About preskill

I am a theoretical physicist at Caltech.

Inflation on the back of an envelope

Last Monday was an exciting day!

After following the BICEP2 announcement via Twitter, I had to board a transcontinental flight, so I had 5 uninterrupted hours to think about what it all meant. Without Internet access or references, and having not thought seriously about inflation for decades, I wanted to reconstruct a few scraps of knowledge needed to interpret the implications of r ~ 0.2.

I did what any physicist would have done … I derived the basic equations without worrying about niceties such as factors of 3 or 2 \pi. None of what I derived was at all original —  the theory has been known for 30 years — but I’ve decided to turn my in-flight notes into a blog post. Experts may cringe at the crude approximations and overlooked conceptual nuances, not to mention the missing references. But some mathematically literate readers who are curious about the implications of the BICEP2 findings may find these notes helpful. I should emphasize that I am not an expert on this stuff (anymore), and if there are serious errors I hope better informed readers will point them out.

By tradition, careless estimates like these are called “back-of-the-envelope” calculations. There have been times when I have made notes on the back of an envelope, or a napkin or place mat. But in this case I had the presence of mind to bring a notepad with me.

Notes from a plane ride

Notes from a plane ride

According to inflation theory, a nearly homogeneous scalar field called the inflaton (denoted by \phi)  filled the very early universe. The value of \phi varied with time, as determined by a potential function V(\phi). The inflaton rolled slowly for a while, while the dark energy stored in V(\phi) caused the universe to expand exponentially. This rapid cosmic inflation lasted long enough that previously existing inhomogeneities in our currently visible universe were nearly smoothed out. What inhomogeneities remained arose from quantum fluctuations in the inflaton and the spacetime geometry occurring during the inflationary period.

Gradually, the rolling inflaton picked up speed. When its kinetic energy became comparable to its potential energy, inflation ended, and the universe “reheated” — the energy previously stored in the potential V(\phi) was converted to hot radiation, instigating a “hot big bang”. As the universe continued to expand, the radiation cooled. Eventually, the energy density in the universe came to be dominated by cold matter, and the relic fluctuations of the inflaton became perturbations in the matter density. Regions that were more dense than average grew even more dense due to their gravitational pull, eventually collapsing into the galaxies and clusters of galaxies that fill the universe today. Relic fluctuations in the geometry became gravitational waves, which BICEP2 seems to have detected.

Both the density perturbations and the gravitational waves have been detected via their influence on the inhomogeneities in the cosmic microwave background. The 2.726 K photons left over from the big bang have a nearly uniform temperature as we scan across the sky, but there are small deviations from perfect uniformity that have been precisely measured. We won’t worry about the details of how the size of the perturbations is inferred from the data. Our goal is to achieve a crude understanding of how the density perturbations and gravitational waves are related, which is what the BICEP2 results are telling us about. We also won’t worry about the details of the shape of the potential function V(\phi), though it’s very interesting that we might learn a lot about that from the data.

Exponential expansion

Einstein’s field equations tell us how the rate at which the universe expands during inflation is related to energy density stored in the scalar field potential. If a(t) is the “scale factor” which describes how lengths grow with time, then roughly

\left(\frac{\dot a}{a}\right)^2 \sim \frac{V}{m_P^2}.

Here \dot a means the time derivative of the scale factor, and m_P = 1/\sqrt{8 \pi G} \approx 2.4 \times 10^{18} GeV is the Planck scale associated with quantum gravity. (G is Newton’s gravitational constant.) I’ve left our a factor of 3 on purpose, and I used the symbol ~ rather than = to emphasize that we are just trying to get a feel for the order of magnitude of things. I’m using units in which Planck’s constant \hbar and the speed of light c are set to one, so mass, energy, and inverse length (or inverse time) all have the same dimensions. 1 GeV means one billion electron volts, about the mass of a proton.

(To persuade yourself that this is at least roughly the right equation, you should note that a similar equation applies to an expanding spherical ball of radius a(t) with uniform mass density V. But in the case of the ball, the mass density would decrease as the ball expands. The universe is different — it can expand without diluting its mass density, so the rate of expansion \dot a / a does not slow down as the expansion proceeds.)

During inflation, the scalar field \phi and therefore the potential energy V(\phi) were changing slowly; it’s a good approximation to assume V is constant. Then the solution is

a(t) \sim a(0) e^{Ht},

where H, the Hubble constant during inflation, is

H \sim \frac{\sqrt{V}}{m_P}.

To explain the smoothness of the observed universe, we require at least 50 “e-foldings” of inflation before the universe reheated — that is, inflation should have lasted for a time at least 50 H^{-1}.

Slow rolling

During inflation the inflaton \phi rolls slowly, so slowly that friction dominates inertia — this friction results from the cosmic expansion. The speed of rolling \dot \phi is determined by

H \dot \phi \sim -V'(\phi).

Here V'(\phi) is the slope of the potential, so the right-hand side is the force exerted by the potential, which matches the frictional force on the left-hand side. The coefficient of \dot \phi has to be H on dimensional grounds. (Here I have blown another factor of 3, but let’s not worry about that.)

Density perturbations

The trickiest thing we need to understand is how inflation produced the density perturbations which later seeded the formation of galaxies. There are several steps to the argument.

Quantum fluctuations of the inflaton

As the universe inflates, the inflaton field is subject to quantum fluctuations, where the size of the fluctuation depends on its wavelength. Due to inflation, the wavelength increases rapidly, like e^{Ht}, and once the wavelength gets large compared to H^{-1}, there isn’t enough time for the fluctuation to wiggle — it gets “frozen in.” Much later, long after the reheating of the universe, the oscillation period of the wave becomes comparable to the age of the universe, and then it can wiggle again. (We say that the fluctuations “cross the horizon” at that stage.) Observations of the anisotropy of the microwave background have determined how big the fluctuations are at the time of horizon crossing. What does inflation theory say about that?

Well, first of all, how big are the fluctuations when they leave the horizon during inflation? Then the wavelength is H^{-1} and the universe is expanding at the rate H, so H is the only thing the magnitude of the fluctuations could depend on. Since the field \phi has the same dimensions as H, we conclude that fluctuations have magnitude

\delta \phi \sim H.

From inflaton fluctuations to density perturbations

Reheating occurs abruptly when the inflaton field reaches a particular value. Because of the quantum fluctuations, some horizon volumes have larger than average values of \phi and some have smaller than average values; hence different regions reheat at slightly different times. The energy density in regions that reheat earlier starts to be reduced by expansion (“red shifted”) earlier, so these regions have a smaller than average energy density. Likewise, regions that reheat later start to red shift later, and wind up having larger than average density.

When we compare different regions of comparable size, we can find the typical (root-mean-square) fluctuations \delta t in the reheating time, knowing the fluctuations in \phi and the rolling speed \dot \phi:

\delta t \sim \frac{\delta \phi}{\dot \phi} \sim \frac{H}{\dot\phi}.

Small fractional fluctuations in the scale factor a right after reheating produce comparable small fractional fluctuations in the energy density \rho. The expansion rate right after reheating roughly matches the expansion rate H right before reheating, and so we find that the characteristic size of the density perturbations is

\delta_S\equiv\left(\frac{\delta \rho}{\rho}\right)_{hor} \sim \frac{\delta a}{a} \sim \frac{\dot a}{a} \delta t\sim \frac{H^2}{\dot \phi}.

