About preskill

I am a theoretical physicist at Caltech, and the Director of the Institute for Quantum Information and Matter. Follow me on Twitter @preskill.

Bell’s inequality 50 years later

This is a jubilee year.* In November 1964, John Bell submitted a paper to the obscure (and now defunct) journal Physics. That paper, entitled “On the Einstein Podolsky Rosen Paradox,” changed how we think about quantum physics.

The paper was about quantum entanglement, the characteristic correlations among parts of a quantum system that are profoundly different than correlations in classical systems. Quantum entanglement had first been explicitly discussed in a 1935 paper by Einstein, Podolsky, and Rosen (hence Bell’s title). Later that same year, the essence of entanglement was nicely and succinctly captured by Schrödinger, who said, “the best possible knowledge of a whole does not necessarily include the best possible knowledge of its parts.” Schrödinger meant that even if we have the most complete knowledge Nature will allow about the state of a highly entangled quantum system, we are still powerless to predict what we’ll see if we look at a small part of the full system. Classical systems aren’t like that — if we know everything about the whole system then we know everything about all the parts as well. I think Schrödinger’s statement is still the best way to explain quantum entanglement in a single vigorous sentence.

To Einstein, quantum entanglement was unsettling, indicating that something is missing from our understanding of the quantum world. Bell proposed thinking about quantum entanglement in a different way, not just as something weird and counter-intuitive, but as a resource that might be employed to perform useful tasks. Bell described a game that can be played by two parties, Alice and Bob. It is a cooperative game, meaning that Alice and Bob are both on the same side, trying to help one another win. In the game, Alice and Bob receive inputs from a referee, and they send outputs to the referee, winning if their outputs are correlated in a particular way which depends on the inputs they receive.

But under the rules of the game, Alice and Bob are not allowed to communicate with one another between when they receive their inputs and when they send their outputs, though they are allowed to use correlated classical bits which might have been distributed to them before the game began. For a particular version of Bell’s game, if Alice and Bob play their best possible strategy then they can win the game with a probability of success no higher than 75%, averaged uniformly over the inputs they could receive. This upper bound on the success probability is Bell’s famous inequality.**

Classical and quantum versions of Bell's game. If Alice and Bob share entangled qubits rather than classical bits, then they can win the game with a higher success probability.

Classical and quantum versions of Bell’s game. If Alice and Bob share entangled qubits rather than classical bits, then they can win the game with a higher success probability.

There is also a quantum version of the game, in which the rules are the same except that Alice and Bob are now permitted to use entangled quantum bits (“qubits”)  which were distributed before the game began. By exploiting their shared entanglement, they can play a better quantum strategy and win the game with a higher success probability, better than 85%. Thus quantum entanglement is a useful resource, enabling Alice and Bob to play the game better than if they shared only classical correlations instead of quantum correlations.

And experimental physicists have been playing the game for decades, winning with a success probability that violates Bell’s inequality. The experiments indicate that quantum correlations really are fundamentally different than, and stronger than, classical correlations.

Why is that such a big deal? Bell showed that a quantum system is more than just a probabilistic classical system, which eventually led to the realization (now widely believed though still not rigorously proven) that accurately predicting the behavior of highly entangled quantum systems is beyond the capacity of ordinary digital computers. Therefore physicists are now striving to scale up the weirdness of the microscopic world to larger and larger scales, eagerly seeking new phenomena and unprecedented technological capabilities.

1964 was a good year. Higgs and others described the Higgs mechanism, Gell-Mann and Zweig proposed the quark model, Penzias and Wilson discovered the cosmic microwave background, and I saw the Beatles on the Ed Sullivan show. Those developments continue to reverberate 50 years later. We’re still looking for evidence of new particle physics beyond the standard model, we’re still trying to unravel the large scale structure of the universe, and I still like listening to the Beatles.

Bell’s legacy is that quantum entanglement is becoming an increasingly pervasive theme of contemporary physics, important not just as the source of a quantum computer’s awesome power, but also as a crucial feature of exotic quantum phases of matter, and even as a vital element of the quantum structure of spacetime itself. 21st century physics will advance not only by probing the short-distance frontier of particle physics and the long-distance frontier of cosmology, but also by exploring the entanglement frontier, by elucidating and exploiting the properties of increasingly complex quantum states.

frontiersSometimes I wonder how the history of physics might have been different if there had been no John Bell. Without Higgs, Brout and Englert and others would have elucidated the spontaneous breakdown of gauge symmetry in 1964. Without Gell-Mann, Zweig could have formulated the quark model. Without Penzias and Wilson, Dicke and collaborators would have discovered the primordial black-body radiation at around the same time.

