule, and so on. One tends to think that technical problems can be solved by hard work, while metaphysical ones might prove intractable, which is why I come down the way I do on this particular question.

But the hidden-variables possibility is still definitely alive and well. And the general program of “trying to invent a better theory than quantum mechanics which would make all these distasteful philosophical implications go away” is certainly a worthwhile one. If anyone wants to suggest their favorite defenses of epistemic or dynamical-collapse approaches, feel free to leave them in comments.

]]>Obviously the infamous 140-character limit of Twitter doesn’t allow the level of precision and subtlety one would always like to achieve when talking about difficult topics. But restrictions lead to creativity, and the results can actually be a bit more accessible than unfettered prose might have been.

Anyway, spacetime isn’t fundamental, it’s just a useful approximation in certain regimes. Someday we hope to know what it’s an approximation to.

]]>Now two students and I have offered a small contribution to this effort. Aidan Chatwin-Davies is a grad student here at Caltech, while Adam Jermyn was an undergraduate who has now gone on to do graduate work at Cambridge. Aidan came up with a simple method for getting out one “quantum bit” (qubit) of information from a black hole, using a strategy similar to “quantum teleportation.” Here’s our paper that just appeared on arxiv:

How to Recover a Qubit That Has Fallen Into a Black Hole

Aidan Chatwin-Davies, Adam S. Jermyn, Sean M. CarrollWe demonstrate an algorithm for the retrieval of a qubit, encoded in spin angular momentum, that has been dropped into a no-firewall unitary black hole. Retrieval is achieved analogously to quantum teleportation by collecting Hawking radiation and performing measurements on the black hole. Importantly, these methods only require the ability to perform measurements from outside the event horizon and to collect the Hawking radiation emitted after the state of interest is dropped into the black hole.

It’s a very specific — i.e. not very general — method: you have to have done measurements on the black hole ahead of time, and then drop in one qubit, and we show how to get it back out. Sadly it doesn’t work for *two* qubits (or more), so there’s no obvious way to generalize the procedure. But maybe the imagination of some clever person will be inspired by this particular thought experiment to come up with a way to get out two qubits, and we’ll be off.

I’m happy to host this guest post by Aidan, explaining the general method behind our madness.

If you were to ask someone on the bus which of Stephen Hawking’s contributions to physics he or she thought was most notable, the answer that you would almost certainly get is his prediction that a black hole should glow as if it were an object with some temperature. This glow is made up of thermal radiation which, unsurprisingly, we call Hawking radiation. As the black hole radiates, its mass slowly decreases and the black hole decreases in size. So, if you waited long enough and were careful not to enlarge the black hole by throwing stuff back in, then eventually it would completely evaporate away, leaving behind nothing but a bunch of Hawking radiation.

At a first glance, this phenomenon of black hole evaporation challenges a central notion in quantum theory, which is that it should not be possible to destroy information. Suppose, for example, that you were to toss a book, or a handful of atoms in a particular quantum state into the black hole. As the black hole evaporates into a collection of thermal Hawking particles, what happens to the information that was contained in that book or in the state of (what were formerly) your atoms? One possibility is that the information actually is destroyed, but then we would have to contend with some pretty ugly foundational consequences for quantum theory. Instead, it could be that the information is preserved in the state of the leftover Hawking radiation, albeit highly scrambled and difficult to distinguish from a thermal state. Besides being very pleasing on philosophical grounds, we also have evidence for the latter possibility from the AdS/CFT correspondence. Moreover, if the process of converting a black hole to Hawking radiation conserves information, then a stunning result of Hayden and Preskill says that for sufficiently old black holes, any information that you toss in comes back out almost a fast as possible!

Even so, exactly how information leaks out of a black hole and how one would go about converting a bunch of Hawking radiation to a useful state is quite mysterious. On that note, what we did in a recent piece of work was to propose a protocol whereby, under very modest and special circumstances, you can toss one qubit (a single unit of quantum information) into a black hole and then recover its state, and hence the information that it carried.