The subscript hor serves to remind us that this is the size of density perturbations as they cross the horizon, before they get a chance to grow due to gravitational instabilities. We have found our first important conclusion: The density perturbations have a size determined by the Hubble constant H and the rolling speed \dot \phi of the inflaton, up to a factor of order one which we have not tried to keep track of. Insofar as the Hubble constant and rolling speed change slowly during inflation, these density perturbations have a strength which is nearly independent of the length scale of the perturbation. From here on we will denote this dimensionless scale of the fluctuations by \delta_S, where the subscript S stands for “scalar”.

Perturbations in terms of the potential

Putting together \dot \phi \sim -V' / H and H^2 \sim V/{m_P}^2 with our expression for \delta_S, we find

\delta_S^2 \sim \frac{H^4}{\dot\phi^2}\sim \frac{H^6}{V'^2} \sim \frac{1}{{m_P}^6}\frac{V^3}{V'^2}.

The observed density perturbations are telling us something interesting about the scalar field potential during inflation.

Gravitational waves and the meaning of r

The gravitational field as well as the inflaton field is subject to quantum fluctuations during inflation. We call these tensor fluctuations to distinguish them from the scalar fluctuations in the energy density. The tensor fluctuations have an effect on the microwave anisotropy which can be distinguished in principle from the scalar fluctuations. We’ll just take that for granted here, without worrying about the details of how it’s done.

While a scalar field fluctuation with wavelength \lambda and strength \delta \phi carries energy density \sim \delta\phi^2 / \lambda^2, a fluctuation of the dimensionless gravitation field h with wavelength \lambda and strength \delta h carries energy density \sim m_P^2 \delta h^2 / \lambda^2. Applying the same dimensional analysis we used to estimate \delta \phi at horizon crossing to the rescaled field h/m_P, we estimate the strength \delta_T of the tensor fluctuations as

\delta_T^2 \sim \frac{H^2}{m_P^2}\sim \frac{V}{m_P^4}.

From observations of the CMB anisotropy we know that \delta_S\sim 10^{-5}, and now BICEP2 claims that the ratio

r = \frac{\delta_T^2}{\delta_S^2}

is about r\sim 0.2 at an angular scale on the sky of about one degree. The conclusion (being a little more careful about the O(1) factors this time) is

V^{1/4} \sim 2 \times 10^{16}~GeV \left(\frac{r}{0.2}\right)^{1/4}.

This is our second important conclusion: The energy density during inflation defines a mass scale, which turns our to be 2 \times 10^{16}~GeV for the observed value of r. This is a very interesting finding because this mass scale is not so far below the Planck scale, where quantum gravity kicks in, and is in fact pretty close to theoretical estimates of the unification scale in supersymmetric grand unified theories. If this mass scale were a factor of 2 smaller, then r would be smaller by a factor of 16, and hence much harder to detect.

Rolling, rolling, rolling, …

Using \delta_S^2 \sim H^4/\dot\phi^2, we can express r as

r = \frac{\delta_T^2}{\delta_S^2}\sim \frac{\dot\phi^2}{m_P^2 H^2}.

It is convenient to measure time in units of the number N = H t of e-foldings of inflation, in terms of which we find

\frac{1}{m_P^2} \left(\frac{d\phi}{dN}\right)^2\sim r;

Now, we know that for inflation to explain the smoothness of the universe we need N larger than 50, and if we assume that the inflaton rolls at a roughly constant rate during N e-foldings, we conclude that, while rolling, the change in the inflaton field is

\frac{\Delta \phi}{m_P} \sim N \sqrt{r}.

This is our third important conclusion — the inflaton field had to roll a long, long, way during inflation — it changed by much more than the Planck scale! Putting in the O(1) factors we have left out reduces the required amount of rolling by about a factor of 3, but we still conclude that the rolling was super-Planckian if r\sim 0.2. That’s curious, because when the scalar field strength is super-Planckian, we expect the kind of effective field theory we have been implicitly using to be a poor approximation because quantum gravity corrections are large. One possible way out is that the inflaton might have rolled round and round in a circle instead of in a straight line, so the field strength stayed sub-Planckian even though the distance traveled was super-Planckian.

Spectral tilt

As the inflaton rolls, the potential energy, and hence also the Hubble constant H, change during inflation. That means that both the scalar and tensor fluctuations have a strength which is not quite independent of length scale. We can parametrize the scale dependence in terms of how the fluctuations change per e-folding of inflation, which is equivalent to the change per logarithmic length scale and is called the “spectral tilt.”

To keep things simple, let’s suppose that the rate of rolling is constant during inflation, at least over the length scales for which we have data. Using \delta_S^2 \sim H^4/\dot\phi^2, and assuming \dot\phi is constant, we estimate the scalar spectral tilt as

-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim - \frac{4 \dot H}{H^2}.

Using \delta_T^2 \sim H^2/m_P^2, we conclude that the tensor spectral tilt is half as big.

From H^2 \sim V/m_P^2, we find

\dot H \sim \frac{1}{2} \dot \phi \frac{V'}{V} H,

and using \dot \phi \sim -V'/H we find

-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim \frac{V'^2}{H^2V}\sim m_P^2\left(\frac{V'}{V}\right)^2\sim \left(\frac{V}{m_P^4}\right)\left(\frac{m_P^6 V'^2}{V^3}\right)\sim \delta_T^2 \delta_S^{-2}\sim r.

Putting in the numbers more carefully we find a scalar spectral tilt of r/4 and a tensor spectral tilt of r/8.

This is our last important conclusion: A relatively large value of r means a significant spectral tilt. In fact, even before the BICEP2 results, the CMB anisotropy data already supported a scalar spectral tilt of about .04, which suggested something like r \sim .16. The BICEP2 detection of the tensor fluctuations (if correct) has confirmed that suspicion.

Summing up

If you have stuck with me this far, and you haven’t seen this stuff before, I hope you’re impressed. Of course, everything I’ve described can be done much more carefully. I’ve tried to convey, though, that the emerging story seems to hold together pretty well. Compared to last week, we have stronger evidence now that inflation occurred, that the mass scale of inflation is high, and that the scalar and tensor fluctuations produced during inflation have been detected. One prediction is that the tensor fluctuations, like the scalar ones, should have a notable spectral tilt, though a lot more data will be needed to pin that down.

I apologize to the experts again, for the sloppiness of these arguments. I hope that I have at least faithfully conveyed some of the spirit of inflation theory in a way that seems somewhat accessible to the uninitiated. And I’m sorry there are no references, but I wasn’t sure which ones to include (and I was too lazy to track them down).

It should also be clear that much can be done to sharpen the confrontation between theory and experiment. A whole lot of fun lies ahead.

Added notes (3/25/2014):

Okay, here’s a good reference, a useful review article by Baumann. (I found out about it on Twitter!)

From Baumann’s lectures I learned a convenient notation. The rolling of the inflaton can be characterized by two “potential slow-roll parameters” defined by

\epsilon = \frac{m_p^2}{2}\left(\frac{V'}{V}\right)^2,\quad \eta = m_p^2\left(\frac{V''}{V}\right).

Both parameters are small during slow rolling, but the relationship between them depends on the shape of the potential. My crude approximation (\epsilon = \eta) would hold for a quadratic potential.

We can express the spectral tilt (as I defined it) in terms of these parameters, finding 2\epsilon for the tensor tilt, and 6 \epsilon - 2\eta for the scalar tilt. To derive these formulas it suffices to know that \delta_S^2 is proportional to V^3/V'^2, and that \delta_T^2 is proportional to H^2; we also use

3H\dot \phi = -V', \quad 3H^2 = V/m_P^2,

keeping factors of 3 that I left out before. (As a homework exercise, check these formulas for the tensor and scalar tilt.)