But it’s not obvious which contemporary of Bell, if any, would have discovered his inequality in Bell’s absence. Not so many good physicists were thinking about quantum entanglement and hidden variables at the time (though David Bohm may have been one notable exception, and his work deeply influenced Bell.) Without Bell, the broader significance of quantum entanglement would have unfolded quite differently and perhaps not until much later. We really owe Bell a great debt.

*I’m stealing the title and opening sentence of this post from Sidney Coleman’s great 1981 lectures on “The magnetic monopole 50 years later.” (I’ve waited a long time for the right opportunity.)

**I’m abusing history somewhat. Bell did not use the language of games, and this particular version of the inequality, which has since been extensively tested in experiments, was derived by Clauser, Horne, Shimony, and Holt in 1969.

When I met with Steven Spielberg to talk about Interstellar

Today I had the awesome and eagerly anticipated privilege of attending a screening of the new film Interstellar, directed by Christopher Nolan. One can’t help but be impressed by Nolan’s fertile visual imagination. But you should know that Caltech’s own Kip Thorne also had a vital role in this project. Indeed, were there no Kip Thorne, Interstellar would never have happened.

On June 2, 2006, I participated in an unusual one-day meeting at Caltech, organized by Kip and the movie producer Lynda Obst (Sleepless in Seattle, Contact, The Invention of Lying, …). Lynda and Kip, who have been close since being introduced by their mutual friend Carl Sagan decades ago, had conceived a movie project together, and had collaborated on a “treatment” outlining the story idea. The treatment adhered to a core principle that was very important to Kip — that the movie be scientifically accurate. Though the story indulged in some wild speculations, at Kip’s insistence it skirted away from any flagrant violation of the firmly established laws of Nature. This principle of scientifically constrained speculation intrigued Steven Spielberg, who was interested in directing.

The purpose of the meeting was to brainstorm about the story and the science behind it with Spielberg, Obst, and Thorne. A remarkable group assembled, including physicists (Andrei Linde, Lisa Randall, Savas Dimopoulos, Mark Wise, as well as Kip), astrobiologists (Frank Drake, David Grinspoon), planetary scientists (Alan Boss, John Spencer, Dave Stevenson), and psychologists (Jay Buckey, James Carter, David Musson). As we all chatted and got acquainted, I couldn’t help but feel that we were taking part in the opening scene of a movie about making a movie. Spielberg came late and left early, but spent about three hours with us; he even brought along his Dad (an engineer).

Time_cover_interstellarThough the official release of Interstellar is still a few days away, you may already know from numerous media reports (including the cover story in this week’s Time Magazine) the essential elements of the story, which involves traveling through a wormhole seeking a new planet for humankind, a replacement for the hopelessly ravaged earth. The narrative evolved substantially as the project progressed, but traveling through a wormhole to visit a distant planet was already central to the original story.

Inevitably, some elements of the Obst/Thorne treatment did not survive in the final film. For one, Stephen Hawking was a prominent character in the original story; he joined the mission because of his unparalleled expertise at wormhole transversal, and Stephen’s ALS symptoms eased during prolonged weightlessness, only to recur upon return to earth gravity. Also, gravitational waves played a big part in the treatment; in particular the opening scene depicted LIGO scientists discovering the wormhole by detecting the gravitational waves emanating from it.

There was plenty to discuss to fill our one-day workshop, including: the rocket technology needed for the trip, the strong but stretchy materials that would allow the ship to pass through the wormhole without being torn apart by tidal gravity, how to select a crew psychologically fit for such a dangerous mission, what exotic life forms might be found on other worlds, how to communicate with an advanced civilization which resides in a higher dimensional bulk rather than the three-dimensional brane to which we’re confined, how to build a wormhole that stays open rather than pinching off and crushing those who attempt to pass through, and whether a wormhole could enable travel backward in time.