More precisely, the protocol describes how to recover a single qubit that is encoded in the spin angular momentum of a particle, *i.e.*, a spin qubit. Spin is a property that any given particle possesses, just like mass or electric charge. For particles that have spin equal to 1/2 (like those that we consider in our protocol), at least classically, you can think of spin as a little arrow which points up or down and says whether the particle is spinning clockwise or counterclockwise about a line drawn through the arrow. In this classical picture, whether the arrow points up or down constitutes one classical bit of information. According to quantum mechanics, however, spin can actually exist in a superposition of being part up and part down; these proportions constitute one qubit of quantum information.

That’s the protocol in a nutshell. If the words “quantum teleportation” mean anything to you, then you can think of the protocol as a variation on the quantum teleportation protocol where the transmitting party is the black hole and measurement is performed in the total angular momentum basis instead of the Bell basis. Of course, this is far from a resolution of the information problem for black holes. However, it is certainly a neat trick which shows, in a special set of circumstances, how to “bounce” a qubit of quantum information off of a black hole.

]]>You can find the fruits of George’s labors at this io9 post. But my own answer went on at sufficient length that I might as well put it up here as well. Of course, as with any “Why?” question, we need to keep in mind that the answer might simply be “Because that’s the way it is.”

Whenever we seem surprised or confused about some aspect of the universe, it’s because we have some pre-existing expectation for what it “should” be like, or what a “natural” universe might be. But the universe doesn’t have a purpose, and there’s nothing more natural than Nature itself — so what we’re really trying to do is figure out what our expectations should be.

The universe is big on human scales, but that doesn’t mean very much. It’s not surprising that humans are small compared to the universe, but big compared to atoms. That feature does have an obvious anthropic explanation — complex structures can only form on in-between scales, not at the very largest or very smallest sizes. Given that living organisms are going to be complex, it’s no surprise that we find ourselves at an in-between size compared to the universe and compared to elementary particles.

What is arguably more interesting is that the universe is so big compared to particle-physics scales. The Planck length, from quantum gravity, is 10^{-33} centimeters, and the size of an atom is roughly 10^{-8} centimeters. The difference between these two numbers is already puzzling — that’s related to the “hierarchy problem” of particle physics. (The size of atoms is fixed by the length scale set by electroweak interactions, while the Planck length is set by Newton’s constant; the two distances are extremely different, and we’re not sure why.) But the scale of the universe is roughly 10^29 centimeters across, which is enormous by any scale of microphysics. It’s perfectly reasonable to ask why.

Part of the answer is that “typical” configurations of stuff, given the laws of physics as we know them, tend to be very close to empty space. (“Typical” means “high entropy” in this context.) That’s a feature of general relativity, which says that space is dynamical, and can expand and contract. So you give me any particular configuration of matter in space, and I can find a lot more configurations where the same collection of matter is spread out over a much larger volume of space. So if we were to “pick a random collection of stuff” obeying the laws of physics, it would be mostly empty space. Which our universe is, kind of.

Two big problems with that. First, even empty space has a natural length scale, which is set by the cosmological constant (energy of the vacuum). In 1998 we discovered that the cosmological constant is not quite zero, although it’s very small. The length scale that it sets (roughly, the distance over which the curvature of space due to the cosmological constant becomes appreciable) is indeed the size of the universe today — about 10^26 centimeters. (Note that the cosmological constant itself is inversely proportional to this length scale — so the question “Why is the cosmological-constant length scale so large?” is the same as “Why is the cosmological constant so small?”)

This raises two big questions. The first is the “coincidence problem”: the universe is expanding, but the length scale associated with the cosmological constant is a constant, so why are they approximately equal *today*? The second is simply the “cosmological constant problem”: why is the cosmological constant scale so enormously larger than the Planck scale, or event than the atomic scale? It’s safe to say that right now there are no widely-accepted answers to either of these questions.

So roughly: the answer to “Why is the universe so big?” is “Because the cosmological constant is so small.” And the answer to “Why is the cosmological constant so small?” is “Nobody knows.”