It is also easy to see that r is proportional to \epsilon; it turns out that r = 16 \epsilon. To get that factor of 16 we need more detailed information about the relative size of the tensor and scalar fluctuations than I explained in the post; I can’t think of a handwaving way to derive it.

We see, though, that the conclusion that the tensor tilt is r/8 does not depend on the details of the potential, while the relation between the scalar tilt and r does depend on the details. Nevertheless, it seems fair to claim (as I did) that, already before we knew the BICEP2 results, the measured nonzero scalar spectral tilt indicated a reasonably large value of r.

Once again, we’re lucky. On the one hand, it’s good to have a robust prediction (for the tensor tilt). On the other hand, it’s good to have a handle (the scalar tilt) for distinguishing among different inflationary models.

One last point is worth mentioning. We have set Planck’s constant \hbar equal to one so far, but it is easy to put the powers of \hbar back in using dimensional analysis (we’ll continue to assume the speed of light c is one). Since Newton’s constant G has the dimensions of length/energy, and the potential V has the dimensions of energy/volume, while \hbar has the dimensions of energy times length, we see that

\delta_T^2 \sim \hbar G^2V.

Thus the production of gravitational waves during inflation is a quantum effect, which would disappear in the limit \hbar \to 0. Likewise, the scalar fluctuation strength \delta_S^2 is also O(\hbar), and hence also a quantum effect.

Therefore the detection of primordial gravitational waves by BICEP2, if correct, confirms that gravity is quantized just like the other fundamental forces. That shouldn’t be a surprise, but it’s nice to know.

My 10 biggest thrills

Wow!

BICEP2 results for the ratio r of gravitational wave perturbations to density perturbations, and the density perturbation spectral tilt n.

Evidence for gravitational waves produced during cosmic inflation. BICEP2 results for the ratio r of gravitational wave perturbations to density perturbations, and the density perturbation spectral tilt n.

Like many physicists, I have been reflecting a lot the past few days about the BICEP2 results, trying to put them in context. Other bloggers have been telling you all about it (here, here, and here, for example); what can I possibly add?

The hoopla this week reminds me of other times I have been really excited about scientific advances. And I recall some wise advice I received from Sean Carroll: blog readers like lists.  So here are (in chronological order)…

My 10 biggest thrills (in science)

This is a very personal list — your results may vary. I’m not saying these are necessarily the most important discoveries of my lifetime (there are conspicuous omissions), just that, as best I can recall, these are the developments that really started my heart pounding at the time.

1) The J/Psi from below (1974)

I was a senior at Princeton during the November Revolution. I was too young to appreciate fully what it was all about — having just learned about the Weinberg-Salam model, I thought at first that the Z boson had been discovered. But by stalking the third floor of Jadwin I picked up the buzz. No, it was charm! The discovery of a very narrow charmonium resonance meant we were on the right track in two ways — charm itself confirmed ideas about the electroweak gauge theory, and the narrowness of the resonance fit in with the then recent idea of asymptotic freedom. Theory triumphant!

2) A magnetic monopole in Palo Alto (1982)

By 1982 I had been thinking about the magnetic monopoles in grand unified theories for a few years. We thought we understood why no monopoles seem to be around. Sure, monopoles would be copiously produced in the very early universe, but then cosmic inflation would blow them away, diluting their density to a hopelessly undetectable value. Then somebody saw one …. a magnetic monopole obediently passed through Blas Cabrera’s loop of superconducting wire, producing a sudden jump in the persistent current. On Valentine’s Day!

According to then current theory, the monopole mass was expected to be about 10^16 GeV (10 million billion times heavier than a proton). Had Nature really been so kind as the bless us with this spectacular message from an staggeringly high energy scale? It seemed too good to be true.

It was. Blas never detected another monopole. As far as I know he never understood what glitch had caused the aberrant signal in his device.

3) “They’re green!” High-temperature superconductivity (1987)

High-temperature superconductors were discovered in 1986 by Bednorz and Mueller, but I did not pay much attention until Paul Chu found one in early 1987 with a critical temperature of 77 K. Then for a while the critical temperature seemed to be creeping higher and higher on an almost daily basis, eventually topping 130K …. one wondered whether it might go up, up, up forever.

It didn’t. Today 138K still seems to be the record.

My most vivid memory is that David Politzer stormed into my office one day with a big grin. “They’re green!” he squealed. David did not mean that high-temperature superconductors would be good for the environment. He was passing on information he had just learned from Phil Anderson, who happened to be visiting Caltech: Chu’s samples were copper oxides.

4) “Now I have mine” Supernova 1987A (1987)

What was most remarkable and satisfying about the 1987 supernova in the nearby Large Magellanic Cloud was that the neutrinos released in a ten second burst during the stellar core collapse were detected here on earth, by gigantic water Cerenkov detectors that had been built to test grand unified theories by looking for proton decay! Not a truly fundamental discovery, but very cool nonetheless.

Soon after it happened some of us were loafing in the Lauritsen seminar room, relishing the good luck that had made the detection possible. Then Feynman piped up: “Tycho Brahe had his supernova, Kepler had his, … and now I have mine!” We were all silent for a few seconds, and then everyone burst out laughing, with Feynman laughing the hardest. It was funny because Feynman was making fun of his own gargantuan ego. Feynman knew a good gag, and I heard him use this line at a few other opportune times thereafter.

5) Science by press conference: Cold fusion (1989)

The New York Times was my source for the news that two chemists claimed to have produced nuclear fusion in heavy water using an electrochemical cell on a tabletop. I was interested enough to consult that day with our local nuclear experts Charlie Barnes, Bob McKeown, and Steve Koonin, none of whom believed it. Still, could it be true?

I decided to spend a quiet day in my office, trying to imagine ways to induce nuclear fusion by stuffing deuterium into a palladium electrode. I came up empty.

My interest dimmed when I heard that they had done a “control” experiment using ordinary water, had observed the same excess heat as with heavy water, and remained just as convinced as before that they were observing fusion. Later, Caltech chemist Nate Lewis gave a clear and convincing talk to the campus community debunking the original experiment.

6) “The face of God” COBE (1992)

I’m often too skeptical. When I first heard in the early 1980s about proposals to detect the anisotropy in the cosmic microwave background, I doubted it would be possible. The signal is so small! It will be blurred by reionization of the universe! What about the galaxy! What about the dust! Blah, blah, blah, …

The COBE DMR instrument showed it could be done, at least at large angular scales, and set the stage for the spectacular advances in observational cosmology we’ve witnessed over the past 20 years. George Smoot infamously declared that he had glimpsed “the face of God.” Overly dramatic, perhaps, but he was excited! And so was I.

7) “83 SNU” Gallex solar neutrinos (1992)

Until 1992 the only neutrinos from the sun ever detected were the relatively high energy neutrinos produced by nuclear reactions involving boron and beryllium — these account for just a tiny fraction of all neutrinos emitted. Fewer than expected were seen, a puzzle that could be resolved if neutrinos have mass and oscillate to another flavor before reaching earth. But it made me uncomfortable that the evidence for solar neutrino oscillations was based on the boron-beryllium side show, and might conceivably be explained just by tweaking the astrophysics of the sun’s core.

The Gallex experiment was the first to detect the lower energy pp neutrinos, the predominant type coming from the sun. The results seemed to confirm that we really did understand the sun and that solar neutrinos really oscillate. (More compelling evidence, from SNO, came later.) I stayed up late the night I heard about the Gallex result, and gave a talk the next day to our particle theory group explaining its significance. The talk title was “83 SNU” — that was the initially reported neutrino flux in Solar Neutrino Units, later revised downward somewhat.