Spielberg was quite engaged in our discussions. Upon his arrival I immediately shot off a text to my daughter Carina: “Steven Spielberg is wearing a Brown University cap!” (Carina was a Brown student at the time, as Spielberg’s daughter had been.) Steven assured us of his keen interest in the project, noting wryly that “Aliens have been very good to me,” and he mentioned some of his favorite space movies, which included some I had also enjoyed as a kid, like Forbidden Planet and (the original) The Day the Earth Stood Still. In one notable moment, Spielberg asked the group “Who believes that intelligent life exists elsewhere in the universe?” We all raised our hands. “And who believes that the earth has been visited by extraterrestrial civilizations?” No one raised a hand. Steven seemed struck by our unanimity, on both questions.

I remember tentatively suggesting that the extraterrestrials had mastered M-theory, thus attaining computational power far beyond the comprehension of earthlings, and that they themselves were really advanced robots, constructed by an earlier generation of computers. Like many of the fun story ideas floated that day, this one had no apparent impact on the final version of the film.

Spielberg later brought in Jonah Nolan to write the screenplay. When Spielberg had to abandon the project because his DreamWorks production company broke up with Paramount Pictures (which owned the story), Jonah’s brother Chris Nolan eventually took over the project. Jonah and Chris Nolan transformed the story, but continued to consult extensively with Kip, who became an Executive Producer and says he is pleased with the final result.

Of the many recent articles about Interstellar, one of the most interesting is this one in Wired by Adam Rogers, which describes how Kip worked closely with the visual effects team at Double Negative to ensure that wormholes and rapidly rotating black holes are accurately depicted in the film (though liberties were taken to avoid confusing the audience). The images produced by sophisticated ray tracing computations were so surprising that at first Kip thought there must be a bug in the software, though eventually he accepted that the calculations are correct, and he is still working hard to more fully understand the results.

ScienceofInterstellarMech.inddI can’t give away the ending of the movie, but I can safely say this: When it’s over you’re going to have a lot of questions. Fortunately for all of us, Kip’s book The Science of Interstellar will be available the same day the movie goes into wide release (November 7), so we’ll all know where to seek enlightenment.

In fact on that very same day we’ll be treated to the release of The Theory of Everything, a biopic about Stephen and Jane Hawking. So November 7 is going to be an unforgettable Black Hole Day. Enjoy!

Where are you, Dr. Frank Baxter?

This year marks the 50th anniversary of my first publication. In 1964, when we were eleven-year-old fifth graders, my best friend Mace Rosenstein and I launched The Pres-stein Gazette, a not-for-profit monthly. Though the first issue sold well, the second issue never appeared.

Front page of the inaugural issue of the Pres-stein Gazette

Front page of the inaugural issue of the Pres-stein Gazette. Faded but still legible, it was produced using a mimeograph machine, a low-cost printing press which was popular in the pre-Xerox era.

One of my contributions to the inaugural  issue was a feature article on solar energy, which concluded that fossil fuel “isn’t of such terrific abundance and it cannot support the world for very long. We must come up with some other source of energy. Solar energy is that source …  when developed solar energy will be a cheap powerful “fuel” serving the entire world generously forever.”

This statement holds up reasonably well 50 years later. You might wonder how an eleven-year-old in 1964 would know something like that. I can explain …

In the 1950s and early 1960s, AT&T and the Bell Telephone System produced nine films about science, which were broadcast on prime-time network television and attracted a substantial audience. After broadcast, the films were distributed to schools as 16 mm prints and frequently shown to students for many years afterward. I don’t remember seeing any of the films on TV, but I eventually saw all nine in school. It was always a treat to watch one of the “Bell Telephone Movies” instead of hearing another boring lecture.

For educational films, the production values were uncommonly high. Remarkably, the first four were all written and directed by the legendary Frank Capra (a Caltech alum), in consultation with a scientific advisory board provided by Bell Labs.  Those four (Our Mr. Sun, Hemo the Magnificent, The Strange Case of the Cosmic Rays, and Unchained Goddess, originally broadcast in 1956-58) are the ones I remember most vividly. DVDs of these films exist, but I have not watched any of them since I was a kid.