But there’s yet another wrinkle. Typical configurations of stuff tend to look like empty space. But our universe, while relatively empty, isn’t *that* empty. It has over a hundred billion galaxies, with a hundred billion stars each, and over 10^50 atoms per star. Worse, there are maybe 10^88 particles (mostly photons and neutrinos) within the observable universe. That’s a lot of particles! A much more natural state of the universe would be enormously emptier than that. Indeed, as space expands the density of particles dilutes away — we’re headed toward a much more natural state, which will be much emptier than the universe we see today.

So, given what we know about physics, the real question is “Why are there so many particles in the observable universe?” That’s one angle on the question “Why is the entropy of the observable universe so small?” And of course the density of particles was much higher, and the entropy much lower, at early times. These questions are also ones to which we have no good answers at the moment.

]]>Nambu was one of the greatest theoretical physicists of the 20th century, although not one with a high public profile. Among his contributions:

- Being the first to really understand spontaneous symmetry breaking in quantum field theory, work for which he won a (very belated) Nobel Prize in 2008. We now understand the pion as a (pseudo-) “Nambu-Goldstone boson.”
- Suggesting that quarks might come in three colors, and those colors might be charges for an SU(3) gauge symmetry, giving rise to force-carrying particles called gluons.
- Proposing the first relativistic string theory, based on what is now called the Nambu-Goto action.

So — not too shabby.

But despite his outsized accomplishments, Nambu was quiet, proper, it’s even fair to say “shy.” He was one of those physicists who talked very little, and was often difficult to understand when he does talk, but if you put in the effort to follow him you would invariably be rewarded. One of his colleagues at the University of Chicago, Bruce Winstein, was charmed by the fact that Nambu was an experimentalist at heart; at home, apparently, he kept a little lab, where he would tinker with electronics to take a break from solving equations.

Any young person in science might want to read this profile of Nambu by his former student Madhusree Mukerjee. In it, Nambu tells of when he first came to the US from Japan, to be a postdoctoral researcher at the Institute for Advanced Study in Princeton. “Everyone seemed smarter than I,” Nambu recalls. “I could not accomplish what I wanted to and had a nervous breakdown.”

If Yoichiro Nambu can have a nervous breakdown because he didn’t feel smart enough, what hope is there for the rest of us?

Here are a few paragraphs I wrote about Nambu and spontaneous symmetry breaking in *The Particle at the End of the Universe*.

A puzzle remained: how do we reconcile the idea that photons have mass inside a superconductor with the conviction that the underlying symmetry of electromagnetism forces the photon to be massless?

This problem was tackled by a number of people, including American physicist Philip Anderson, Soviet physicist Nikolai Bogoliubov, and Japanese-American physicist Yoichiro Nambu. The key turned out to be that the symmetry was indeed there, but that it was hidden by a field with that took on a nonzero value in the superconductor. According to the jargon that accompanies this phenomenon, we say the symmetry is “spontaneously broken”: the symmetry is there in the underlying equations, but the particular solution to those equations in which we are interested doesn’t look very symmetrical.

Yoichiro Nambu, despite the fact that he won the Nobel Prize in 2008 and has garnered numerous other honors over the years, remains relatively unknown outside physics. That’s a shame, as his contributions are comparable to those of better-known colleagues. Not only was he one of the first to understand spontaneous symmetry breaking in particle physics, but he was also the first to propose that quarks carry color, to suggest the existence of gluons, and to point out that certain particle properties could be explained by imagining that the particles were really tiny strings, thus launching string theory. Theoretical physicists admire Nambu’s accomplishments, but his inclination is to avoid the limelight.

For several years in the early 2000’s I was a faculty member at the University of Chicago, with an office across the hall from Nambu’s. We didn’t interact much, but when we did he was unfailingly gracious and polite. My major encounter with him was one time when he knocked on my door, hoping that I could help him with the email system on the theory group computers, which tended to take time off at unpredictable intervals. I wasn’t much help, but he took it philosophically. Peter Freund, another theorist at Chicago, describes Nambu as a “magician”: “He suddenly pulls a whole array of rabbits out of his hat and, before you know it, the rabbits reassemble in an entirely novel formation and by God, they balance the impossible on their fluffy cottontails.” His highly developed sense of etiquette, however, failed him when he was briefly appointed as department chair: reluctant to explicitly say “no” to any question, he would indicate disapproval by pausing before saying “yes.” This led to a certain amount of consternation among his colleagues, once they realized that their requests hadn’t actually been granted.