8) Awestruck: Shor’s algorithm (1994)

I’ve written before about how Peter Shor’s discovery of an efficient quantum algorithm for factoring numbers changed my life. This came at a pivotal time for me, as the SSC had been cancelled six months earlier, and I was growing pessimistic about the future of particle physics. I realized that observational cosmology would have a bright future, but I sensed that theoretical cosmology would be dominated by data analysis, where I would have little comparative advantage. So I became a quantum informationist, and have not regretted it.

9) The Higgs boson at last (2012)

The discovery of the Higgs boson was exciting because we had been waiting soooo long for it to happen. Unable to stream the live feed of the announcement, I followed developments via Twitter. That was the first time I appreciated the potential value of Twitter for scientific communication, and soon after I started to tweet.

10) A lucky universe: BICEP2 (2014)

Many past experiences prepared me to appreciate the BICEP2 announcement this past Monday.

I first came to admire Alan Guth‘s distinctive clarity of thought in the fall of 1973 when he was the instructor for my classical mechanics course at Princeton (one of the best classes I ever took). I got to know him better in the summer of 1979 when I was a graduate student, and Alan invited me to visit Cornell because we were both interested in magnetic monopole production  in the very early universe. Months later Alan realized that cosmic inflation could explain the isotropy and flatness of the universe, as well as the dearth of magnetic monopoles. I recall his first seminar at Harvard explaining his discovery. Steve Weinberg had to leave before the seminar was over, and Alan called as Steve walked out, “I was hoping to hear your reaction.” Steve replied, “My reaction is applause.” We all felt that way.

I was at a wonderful workshop in Cambridge during the summer of 1982, where Alan and others made great progress in understanding the origin of primordial density perturbations produced from quantum fluctuations during inflation (Bardeen, Steinhardt, Turner, Starobinsky, and Hawking were also working on that problem, and they all reached a consensus by the end of the three-week workshop … meanwhile I was thinking about the cosmological implications of axions).

I also met Andrei Linde at that same workshop, my first encounter with his mischievous grin and deadpan wit. (There was a delegation of Russians, who split their time between Xeroxing papers and watching the World Cup on TV.) When Andrei visited Caltech in 1987, I took him to Disneyland, and he had even more fun than my two-year-old daughter.

During my first year at Caltech in 1984, Mark Wise and Larry Abbott told me about their calculations of the gravitational waves produced during inflation, which they used to derive a bound on the characteristic energy scale driving inflation, a few times 10^16 GeV. We mused about whether the signal might turn out to be detectable someday. Would Nature really be so kind as to place that mass scale below the Abbott-Wise bound, yet high enough (above 10^16 GeV) to be detectable? It seemed unlikely.

Last week I caught up with the rumors about the BICEP2 results by scanning my Twitter feed on my iPad, while still lying in bed during the early morning. I immediately leapt up and stumbled around the house in the dark, mumbling to myself over and over again, “Holy Shit! … Holy Shit! …” The dog cast a curious glance my way, then went back to sleep.

Like millions of others, I was frustrated Monday morning, trying to follow the live feed of the discovery announcement broadcast from the hopelessly overtaxed Center for Astrophysics website. I was able to join in the moment, though, by following on Twitter, and I indulged in a few breathless tweets of my own.

Many of his friends have been thinking a lot these past few days about Andrew Lange, who had been the leader of the BICEP team (current senior team members John Kovac and Chao-Lin Kuo were Caltech postdocs under Andrew in the mid-2000s). One day in September 2007 he sent me an unexpected email, with the subject heading “the bard of cosmology.” Having discovered on the Internet a poem I had written to introduce a seminar by Craig Hogan, Andrew wrote:

“John,

just came across this – I must have been out of town for the event.

l love it.

it will be posted prominently in our lab today (with “LISA” replaced by “BICEP”, and remain our rallying cry till we detect the B-mode.

have you set it to music yet?

a”

I lifted a couplet from that poem for one of my tweets (while rumors were swirling prior to the official announcement):

We’ll finally know how the cosmos behaves
If we can detect gravitational waves.

Assuming the BICEP2 measurement r ~ 0.2 is really a detection of primordial gravitational waves, we have learned that the characteristic mass scale during inflation is an astonishingly high 2 X 10^16 GeV. Were it a factor of 2 smaller, the signal would have been far too small to detect in current experiments. This time, Nature really is on our side, eagerly revealing secrets about physics at a scale far, far beyond what we will every explore using particle accelerators. We feel lucky.

We physicists can never quite believe that the equations we scrawl on a notepad actually have something to do with the real universe. You would think we’d be used to that by now, but we’re not — when it happens we’re amazed. In my case, never more so than this time.

The BICEP2 paper, a historic document (if the result holds up), ends just the way it should:

“We dedicate this paper to the memory of Andrew Lange, whom we sorely miss.”

Fundamental Physics Prize Prediction: Green and Schwarz

Michael Green

Michael Green

John Schwarz

John Schwarz

The big news today is the announcement of the nominees for the 2014 Fundamental Physics Prize: (1) Michael Green and John Schwarz, for pioneering contributions to string theory, (2) Joseph Polchinski, for discovering the central role of D-branes in string theory, and (3) Andrew Strominger and Cumrun Vafa, for discovering (using D-branes) the microscopic origin of black hole entropy in string theory. As in past years, all the nominees are marvelously deserving. The winner of the $3 million prize will be announced in San Francisco on December 12; the others will receive the $300,000 Physics Frontiers Prize.

I wrote about my admiration for Joe Polchinski when he was nominated last year, and I have also greatly admired the work of Strominger and Vafa for many years. But the story of Green and Schwarz is especially compelling. String theory, which was originally proposed as a theory of the strong interaction, had been an active research area from 1968 through the early 70s. But when asymptotic freedom was discovered in 1973, and quantum chromodynamics became clearly established as the right theory of the strong interaction, interest in string theory collapsed. Even the 1974 proposal by Scherk and Schwarz that string theory is really a compelling candidate for a quantum theory of gravity failed to generate much excitement.

A faithful few continued to develop string theory through the late 70s and early 80’s, particularly Green and Schwarz, who began collaborating in 1979. Together they clarified the different variants of the theory, which they named Types I, IIA, and IIB, and which were later recognized as different solutions to a single underlying theory (sometimes called M-theory). In retrospect, Green and Schwarz were making remarkable progress, but were still largely ignored.

In 1983, Luis Alvarez-Gaume and Edward Witten analyzed the gravitational anomalies that afflict higher dimensional “chiral” theories (in which left-handed and right-handed particles behave differently), and discovered a beautiful cancellation of these anomalies in the Type IIB string theory. But anomalies, which render a theory inconsistent, seemed to be a nail in the coffin of Type I theory, at that time the best hope for uniting gravitation with the other fundamental (gauge) interactions.

Then, working together at the Aspen Center for Physics during the summer of 1984, Green and Schwarz discovered an even more miraculous cancellation of anomalies in Type I string theory, which worked for only one possible gauge group: SO(32). (Within days they and others found that anomalies cancel for E8 X E8 as well, which provided the impetus for the invention of the heterotic string theory.) The anomaly cancellation drove a surge of enthusiasm for string theory as a unified theory of fundamental physics. The transformation of string theory from a backwater to the hottest topic in physics occurred virtually overnight. It was an exciting time.

When John turned 60 in 2001, I contributed a poem to a book assembled in his honor, hoping to capture in the poem the transformation that Green and Schwarz fomented (and also to express irritation about the widespread misspelling of “Schwarz”). I have appended the poem below, along with the photo of myself I included at the time to express my appreciation for strings.