The star of the first eight films was Dr. Frank Baxter, who played Dr. Research, the science expert. Baxter was actually an English professor at USC who had previous television experience as the popular host of a show about Shakespeare, but he made a convincing and pleasingly avuncular scientist. (The ninth film, Restless Sea, was produced by Disney, and Walt Disney himself served as host.) The other lead role was Mr. Writer, a skeptical and likeable Everyman who learned from Dr. Research’s clear explanations and sometimes translated them into vernacular.

The first film, Our Mr. Sun, debuted in 1956 (broadcast in color, a rarity at that time) and was seen by 24 million prime-time viewers. Mr. Writer was Eddie Albert, a well-known screen actor who later achieved greater fame as the lead on the 1960s TV situation comedy Green Acres. Lionel Barrymore appeared in a supporting role.

Dr. Frank Baxter and Eddie Albert in Our Mr. Sun.

Dr. Frank Baxter and Eddie Albert in Our Mr. Sun. (Source: Wikipedia)

Our Mr. Sun must have been the primary (unacknowledged) source for my article in the Pres-stein Gazette.  Though I learned from Wikipedia that Capra insisted (to the chagrin of some of his scientific advisers) on injecting some religious themes into the film, I don’t remember that aspect at all. The scientific content was remarkably sophisticated for a film that could be readily enjoyed by elementary school students, and I remember (or think I do) clever animations teaching me about the carbon cycle in stellar nuclear furnaces and photosynthesis as the ultimate source of all food sustaining life on earth. But I was especially struck by Dr. Baxter’s dire warning that, as the earth’s population grows, our planet will face shortages of food and fuel. On a more upbeat note he suggested that advanced technologies for harnessing the power of the sun would be the key to our survival, which inspired the optimistic conclusion of my article.

A lavishly produced prime-time show about science was a real novelty in 1956, many years before NOVA or the Discovery Channel. I wonder how many readers remember seeing the Dr. Frank Baxter movies when you were kids, either on TV or in school. Or was there another show that inspired you like Our Mr. Sun inspired me? I hope some of you will describe your experiences in the comments.

And I also wonder what resource could have a comparable impact on an eleven-year-old in today’s very different media environment. The obvious comparison is with Neil deGrasse Tyson’s revival of Cosmos, which aired on Fox in 2014. The premiere episode of Cosmos drew 8.5 million viewers on the night it was broadcast, but that is a poor measure of impact nowadays. Each episode has been rebroadcast many times, not just in the US and Canada but internationally as well, and the whole series is now available in DVD and Blu-ray. Will lots of kids in the coming years own it and watch it? Is Cosmos likely to be shown in classrooms as well?

Science is accessible to the curious through many other avenues today, particularly on YouTube. One can watch TED talks, or Minute Physics, or Veritasium, or Khan Academy, or Lenny Susskind’s lectures, not to mention our own IQIM videos on PHD Comics. And there are many other options. Maybe too many?

But do kids watch this stuff? If not, what online sources inspire them? Do they get as excited as I did when I watched Dr. Frank Baxter at age 11?

I don’t know. What do you think?

Macroscopic quantum teleportation: the story of my chair

In the summer of 2000, a miracle occurred: The National Science Foundation decided to fund a new Institute for Quantum Information at Caltech with a 5 million dollar award from their Information Technology Research program. I was to be the founding director of the IQI.

Jeff Kimble explained to me why we should propose establishing the IQI. He knew I had used my slice of our shared DARPA grant to bring Alexei Kitaev to Caltech as a visiting professor, which had been wonderful. Recalling how much we had both benefited from Kitaev’s visit, Jeff remarked emphatically that “This stuff’s not free.” He had a point. To have more fun we’d need more money. Jeff took the lead in recruiting a large team of Caltech theorists and experimentalists to join the proposal we submitted, but the NSF was primarily interested in supporting the theory of quantum computation rather than the experimental part of the proposal. That was how I wound up in charge, though I continued to rely on Jeff’s advice and support.

This was a new experience for me and I worried a lot about how directing an institute would change my life. But I had one worry above all: space. We envisioned a thriving institute brimming over with talented and enthusiastic young scientists and visitors drawn from the physics, computer science, and engineering communities. But how could we carve out a place on the Caltech campus where they could work and interact?