After the BCS theory of superconductivity was proposed, Nambu began to study the phenomenon from the perspective of a particle physicist. He put his finger on the key role played by spontaneous symmetry breaking, and began to wonder about its wider applicability. One of Nambu’s breakthroughs was to show (partly in collaboration with Italian physicist Giovanni Jona-Lasinio) how spontaneous symmetry breaking could happen even if you weren’t inside a superconductor. It could happen in empty space, in the presence of a field with a nonzero value — a clear precursor to the Higgs field. Interestingly, this theory also showed how a fermion field could start out massless, but gain mass through the process of symmetry breaking.

As brilliant as it was, Nambu’s suggestion of spontaneous symmetry breaking came with a price. While his models gave masses to fermions, they also predicted a new massless boson particle — exactly what particle physicists were trying to avoid, since they didn’t see any such particles created by the nuclear forces. Soon thereafter, Scottish physicist Jeffrey Goldstone argued that this wasn’t just an annoyance: this kind of symmetry breaking necessarily gave rise to massless particles, now called “Nambu-Goldstone bosons.” Pakistani physicist Abdus Salam and American physicist Steven Weinberg then collaborated with Goldstone in promoting this argument to what seemed like an air-tight proof, now called “Goldstone’s theorem.”

One question that must be addressed by any theory of broken symmetry is, what is the field that breaks the symmetry? In a superconductor the role is played by the Cooper pairs, composite states of electrons. In the Nambu/Jona-Lasinio model, a similar effect happens with composite nucleons. Starting with Goldstone’s 1961 paper, however, physicists become comfortable with the idea of simply positing a set of new fundamental boson fields whose job it was to break symmetries by taking on a nonzero value in empty space. The kind of fields required are known as a “scalar” fields, which is a way of saying they have no intrinsic spin. The gauge fields that carry forces, although they are also bosons, have spin equal to one.

If the symmetry weren’t broken, all the fields in Goldstone’s model would behave in exactly the same way, as massive scalar bosons, due to the requirements of the symmetry. When the symmetry is broken, the fields differentiate themselves. In the case of a global symmetry (a single transformation all throughout space), which is what Goldstone considered, one field remains massive, while the others become massless Nambu-Goldstone bosons — that’s Goldstone’s theorem.

]]>So it was a great honor for me to appear as one of the guests when the show breezed through LA back in March. It was a terrific event, as you might guess from the other guests: comedian Joe Rogan, TV writer David X. Cohen, and one Eric Idle, who used to play in the Rutles. And now selected bits of the program can be listened to at home, courtesy of this handy podcast link, or directly on iTunes.

Be sure to check out the other stops on the IMC tour of the US, which included visits to NYC, Chicago, and San Francisco, featuring many friends-of-the-blog along the way.

These guys, of course, are heavy hitters, so you never know who is going to show up at one of these things. Their relationship with Eric Idle goes back quite a ways, and he actually composed and performed a theme song for the show (below). Naturally, since he was on stage in LA, they asked him to do a live version, which was a big hit. And there in the band, performing on ukulele for just that one song, was Jeff Lynne, of the Electric Light Orchestra. Maybe a bit under-utilized in this context, but why not get the best when you can?

]]>I was reminded of this when I saw a Quantum Diaries post by Alex Millar, entitled “Why Dark Matter Exists.” Why do we live in a universe with five times as much dark matter as ordinary matter, anyway? As it turns out, the post was more about explaining all of the wonderful evidence we have that there is so much dark matter. That’s a very respectable question, one that I’ve covered now and again. The less-respectable (but still interesting to me) question is, Why is the universe like that? Is the existence of dark matter indeed unsurprising, or is it an unusual feature that we should take as an important clue as to the nature of our world?