I’ll be delighted if Polchinski, or Strominger and Vafa win the prize — they deserve it. But it will be especially satisfying if Green and Schwarz win. They started it all, and refused to give up.

To John Schwarz

Thirty years ago or more
John saw what physics had in store.
He had a vision of a string
And focused on that one big thing.

But then in nineteen-seven-three
Most physicists had to agree
That hadrons blasted to debris
Were well described by QCD.

The string, it seemed, by then was dead.
But John said: “It’s space-time instead!
The string can be revived again.
Give masses twenty powers of ten!”

Then Dr. Green and Dr. Black,
Writing papers by the stack,
Made One, Two-A, and Two-B glisten.
Why is it none of us would listen?

We said, “Who cares if super tricks
Bring D to ten from twenty-six?
Your theory must have fatal flaws.
Anomalies will doom your cause.”

If you weren’t there you couldn’t know
The impact of that mightly blow:
“The Green-Schwarz theory could be true —
It works for S-O-thirty-two!”

Then strings of course became the rage
And young folks of a certain age
Could not resist their siren call:
One theory that explains it all.

Because he never would give in,
Pursued his dream with discipline,
John Schwarz has been a hero to me.
So please, don’t spell it with a “t”!

Expressing my admiration for strings in 2001

Expressing my admiration for strings in 2001.

Frontiers of Quantum Information Science

Just a few years ago, if you wanted to look for recent research articles about quantum entanglement, you would check out the quantum physics [quant-ph] archive at arXiv.org. Since 1994, quant-ph has been the central repository for papers about quantum computing and the broader field of quantum information science. But over the past few years there has been a notable change. Increasingly, exciting papers about quantum entanglement are found at the condensed matter [cond-mat] and high energy physics – theory [hep-th] archives.

I don’t know for sure, but that trend may have had something to do with an invitation I received a few months ago from David Gross, to organize the next Jerusalem Winter School in Theoretical Physics. David has been the General Director of the School for, well, I’m not sure how long, but it must be a long time. In the past, the topic of the school has rotated between particle physics, condensed matter physics, and astrophysics. Every year, a group of world-class scientists gives lectures on cutting-edge research for an enthusiastic audience of postdoctoral scholars and advanced graduate students.

David suggested that a good topic for the next school would be “quantum information, broadly envisaged — from quantum computing to strongly correlated electrons.” After some hesitation for family reasons, I embraced this opportunity to amplify David’s message: quantum information has arrived as a major subfield of physics, and its relevance to other areas of physics is becoming broadly appreciated.

I’m not good at organizing things myself, so I recruited two friends who are very good at it to help me: Michael Ben-Or and Patrick Hayden. As the local organizer at The Hebrew University, Michael has to do a lot of the hard work that I’m glad to avoid. We decided to call the school “Frontiers of Quantum Information Science,” and put together a slate of 10 lecturers, which I’m very excited about. The lectures will cover the core areas of quantum information, as well as some of the important ways in which quantum information relates to quantum matter, quantum field theory, and quantum gravity. Each lecturer will give three or four ninety-minute lectures, on these topics:

Scott Aaronson (MIT), Quantum complexity and quantum optics
David DiVincenzo (Aachen), Quantum computing with superconducting circuits
Daniel Harlow (Princeton), Black holes and quantum information
Michal Horodecki (Gdansk), Quantum information and thermodynamics
Stephen Jordan (NIST), Quantum algorithms
Rob Myers (Perimeter), Entanglement in quantum field theory
Renato Renner (ETH), Quantum foundations
Ady Stern (Weizmann), Topological quantum computing
Barbara Terhal (Aachen), Quantum error correction
Frank Verstraete (Vienna), Quantum information and quantum matter

The school will run from 30 December 2013 to 9 January 2014 at the Israel Institute for Advanced Studies at The Hebrew University in Jerusalem. If you are interested in attending, please visit the website for more information and fill out the registration form by November 1. I hope you can come — it’s going to be a lot of fun.

Rereading the first paragraph of this post, I got slightly nervous about whether the trend I described can be documented, so I have done a little bit of research. Going back to 2005, I plotted the number of papers with the word “entanglement” in the title on quant-ph, cond-mat, hep-th, and also the general relativity and quantum cosmology [gr-qc] archive. For 2013, I rescaled the data for the year up to now, taking into account that Sep. 22 is the 265th day of the year. I didn’t make any adjustment for papers being cross-listed on multiple archives.

Here is the data for quant-ph:quantph-plot-pdfIt’s remarkably flat. Here is the aggregated data for the other three archives:arxiv-plot-pdfIt’s pretty clear that something started to happen around 2010. I realize one could do a much more serious study of this issue, but since I was only willing to spend an hour on it, I feel vindicated.

Free Feynman!

Last Friday the 13th was a lucky day for those who love physics — The online html version of Volume 1 of the Feynman Lectures on Physics (FLP) was released! Now anyone with Internet access and a web browser can enjoy these unique lectures for free. They look beautiful.

Mike Gottlieb at Caltech on 20 September 2013. He's the one on the right.

Mike Gottlieb at Caltech on 20 September 2013. He’s the one on the right.

On the day of release, over 86,000 visitors viewed the website, and the Amazon sales rank of the paperback version of FLP leapt over the weekend from 67,000 to 12,000. My tweet about the release was retweeted over 150 times (my most retweets ever).

Free html versions of Volumes 2 and 3 are in preparation. Soon pdf versions of all three volumes will be offered for sale, each available in both desktop and tablet versions at a price comparable to the cost of the paperback editions. All these happy developments resulted from a lot of effort by many people. You can learn about some of the history and the people involved from Kip Thorne’s 2010 preface to the print edition.

A hero of the story is Mike Gottlieb, who spends most of his time in Costa Rica, but passed through Caltech yesterday for a brief visit. Mike entered the University of Maryland to study mathematics at age 15 and at age 16 began a career as a self-employed computer software consultant. In 1999, when Mike was 39,  a chance meeting with Feynman’s friend and co-author Ralph Leighton changed Mike’s life.

At Ralph’s suggestion, Mike read Feynman’s Lectures on Computation. Impressed by Feynman’s insights and engaging presentation style, Mike became eager to learn more about physics; again following Ralph’s suggestion, he decided to master the Feynman Lectures on Physics. Holed up at a rented farm in Costa Rica without a computer, he pored over the lectures for six months, painstakingly compiling a handwritten list of about 200 errata.

Kip’s preface picks up the story at that stage. I won’t repeat all that, except to note two pivotal developments. Rudi Pfeiffer was a postdoc at the University of Vienna in 2006 when, frustrated by the publisher’s resistance to correcting errata that he and others had found, he (later joined by Gottlieb) began converting FLP to LaTeX, the modern computer system for typesetting mathematics. Eventually, all the figures were redrawn in electronic form as scalable vector graphics, paving the way for a “New Millenium Edition” of FLP (published in 2011), as well as other electronically enhanced editions planned for the future. Except that, before all that could happen, Caltech’s Intellectual Property Counsel Adam Cochran had to untangle a thicket of conflicting publishing rights, which I have never been able to understand in detail and therefore will not attempt to explain.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

Rudi Pfeiffer and Mike Gottlieb at Caltech in 2008.

The proposal to offer an html version for free has been enthusiastically pursued by Caltech and has received essential financial support from Carver Mead. The task of converting Volume 1 from LaTeX to html was carried out for a fee by Caltech alum Michael Hartl; Gottlieb is doing the conversion himself for the other volumes, which are already far along.