To my surprise and delight, Jeff and I soon discovered that someone else at Caltech shared our excitement over the potential of IQI — Richard Murray, who was then the Chair of Caltech’s Division of Engineering and Applied Science. Richard arranged for the IQI to occupy office space in Steele Laboratory and some space we could configure as we pleased in Jorgensen Laboratory. The hub of the IQI became the lounge in Jorgensen, which we used for our seminar receptions, group meetings, and innumerable informal discussions, until our move to the beautiful Annenberg Center when it opened in 2009.

I sketched a rough plan for the Jorgensen layout, including furniture for the lounge. The furniture, I was told, was “NIC”. Though I was too embarrassed to ask, I eventually inferred this meant “Not in Contract” — I would need to go furniture shopping, one of my many burgeoning responsibilities as Director.

By this time, Ann Harvey was in place as IQI administrator, a huge relief. But furniture was something I thought I knew about, because I had designed and furnished a common area for the particle theory group a couple of years earlier. As we had done on that previous occasion, my wife Roberta and I went to Krause’s Sofa Factory to order a custom-made couch, love seat, and lounge chair, in a grayish green leather which we thought would blend well with the carpeting.

Directing an institute is not as simple as it sounds, though. Before the furniture was delivered, Krause’s declared bankruptcy! We had paid in full, but I had some anxious moments wondering whether there would be a place to sit down in the IQI lounge. In the end, after some delay, our furniture was delivered in time for the grand opening of the new space in September 2001. A happy ending, but not really the end of the story.

Before the move to Annenberg in 2009, I ordered furniture to fill our (much smaller) studio space, which became the new IQI common area. The Jorgensen furniture was retired, and everything was new! It was nice … But every once in a while I felt a twinge of sadness. I missed my old leather chair, from which I had pontificated at eight years worth of group meetings. That chair and I had been through a lot together, and I couldn’t help but feel that my chair’s career had been cut short before its time.

I don’t recall mentioning these feelings to anyone, but someone must have sensed by regrets. Because one day not long after the move another miracle occurred … my chair was baaack! Sitting in it again felt … good. For five years now I’ve been pontificating from my old chair in our new studio, just like I used to. No one told me how my chair had been returned to me, and I knew better than to ask.

My chair today. Like me, a bit worn but still far from retirement.

My chair today. Like me, a bit worn but still far from retirement.

Eventually the truth comes out. At my 60th birthday celebration last year, Stephanie Wehner and Darrick Chang admitted to being the perpetrators, and revealed the whole amazing story in their article on “Macroscopic Quantum Teleportation” in a special issue of Nature Relocations. Their breakthrough article was enhanced by Stephanie’s extraordinary artwork, which you really have to see to believe. So if your curiosity is piqued, please follow this link to find out more.

Why, you may wonder, am I reminiscing today about the story of my chair? Well, is an excuse really necessary? But if you must know, it may be because, after two renewals and 14 years of operation, I submitted the IQI Final Report to the NSF this week. Don’t worry — the Report is not really Final, because the IQI has become part of an even grander vision, the IQIM (which has given birth to this blog among other good things). Like my chair, the IQI is not quite what it was, yet it lives on.

The nostalgic feelings aroused by filing the Final Report led me to reread the wonderful volume my colleagues put together for my birthday celebration, which recounts not only the unforgettable exploits of Stephanie and Darrick, but many other stories and testimonials that deeply touched me.

Browsing through that book today, one thing that struck me is the ways we sometimes have impact on others without even being aware of it. For example, Aram Harrow, Debbie Leung, Joe Renes and Stephanie all remember lectures I gave when they were undergraduate students (before I knew them), which might have influenced their later research careers. Knowing this will make it a little harder to say no the next time I’m invited to give a talk. Yaoyun Shi has vivid memories of the time I wore my gorilla mask to the IQI seminar on Halloween, which inspired him to dress up as “a butcher threatening to cut off the ears of my students with a bloody machete if they were not listening,” thus boosting his teaching evaluations. And Alexios Polychronakos, upon hearing that I had left particle theory to pursue quantum computing, felt it “was a bit like watching your father move to Las Vegas and marry a young dancer after you leave for college,” while at the same time he appreciated “that such reinventions are within the spectrum of possibilities for physicists who still have a pulse.”

I’m proud of what the IQI(M) has accomplished, but we’re just getting started. After 14 years, I still have a pulse, and my chair has plenty of wear left. Together we look forward to many more years of pontification.

 

 

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.