Generally, physicists love asking these kinds of questions (“why does the universe look this way, rather than that way?”), and yet are terribly sloppy at answering them. Questions about surprise and probability require a measure: a way of assigning, to each set of possibilities, some kind of probability number. Your answer wholly depends on how you assign that measure. If you have a coin, and your probability measure is “it will be heads half the time and tails half the time,” then getting twenty heads in a row is very surprising. If you have reason to think the coin is loaded, and your measure is “it comes up heads almost every time,” then twenty heads in a row isn’t surprising at all. Yet physicists love to bat around these questions in reference to the universe itself, without really bothering to justify one measure rather than another.

With respect to dark matter, we’re contemplating a measure over all the various ways the universe could be, including both the laws of physics (which tell us what particles there can be) and the initial conditions (which set the stage for the later evolution). Clearly finding the “right” such measure is pretty much hopeless! But we can try to set up some reasonable considerations, and see where that leads us.

Here are the important facts we know about dark matter:

**It’s dark.**Doesn’t interact with electromagnetism, at least not with anywhere near the strength that ordinary charged particles do.**It’s cold.**Individual dark matter particles are moving slowly and have been for a while, otherwise they would have damped perturbations in the early universe.**There’s a goodly amount of it.**About 25% of the energy density of the current universe, compared to only about 5% in the form of ordinary matter.**It’s stable, or nearly so.**The dark matter particle has to be long-lived, or it would have decayed away a long time ago.**It’s dissipationless, or nearly so.**Ordinary matter settles down to make galaxies because it can lose energy through collisions and radiation; dark matter doesn’t seem to do that, giving rise to puffy halos rather than thin galactic disks.

None of these properties is, by itself, very hard to satisfy if we’re just inventing new particles. But if we try to be honest — asking “What would expect to see, if we didn’t know what things actually looked like?” — there is a certain amount of tension involved in satisfying them all at once. Let’s take them in turn.

Having a particle be **dark** isn’t hard at all. All electrically-neutral particles are dark in this sense. Photons, gravitons, neutrinos, neutrons, what have you.

It’s also not hard to imagine particles that are **cold**. The universe is a pretty old place, and things tend to cool off as the universe expands. Massless particles like photons or gravitons never slow down, of course, since they always move at the speed of light, so they don’t lead to dark-matter candidates. Indeed, even particles that are very light, like neutrinos, tend to be moving too quickly to be dark matter. The simplest way to get a good cold dark matter candidate is just to imagine something that is relatively heavy; then it will naturally be cold (slowly-moving) at late times, even it was hot in the very early universe. Ordinary atoms do exactly that. It’s possible to have very low-mass particles that are nevertheless cold; axions are a good example. They were simply never hot, even in the early universe; we say they are non-thermal relics. But in the space of all the particles you can imagine being cold (in some ill-defined measure we are just making up), it seems easiest to consider particles that are heavy enough to cool down over time.

Getting the right **amount** of dark matter is tricky, but certainly not a deal-breaker. Massive particles, generally speaking, will tend to bump into massive anti-particles and annihilate into lower-mass particles (such as photons). If our dark matter candidate interacts too strongly, it will annihilate too readily, and simply be wiped out. Conversely, if it doesn’t interact strongly enough, we will be stuck with too much of it at the end of the day. The sweet spot is approximately the interaction strength of the weak nuclear force, which is why WIMPs (weakly interacting massive particles) are such a popular dark-matter candidate. Again, this certainly isn’t the only kind of possibility, but it seems very natural and robust.

We need our dark matter particles to be **stable**, and now we’re coming up against a bigger issue than before. There are plenty of stable particles in nature — photons, gravitons, neutrinos, electrons, protons. All the ones we know about are either massless (photons, gravitons), or they are the lightest kind of particle that carries some *conserved quantity*. Protons are the lightest particles carrying baryon number; electrons are the lightest particles carrying electric charge; neutrinos are the lightest particle carrying fermion number. All of these conserved quantities are the result of some symmetry of nature, following from Noether’s Theorem. So if you want your dark matter candidate to be relatively massive but also stable (or nearly), the most straightforward route is to have it carry some conserved quantity that follows from a symmetry. What quantity is that supposed to be? Symmetries don’t just grow on trees, you know. The most robust kinds of symmetries are gauge symmetries, which entail long-range forces like electromagnetism (associated with conservation of electric charge). So arguably the easiest way to make a stable, massive dark-matter particle would be to have it carry some analogue of electric charge. (This is exactly what Lotty Ackerman, Matt Buckley, Marc Kamionkowski and I did with our Dark Electromagnetism paper.) Axions, always wanting to be the exception to every rule, get around this by being so low-mass and so weakly-interacting that they do decay, but extremely slowly.