Aside from the pending html editions of Volumes 2 and 3, and the pdf editions of all three volumes, there is another very exciting longer-term project in the works — the html will provide the basis for a Multimedia Edition of FLP. Audio for every one of Feynman’s lectures was recorded, and has been digitally enhanced by Ralph Leighton. In addition, the blackboards were photographed for almost all of the lectures. The audio and photos will be embedded in the Multimedia Edition, possibly accompanied by some additional animations and “Ken Burns style” movies. The audio in particular is great fun, bringing to life Feynman the consummate performer. For the impatient, a multimedia version of six of the lectures is already available as an iBook. To see a quick preview, watch Adam’s TEDxCaltech talk.

Mike Gottlieb has now devoted 13 years of his life to enhancing FLP and bringing the lectures to a broader audience, receiving little monetary compensation. I asked him yesterday about his motivation, and his answer surprised me somewhat. Mike wants to be able to look back at his life feeling that he has made a bigger contribution to the world than merely writing code and making money. He would love to have a role in solving the great open problems in physics, in particular the problem of reconciling general relativity with quantum mechanics, but feels it is beyond his ability to solve those problems himself. Instead, Mike feels he can best facilitate progress in physics by inspiring other very talented young people to become physicists and work on the most important problems. In Mike’s view, there is no better way of inspiring students to pursue physics than broadening access to the Feynman Lectures on Physics!

What’s inside a black hole?

I have a multiple choice question for you.

What’s inside a black hole?

(A) An unlimited amount of stuff.
(B) Nothing at all.
(C) A huge but finite amount of stuff, which is also outside the black hole.
(D) None of the above.

The first three answers all seem absurd, boosting the credibility of (D). Yet … at the “Rapid Response Workshop” on black holes I attended last week at the KITP in Santa Barbara (and which continues this week), most participants were advocating some version of (A), (B), or (C), with varying degrees of conviction.

When physicists get together to talk about black holes, someone is bound to draw a cartoon like this one:

Penrose diagram depicting the causal structure of a black hole spacetime.

Part of a Penrose diagram depicting the causal structure of a black hole spacetime.

I’m sure I’ve drawn and contemplated some version of this diagram hundreds of times over the past 25 years in the privacy of my office, and many times in public discussions (including at least five times during the talk I gave at the KITP). This picture vividly captures the defining property of a black hole, found by solving Einstein’s classical field equations for gravitation: once you go inside there is no way out. Instead you are unavoidably drawn to the dreaded singularity, where known laws of physics break down (and the picture can no longer be trusted). If taken seriously, the picture says that whatever falls into a black hole is gone forever, at least from the perspective of observers who stay outside.

But for nearly 40 years now, we have known that black holes can shed their mass by emitting radiation, and presumably this process continues until the black hole disappears completely. If we choose to, we can maintain the black hole for as long as we please by feeding it new stuff at the same rate that radiation carries energy away. What I mean by option (A) is that  the radiation is completely featureless, carrying no information about what kind of stuff fell in. That means we can hide as much information as we please inside a black hole of a given mass.

On the other hand, the beautiful theory of black hole thermodynamics indicates that the entropy of a black hole is determined by its mass. For all other systems we know of besides black holes, the entropy of the system quantifies how much information we can hide in the system. If (A) is the right answer, then black holes would be fundamentally different in this respect, able to hide an unlimited amount of information even though their entropy is finite. Maybe that’s possible, but it would be rather disgusting, a reason to dislike answer (A).

There is another way to argue that (A) is not the right answer, based on what we call AdS/CFT duality. AdS just describes a consistent way to put a black hole in a “bottle,” so we can regard the black hole together with the radiation outside it as a closed system. Now, in gravitation it is crucial to focus on properties of spacetime that do not depend on the observer’s viewpoint; otherwise we can easily get very confused. The best way to be sure we have a solid way of describing things is to pay attention to what happens at the boundary of the spacetime, the walls of the bottle — that’s what CFT refers to. AdS/CFT provides us with tools for describing what happens when a black hole forms and evaporates, phrased entirely in terms of what happens on the walls of the bottle. If we can describe the physics perfectly by sticking to the walls of the bottle, always staying far away from the black hole, there doesn’t seem to be anyplace to hide an unlimited amount of stuff.

At the KITP, both Bill Unruh and Bob Wald argued forcefully for (A). They acknowledge the challenge of understanding the meaning of black hole entropy and of explaining why the AdS/CFT argument is wrong. But neither is willing to disavow the powerful message conveyed by that telling diagram of the black hole spacetime. As Bill said: “There is all that stuff that fell in and it crashed into the singularity and that’s it. Bye-bye.”

Adherents of (B) and (C) like to think about black hole physics from the perspective of an observer who stays outside the black hole. From that viewpoint, they say, the black hole behaves like any other system with a temperature and a finite entropy. Stuff falling in sticks to the black hole’s outer edge and gets rapidly mixed in with other stuff the black hole absorbed previously. For a black hole of a given mass, though, there is a limit to how much stuff it can hold. Eventually, what fell in comes out again, but in a form so highly scrambled as to be nearly unrecognizable.

Where the (B) and (C) camps differ concerns what happens to a brave observer who falls into a black hole. According to (C), an observer falling in crosses from the outside to the inside of a black hole peacefully, which poses a puzzle I discussed here. The puzzle arises because an uneventful crossing implies strong quantum entanglement between the region A just inside the black hole and region B just outside. On the other hand, as information leaks out of a black hole, region B should be strongly  entangled with the radiation system R emitted by the black hole long ago. Entanglement can’t be shared, so it does not make sense for B to be entangled with both A and R. What’s going on? Answer (C) resolves the puzzle by positing that A and R are not really different systems, but rather two ways to describe the same system, as I discussed here.That seems pretty crazy, because R could be far, far away from the black hole.

Answer (B) resolves the puzzle differently, by positing that region A does not actually exist, because the black hole has no interior. An observer who attempts to fall in gets a very rude surprise, striking a seething “firewall” at the last moment before passing to the inside. That seems pretty crazy, because no firewall is predicted by Einstein’s trusty equations, which are normally very successful at describing spacetime geometry.

At the workshop, Don Marolf and Raphael Bousso gave some new arguments supporting (B). Both acknowledge that we still lack a concrete picture of how firewalls are created as black holes form, but Bousso insisted that “It is time to constrain and construct the dynamics of firewalls.” Joe Polchinski emphasized that, while AdS/CFT provides a very satisfactory description of physics outside a black hole, it has not yet been able to tell us enough about the black hole interior to settle whether there are firewalls or not, at least for generic black holes formed from collapsing matter.

Lenny Susskind, Juan Maldacena, Ted Jacobson, and I all offered different perspectives on how (C) could turn out to be the right answer. We all told different stories, but perhaps each of us had at least part of the right answer. I’m not at KITP this week, but there have been further talks supporting (C) by Raju, Nomura, and the Verlindes.

I had a fun week at the KITP. If you watch the videos of the talks, you might get an occasional glimpse of me typing furiously on my laptop. It looks like I’m doing my email, but actually that’s how I take notes, which helps me to pay attention. Every once in a while I was inspired to tweet.

I have felt for a while that ideas from quantum information can help us to grasp the mysteries of quantum gravity, so I appreciated that quantum information concepts came up in many of the talks. Susskind invoked quantum error-correcting codes in discussing how sensitively the state of the Hawking radiation depends on the information it encodes, and Maldacena used tensor networks to explain how to build spacetime geometry from quantum entanglement. Scott Aaronson proposed the appropriate acronym HARD for HAwking Radiation Decoding, and argued (following Harlow and Hayden) that this task is as hard as inverting an injective one-way function, something we don’t expect quantum computers to be able to do.