Finally, we’d like our dark matter particle to be dissipationless. And here’s the problem: if we follow the logic so far, and end up with a massive neutral particle carrying a new kind of conserved quantity associated with a gauge symmetry and therefore a long-range force, it tends to have dissipation. You can make magnetic fields, you can scatter and emit “dark photons,” you can make “dark atoms,” what have you. It’s not necessarily impossible to make everything work out, but it’s probably safe to say that you would expect dissipation if you knew your particle was coupled to photon-like particles but didn’t know anything else. There’s a straightforward fix, of course: if your particle is stable because of a global symmetry, rather than a gauge symmetry, then it isn’t coupled to any long-range forces. Protons are kept stable by carrying baryon number, which comes from a global symmetry. However, global symmetries are generally thought to be more delicate than gauge symmetries — it’s fairly easy to break them, whereas gauge symmetries are quite robust. Nothing stops us from imagining global symmetries that keep our dark matter particles stable; but it isn’t the first thing you might expect. Indeed, in the most popular WIMP models based on supersymmetry, there is a global symmetry called R-parity that is responsible for keeping the dark matter candidate stable. This is kind of a puzzle for such models; it wouldn’t be hard to imagine that R-parity is somehow broken, allowing the lightest supersymmetric particles to decay.

So should we be surprised that we live in a universe full of dark matter? I’m going to say: yes. The existence of dark matter itself isn’t surprising, but it seems easier to imagine that it would have been hot rather than cold, or dissipative rather than dissipationless. I wouldn’t count this as one of the biggest surprises the universe has given us, since there are so many ways to evade these back-of-the-envelope considerations. But it’s something to think about.

]]>Nearly thirty years later, Witten is still going strong. As evidence, check out this paper that recently appeared on the arxiv, with co-authors Davide Gaiotto and Greg Moore:

Algebra of the Infrared: String Field Theoretic Structures in Massive N=(2,2) Field Theory In Two Dimensions

Davide Gaiotto, Gregory W. Moore, Edward WittenWe introduce a “web-based formalism” for describing the category of half-supersymmetric boundary conditions in 1+1 dimensional massive field theories with N=(2,2) supersymmetry and unbroken U(1)R symmetry. We show that the category can be completely constructed from data available in the far infrared, namely, the vacua, the central charges of soliton sectors, and the spaces of soliton states on ℝ, together with certain “interaction and boundary emission amplitudes”. These amplitudes are shown to satisfy a system of algebraic constraints related to the theory of A∞ and L∞ algebras. The web-based formalism also gives a method of finding the BPS states for the theory on a half-line and on an interval. We investigate half-supersymmetric interfaces between theories and show that they have, in a certain sense, an associative “operator product.” We derive a categorification of wall-crossing formulae. The example of Landau-Ginzburg theories is described in depth drawing on ideas from Morse theory, and its interpretation in terms of supersymmetric quantum mechanics. In this context we show that the web-based category is equivalent to a version of the Fukaya-Seidel A∞-category associated to a holomorphic Lefschetz fibration, and we describe unusual local operators that appear in massive Landau-Ginzburg theories. We indicate potential applications to the theory of surface defects in theories of class S and to the gauge-theoretic approach to knot homology.

I cannot, in good conscience, say that I understand very much about this new paper. It’s a kind of mathematica/formal field theory that is pretty far outside my bailiwick. (This is why scientists roll their eyes when a movie “physicist” is able to invent a unified field theory, build a time machine, and construct nanobots that can cure cancer. Specialization is real, folks!)