In the organizational session that launched the meeting, Polchinski remarked regarding firewalls that “Nobody has the slightest idea what is going on,” and Gary Horowitz commented that “I’m still getting over the shock over how little we’ve learned in the past 30 years.” I guess that’s fair. Understanding what’s inside black holes has turned out to be remarkably subtle, making the problem more and more tantalizing. Maybe the current state of confusion regarding black hole information means that we’re on the verge of important discoveries about quantum gravity, or maybe not. In any case, invigorating discussions like what I heard last week are bound to facilitate progress.

Monopoles passing through Flatland!

Like many mathematically inclined teenagers, I was charmed when I first read the book Flatland by Edwin Abbott Abbott.* It’s a story about a Sphere who visits a two-dimensional world and tries to awaken its inhabitants to the existence of a third dimension. As perceived by Flatlanders, the Sphere is a circle which appears as a point, grows to maximum size, then shrinks and disappears.

My memories of Flatland were aroused as I read a delightful recent paper by Max Metlitski, Charlie Kane, and Matthew Fisher about magnetic monopoles and three-dimensional bosonic topological insulators. To explain why, I’ll need to recall a few elements of the theory of monopoles and of topological insulators, before returning to the connection between the two and why that reminds me of Flatland.

Flatlanders, confined to the surface of a topological insulator, are convinced by a magnetic monopole that there is a third dimension.

Flatlanders, confined to the two-dimensional surface of a topological insulator, are convinced by a magnetic monopole that a third dimension must exist.

Monopoles

Paul Dirac was no ordinary genius. Aside from formulating relativistic electron theory and predicting the existence of antimatter, Dirac launched the quantum theory of magnetic monopoles in a famous 1931 paper. Dirac envisioned a magnetic monopole as a semi-infinitely long, infinitesimally thin string of magnetic flux, such that the end of the string, where the flux spills out, seems to be a magnetic charge. For this picture to make sense, the string should be invisible. Dirac pointed out that an electron with electric charge e, transported around a string carrying flux \Phi, could detect the string (via what later came to be called the Aharonov-Bohm effect) unless the flux is an integer multiple of 2\pi\hbar /e, where \hbar is Planck’s constant. Conversely, in order for the string to be invisible, if a magnetic monopole exists with magnetic charge g_D = 2\pi\hbar /e, then all electric charges must be integer multiples of e. Thus the existence of magnetic monopoles (which have never been observed) could explain quantization of electric charge (which has been observed).

Captivated by the beauty of his own proposal, Dirac concluded his paper by remarking, “One would be surprised if Nature had made no use of it.”

Our understanding of quantized magnetic monopoles advanced again in 1979 when another extraordinary physicist, Edward Witten, discussed a generalization of Dirac’s quantization condition. Witten noted that the Lagrange density of electrodynamics could contain a term of the form

\frac{\theta e^2\hbar}{4\pi^2}~\vec{E}\cdot\vec{B},

where \vec{E} is the electric field and \vec{B} is the magnetic field. This “\theta term” may also be expressed as

\frac{\theta e^2\hbar}{8\pi^2}~ \partial^\mu\left(\epsilon_{\mu\nu\lambda\sigma}A^\nu\partial^\lambda A^\sigma \right),

where A is the vector potential, and hence is a total derivative which makes no contribution to the classical field equations of electrodynamics. But Witten realized that it can have important consequences for the quantum properties of magnetic monopoles. Specifically, the \theta term modifies the field momentum conjugate to the vector potential, which becomes

\vec{E}+\frac{\theta e^2\hbar}{4\pi^2}\vec{B}.

Because the Gauss law condition satisfied by physical quantum states is altered, for a monopole with magnetic charge m g_D , where g_D is Dirac’s minimal charge 2\pi\hbar /e and m is an integer, the allowed values of the electric charge become

q = e\left( n - \frac{\theta m}{2\pi}\right),

where n is an integer. This spectrum of allowed charges remains invariant if \theta advances by 2\pi, suggesting that the parameter \theta is actually an angular variable with period 2\pi. This periodicity of \theta can be readily verified in a theory admitting fermions with the minimal charge e. But if the charged particles are bosons then \theta turns out to be a periodic variable with period 4\pi instead.

That \theta has a different period for a bosonic theory than a fermionic one has an interesting interpretation. As Goldhaber noticed in 1976, dyons carrying both magnetic and electric charge can exhibit statistical transmutation. That is, in a purely bosonic theory, a dyon with magnetic charge g_D= 2\pi\hbar/e and electric charge ne is a fermion if n is an odd integer — when two dyons are exchanged, transport of each dyon’s electric charge in the magnetic field of the other dyon induces a sign change in the wave function. In a fermionic theory the story is different; now we can think of the dyon as a fermionic electric charge bound to a bosonic monopole. There are two canceling contributions to the exchange phase of the dyon, which is therefore a boson for any integer value of n, whether even or odd.

As \theta smoothly increases from 0 to 2\pi, the statistics (whether bosonic or fermionic) of a dyon remains fixed even as the dyon’s electric charge increases by e. For the bosonic theory with \theta = 2\pi, then, dyons with magnetic charge g_D and electric charge ne are bosons for n odd and fermions for n even, the opposite of what happens when \theta=0. For the bosonic theory, unlike the fermionic theory, we need to increase \theta by 4\pi for the physics of dyons to be fully invariant.

In 1979 Ed Witten was a postdoc at Harvard, where I was a student, though he was visiting CERN for the summer when he wrote his paper about the \theta-dependent monopole charge. I always read Ed’s papers carefully, but I gave special scrutiny to this one because magnetic monopoles were a pet interest of mine. At the time, I wondered whether the Witten effect might clarify how to realize the \theta parameter in a lattice gauge theory. But it certainly did not occur to me that the \theta-dependent electric charge of a magnetic monopole could have important implications for quantum condensed matter physics. Theoretical breakthroughs often have unexpected consequences, which may take decades to emerge.

Symmetry-protected topological phases

Okay, now let’s talk about topological insulators, a very hot topic in condensed matter physics these days. Actually, a topological insulator is a particular instance of a more general concept called a symmetry-protected topological phase of matter (or SPT phase). Consider a d-dimensional hunk of material with a (d-1)-dimensional boundary. If the material is in an SPT phase, then the physics of the d-dimensional bulk is boring — it’s just an insulator with an energy gap, admitting no low-energy propagating excitations. But the physics of the (d-1)-dimensional edge is exotic and exciting — for example the edge might support “gapless” excitations of arbitrarily low energy which can conduct electricity. The exotica exhibited by the edge is a consequence of a symmetry, and is destroyed if the symmetry is broken either explicitly or spontaneously; that is why we say the phase is “symmetry protected.”

The low-energy edge excitations can be described by a (d-1)-dimensional effective field theory. But for a typical SPT phase, this effective field theory is what we call anomalous, which means that for one reason or another the theory does not really make sense. The anomaly tells us something interesting and important, namely that the (d-1)-dimensional theory cannot be really, truly (d-1) dimensional; it can arise only at the edge of a higher-dimensional system.

This phenomenon, in which the edge does not make sense by itself without the bulk, is nicely illustrated by the integer quantum Hall effect, which occurs in a two-dimensional electron system in a high magnetic field and at low temperature, if the sample is sufficiently clean so that the electrons are highly mobile and rarely scattered by impurities. In this case the relevant symmetry is electron number, or equivalently the electric charge. At the one-dimensional edge of a two-dimensional quantum Hall sample, charge carriers move in only one direction — to the right, say, but not to the left. A theory with such chiral electric charges does not really make sense. One problem is that electric charge is not conserved — an electric field along the edge causes charge to be locally created, which makes the theory inconsistent.