But there are two things about the paper that I nevertheless can’t help but remarking on. One is that it’s 429 pages long. I mean, damn. That’s a book, not a paper. Scuttlebutt informs me that the authors had to negotiate specially with the arxiv administrators just to upload the beast. Most amusingly, they knew perfectly well that a 400+ page work might come across as a little intimidating, so they wrote a summary paper!

An Introduction To The Web-Based Formalism

Davide Gaiotto, Gregory W. Moore, Edward WittenThis paper summarizes our rather lengthy paper, “Algebra of the Infrared: String Field Theoretic Structures in Massive N=(2,2) Field Theory In Two Dimensions,” and is meant to be an informal, yet detailed, introduction and summary of that larger work.

This short, user-friendly introduction is a mere 45 pages — still longer than 95% of the papers in this field. After a one-paragraph introduction, the first words of the lighthearted summary paper are “Let X be a Kähler manifold, and W : X → C a holomorphic Morse function.” So maybe it’s not *that* informal.

The second remarkable thing is — hey look, there’s my name! Both of the papers cite one of my old works from when I was a grad student, with Simeon Hellerman and Mark Trodden. (A related paper was written near the same time by Gary Gibbons and Paul Townsend.)

Domain Wall Junctions are 1/4-BPS States

Sean M. Carroll, Simeon Hellerman, Mark TroddenWe study N=1 SUSY theories in four dimensions with multiple discrete vacua, which admit solitonic solutions describing segments of domain walls meeting at one-dimensional junctions. We show that there exist solutions preserving one quarter of the underlying supersymmetry — a single Hermitian supercharge. We derive a BPS bound for the masses of these solutions and construct a solution explicitly in a special case. The relevance to the confining phase of N=1 SUSY Yang-Mills and the M-theory/SYM relationship is discussed.

Simeon, who was a graduate student at UCSB at the time and is now faculty at the Kavli IPMU in Japan, was the driving force behind this paper. Mark and I had recently written a paper on different ways that topological defects could intersect and join together. Simeon, who is an expert in supersymmetry, noticed that there was a natural way to make something like that happen in supersymmetric theories: in particular, you could have domain walls (sheets that stretch through space, separating different possible vacuum states) could intersect at “junctions.” Even better, domain-wall junction configurations would break some of the supersymmetry but not all of it. Setups like that are known as BPS states, and are highly valued and useful to supersymmetry aficionados. In general, solutions to quantum field theories are very difficult to find and characterize with any precision, but the BPS property lets you invoke some of the magic of supersymmetry to prove results that would otherwise be intractable.

Admittedly, the above paragraph is likely to be just as opaque to the person on the street as the Gaiotto/Moore/Witten paper is to me. The point is that we were able to study the behavior of domain walls and how they come together using some simple but elegant techniques in field theory. Think of drawing some configuration of walls as a network of lines in a plane. (All of the configurations we studied were invariant along some “vertical” direction in space, as well as static in time, so all the action happens in a two-dimensional plane.) Then we were able to investigate the set of all possible ways such walls could come together to form allowed solutions. Here’s an example, using walls that separate four different possible vacuum states:

As far as I understand it (remember — not that far!), this is a very baby version of what Gaiotto, Moore, and Witten have done. Like us, they look at a large-distance limit, worrying about how defects come together rather than the detailed profiles of the individual configurations. That’s the “infrared” in their title. Unlike us, they go way farther, down a road known as “categorification” of the solutions. In particular, they use a famous property of BPS states: you can multiply them together to get other BPS states. That’s the “algebra” of their title. To mathematicians, algebras aren’t just ways of “solving for *x*” in equations that tortured you in high school; they are mathematical structures describing sets of vectors that can be multiplied by each other to produce other vectors. (Complex numbers are an algebra; so are ordinary three-dimensional vectors, using the cross product operation.)

At this point you’re allowed to ask: Why should I care? At least, why should I imagine putting in the work to read a 429-page opus about this stuff? For that matter, why did these smart guys put in the work to *write* such an opus?