The way the theory resolves this conundrum is quite remarkable. A two-dimensional strip of quantum Hall fluid has two edges, one at the top, the other at the bottom. While the top edge has only right-moving excitations, the bottom edge has only left-moving excitations. When electric charge appears on the top edge, it is simultaneously removed from the bottom edge. Rather miraculously, charge can be conveyed across the bulk from one edge to the other, even though the bulk does not have any low-energy excitations at all.

I first learned about this interplay of edge and bulk physics from a beautiful 1985 paper by Curt Callan and Jeff Harvey. They explained very lucidly how an edge theory with an anomaly and a bulk theory with an anomaly can fit together, with each solving the other’s problems. Curiously, the authors did not mention any connection with the quantum Hall effect, which had been discovered five years earlier, and I didn’t appreciate the connection myself until years later.

Topological insulators

In the case of topological insulators, the symmetries which protect the gapless edge excitations are time-reversal invariance and conserved particle number, i.e. U(1) symmetry. Though the particle number might not be coupled to an electromagnetic gauge field, it is instructive for the purpose of understanding the properties of the symmetry-protected phase to imagine that the U(1) symmetry is gauged, and then to consider the potential anomalies that could afflict this gauge symmetry. The first topological insulators conceived by theorists were envisioned as systems of non-interacting electrons whose properties were relatively easy to understand using band theory. But it was not so clear at first how interactions among the electrons might alter their exotic behavior. The wonderful thing about anomalies is that they are robust with respect to interactions. In many cases we can infer the features of anomalies by studying a theory of non-interacting particles, assured that these features survive even when the particles interact.

As have many previous authors, Metlitski et al. argue that when we couple the conserved particle number to a U(1) gauge field, the effective theory describing the bulk physics of a topological insulator in three dimensions may contain a \theta term. But wait … since the electric field is even under time reversal and the magnetic field is odd, the \theta term is T-odd; under T, \theta is mapped to -\theta, so T seems to be violated if \theta has any nonzero value. Except … we have to remember that \theta is really a periodic variable. For a fermionic topological insulator the period is 2\pi; therefore the theory with \theta = \pi is time reversal invariant; \theta = \pi maps to \theta = -\pi under T, which is equivalent to a rotation of \theta by 2\pi. For a bosonic topological insulator the period is 4\pi, which means that \theta = 2\pi is the nontrivial T-invariant value.

If we say that a “trivial” insulator (e.g., the vacuum) has \theta = 0, then we may say that a bulk material with \theta = \pi (fermionic case) or \theta = 2\pi (bosonic case) is a “nontrivial” (a.k.a. topological) insulator. At the edge of the sample, where bulk material meets vacuum, \theta must rotate suddenly by \pi (fermions) or by 2\pi (bosons). The exotic edge physics is a consequence of this abrupt change in \theta.

Monopoles in Flatland

To understand the edge physics, and in particular to grasp how fermionic and bosonic topological insulators differ, Metlitski et al. invite us to imagine a magnetic monopole with magnetic charge g_D passing through the boundary between the bulk and the surrounding vacuum. To the Flatlanders confined to the surface of the bulk sample, the passing monopole induces a sudden change in the magnetic flux through the surface by a single flux quantum g_D, which could arise due to a quantum tunneling event. What does the Flatlander see?

In a fermionic topological insulator, there is a monopole that carries charge e/2 when inside the sample (where \theta=-\pi) and charge 0 when outside (where \theta=0). Since electric charge is surely conserved in the full three-dimensional theory, the change in the monopole’s charge must be compensated by a corresponding change in the charge residing on the surface. Flatlanders are puzzled to witness a spontaneously arising excitation with charge e/2. This is an anomaly — electric charge conservation is violated, which can only make sense if Flatlanders are confined to a surface in a higher-dimensional world. Though unable to escape their surface world, the Flatlanders can be convinced by the Monopole that an extra dimension must exist.

In a bosonic topological insulator, the story is somewhat different: there is a monopole that carries electric charge 0 when inside the sample (where \theta=-2\pi) and charge -e when outside (where \theta=0). In this case, though, there are bosonic charge-e particles living on the surface. A monopole can pick up a charged particle as it passes through Flatland, so that its charge is 0 both inside the bulk sample and outside in the vacuum. Flatlanders are happy — electric charge is conserved!

But hold on … there’s still something wrong. Inside the bulk (where \theta= -2\pi) a monopole with electric charge 0 is a fermion, while outside in the vacuum (where \theta = 0) it is a boson. In the three-dimensional theory it is not possible for any local process to create an isolated fermion, so if the fermionic monopole becomes a bosonic monople as it passes through Flatland, it must leave a fermion behind. Flatlanders are puzzled to witness a spontaneously arising fermion. This is an anomaly — conservation of fermionic parity is violated, which can only make sense if Flatlanders are confined to a surface in a higher-dimensional world. Once again, the clever residents of Flatland learn from the Monopole about an extra spatial dimension, without ever venturing outside their two-dimensional home.

Topological order gets edgy

This post is already pretty long and I should wrap it up. Before concluding I’ll remark that the theory of symmetry-protected phases has been developing rapidly in recent months.

In particular, a new idea, introduced last fall by Vishwanath and Senthil, has been attracting increasing attention. While in most previously studied SPT phases the unbroken symmetry protects gapless excitations confined to the edge of the sample, Vishwanath and Senthil pointed out another possibility — a gapped edge exhibiting topological order. The surface can support anyons with exotic braiding statistics.

Here, too, anomalies are central to the discussion. While anyons in two-dimensional media are already a much-studied subject, the anyon models that can be realized at the edges of three-dimensional SPT phases are different than anyon models realized in really, truly two-dimensional systems. What’s new are not the braiding properties of the anyons, but rather how the anyons transform under the symmetry. Flatlanders who study the symmetry realization in their gapped two-dimensional world should be able to infer the existence of the three-dimensional bulk.

The pace of discovery picked up this month when four papers appeared simultaneously on the preprint arXiv, by Metlitski-Kane-Fisher, Chen-Fidkowski-Vishwanath, Bonderson-Nayak-Qi, and Wang-Potter-Senthil, all proposing and analyzing models of SPT phases with gapped edges. It remains to be seen, though, whether this physics will be realized in actual materials.

Are we on the edge?

In Flatland, our two-dimensional friend, finally able to perceive the third dimension thanks to the Sphere’s insistent tutelage, begs to enter a world of still higher dimensions, “where thine own intestines, and those of kindred Spheres, will lie exposed to … view.” The Sphere is baffled by the Flatlander’s request, protesting, “There is no such land. The very idea of it is utterly inconceivable.”

Let’s not be so dogmatic as the Sphere. The lessons learned from the quantum Hall effect and the topological insulator have prepared us to take the next step, envisioning our own three-dimensional world as the edge of a higher-dimensional bulk system. The existence of an unseen bulk may be inferred in the future by us edgelings, if experimental explorations of our three-dimensional effective theory reveal anomalies begging for an explanation.

Perhaps we are on the edge … of a great discovery. At least it’s conceivable.

*Disclaimer: The gender politics of Flatland, to put it mildly, is outdated and offensive. I don’t wish to endorse the idea that women are one dimensional! I included the reference to Flatland because the imagery of two-dimensional beings struggling to imagine the third dimension is a perfect fit to the scientific content of this post.