It’s a reasonable question, but there’s also a reasonable answer. In theoretical physics there are a number of puzzles and unanswered questions that we are faced with, from “Why is the mass of the Higgs 125 GeV?” to “How does information escape from black holes?” Really these are all different sub-questions of the big one, “How does Nature work?” By construction, we don’t know the answer to these questions — if we did, we’d move onto other ones. But we don’t even know the right way to go about getting the answers. When Einstein started thinking about fitting gravity into the framework of special relativity, Riemannian geometry was absolutely the last thing on his mind. It’s hard to tell what paths you’ll have to walk down to get to the final answer.

So there are different techniques. Some people will try a direct approach: if you want to know how information comes out of a black hole, think as hard as you can about what happens when black holes radiate. If you want to know why the Higgs mass is what it is, think as hard as you can about the Higgs field and other possible fields we haven’t yet found.

But there’s also a more patient, foundational approach. Quantum field theory is hard; to be honest, we don’t understand it all that well. There’s little question that there’s a lot to be learned by studying the fundamental behavior of quantum field theories in highly idealized contexts, if only to better understand the space of things that can possibly happen with an eye to eventually applying them to the real world. That, I suspect, is the kind of motivation behind a massive undertaking like this. I don’t want to speak for the authors; maybe they just thought the math was cool and had fun learning about these highly unrealistic (but still extremely rich) toy models. But the ultimate goal is to learn some basic wisdom that we will someday put to use in answering that underlying question: How does Nature work?

As I said, it’s not really my bag. I don’t have nearly the patience nor that mathematical aptitude that is required to make real progress in this kind of way. I’d rather try to work out on general principles what could have happened near the Big Bang, or how our classical world emerges out of the quantum wave function.

But, let a thousand flowers bloom! Gaiotto, Moore, and Witten certainly know what they’re doing, and hardly need to look for my approval. It’s one strategy among many, and as a community we’re smart enough to probe in a number of different directions. Hopefully this approach will revolutionize our understanding of quantum field theory — and at my retirement party everyone will be asking me why I didn’t stick to working on domain-wall junctions.

]]>As with teenagers, conference attendees secretly and falsely believe that other groups are having a much better time… Your conference strategies should therefore be geared towards counteracting the tendency to re-live your teenage years.

It’s surprising to realize how much “smart ways to behave at conferences” are really just “smart ways to behave in life.” (Though probably it shouldn’t be that surprising — academics aren’t the special flowers we like to think we are.) This bit of advice in particular struck me as useful:

[If] you worry someone will ask you what you work on, have something to say that’s three sentences long and takes fifteen seconds to get through. Write it down and practice it if you like.

I think every person should do that all the time. As you go through life, there will be multiple occasions on which people ask you “What do you do?” (If you’re in academia or an otherwise creative field, it will be “What are you working on?”) A high percentage of the time, questions like that elicit an awkwardly long pause, or something deflecting like “Oh, you know, lots of things.”

It makes sense. In your mind, “what you do” or “what you’re working on” is this incredibly rich, diverse, tightly interconnected set of things, and here is someone you don’t know asking you to instantly distill it down to a pithy phrase. Outrageous! And what’s worse, if you actually give a substantive answer, you’ll inevitably be leaving something out. You think, “Well I could mention this one thing that I’m mostly thinking about, but it’s not really representative of what I’m usually doing, so maybe I should mention this other thing…”

It’s not really a good look. Think about your own feelings when you ask someone what they do, and they respond with “Oh, I don’t know” or “Oh, lots of things.” Really? You don’t know what you do? Are you a spy whose memories are wiped at the conclusion of each mission? I’m sure you do many things, but perhaps picking out one would provide me with more useful information? At a conference in particular, it’s not the best first impression.

It’s important to come to terms with the fact that there are no perfect answers to the whatdoyoudo/whatareyouworkingon kinds of questions — and yet, we should have answers ready. Ones that are confident, short, and convey just a bit of the necessary flavor, so that more detail can emerge over the course of further conversation, which after all is the point of these well-meaning interrogations. This is especially true if you’re going to academic meetings, but it holds for life more generally. As adults, we should be better at these everyday skills than we were as teenagers.

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