Quantum Mechanics

By Freeman Dyson 

Freeman Dyson was awarded the 2012 Henri Poincaré Prize at the International Mathematical Physics Congress in August. On this occasion, he delivered the lecture “Is a Graviton Detectable?” a PDF of which is available at http://publications.ias.edu/poincare2012/dyson.pdf.

John Brockman, founder and proprietor of the Edge website, asks a question every New Year and invites the public to answer it. THE EDGE QUESTION 2012 was, “What is your favorite deep, elegant, or beautiful
explanation?” He got 150 answers that are published in a book, This Explains Everything (Harper Collins, 2013). Here is my contribution.

The situation that I am trying to explain is the existence side by side of two apparently incompatible pictures of the universe. One is the classical picture of our world as a collection of things and facts that we can see and feel, dominated by universal gravitation. The other is the quantum picture of atoms and radiation that behave in an unpredictable fashion, dominated by probabilities and uncertainties. Both pictures appear to be true, but the relationship between them is a mystery.

The orthodox view among physicists is that we must find a unified theory that includes both pictures as special cases. The unified theory must include a quantum theory of gravitation, so that particles called gravitons must exist, combining the properties of gravitation with quantum uncertainties.

Slide from Nima Arkani-Hamed’s lecture, “The Inevitability of Physical Laws: Why the Higgs Has to Exist.”

Following the discovery in July of a Higgs-like boson—an effort that took more than fifty years of experimental work and more than 10,000 scientists and engineers working on the Large Hadron Collider—Juan Maldacena and Nima Arkani-Hamed, two Professors in the School of Natural Sciences, gave separate public lectures on the symmetry and simplicity of the laws of physics, and why the discovery of the Higgs was inevitable.

Peter Higgs, who predicted the existence of the particle, gave one of his first seminars on the topic at the Institute in 1966, at the invitation of Freeman Dyson. “The discovery attests to the enormous importance of fundamental, deep ideas, the substantial length of time these ideas can take to come to fruition, and the enormous impact they have on the world,” said Robbert Dijkgraaf, Director and Leon Levy Professor.

In their lectures “The Symmetry and Simplicity of the Laws of Nature and the Higgs Boson” and “The Inevitability of Physical Laws:
Why the Higgs Has to Exist,” Maldacena and Arkani-Hamed described the theoretical ideas that were developed in the 1960s and 70s, leading to our current understanding of the Standard Model of particle physics and the recent discovery of the Higgs-like boson. Arkani-Hamed framed the hunt for the Higgs as a detective story with an inevitable ending. Maldacena compared our understanding of nature to the fairytale Beauty and the Beast.

“What we know already is incredibly rigid. The laws are very rigid within the structure we have, and they are very fragile to monkeying with the structure,” said Arkani-Hamed. “Often in physics and mathematics, people will talk about beauty. Things that are beautiful, ideas that are beautiful, theoretical structures that are beautiful, have this feeling of inevitability, and this flip side of rigidity and fragility about them.”

After his teatime conversation with Hugh Montgomery, Freeman Dyson wrote this letter to Atle Selberg with references showing that the pair-correlation of the zeros of the zeta function is identical to that of the eigenvalues of a random matrix.

In early April 1972, Hugh Montgomery, who had been a Member in the School of Mathematics the previous year, stopped by the Institute to share a new result with Atle Selberg, a Professor in the School. The discussion between Montgomery and Selberg involved Montgomery’s work on the zeros of the Riemann zeta function, which is connected to the pattern of the prime numbers in number theory. Generations of mathematicians at the Institute and elsewhere have tried to prove the Riemann Hypothesis, which conjectures that the non-trivial zeros (those that are not easy to find) of the Riemann zeta function lie on the critical line with real part equal to 1⁄2.

Montgomery had found that the statistical distribution of the zeros on the critical line of the Riemann zeta function has a certain property, now called Montgomery’s pair correlation conjecture. He explained that the zeros tend to repel between neighboring levels. At teatime, Montgomery mentioned his result to Freeman Dyson, Professor in the School of Natural Sciences.

In the 1960s, Dyson had worked on random matrix theory, which was proposed by physicist Eugene Wigner in 1951 to describe nuclear physics. The quantum mechanics of a heavy nucleus is complex and poorly understood. Wigner made a bold conjecture that the statistics of the energy levels could be captured by random matrices. Because of Dyson’s work on random matrices, the distribution or the statistical behavior of the eigenvalues of these matrices has been understood since the 1960s.

By Robbert Dijkgraaf 

Robbert Dijkgraaf, IAS Director and Leon Levy Professor, in the Mathematics–Natural Sciences Library in Fuld Hall

I am honored and heartened to have joined the Institute for Advanced Study this summer as its ninth Director. The warmness of the welcome that my family and I have felt has surpassed our highest expectations. The Institute certainly has mastered the art of induction.

The start of my Directorship has been highly fortuitous. On July 4, I popped champagne during a 3 a.m. party to celebrate the LHC’s discovery of a particle that looks very much like the Higgs boson—the final element of the Standard Model, to which Institute Faculty and Members have contributed many of the theoretical foundations. I also became the first Leon Levy Professor at the Institute due to the great generosity of the Leon Levy Foundation, founded by Trustee Shelby White and her late husband Leon Levy, which has endowed the Directorship. Additionally, four of our Professors in the School of Natural Sciences—Nima Arkani-Hamed, Juan Maldacena, Nathan Seiberg, and Edward Witten—were awarded the inaugural Fundamental Physics Prize of the Milner Foundation for their path-breaking contributions to fundamental physics. And that was just the first month.

Nearly a century ago, Abraham Flexner, the founding Director of the Institute, introduced the essay “The Usefulness of Useless Knowledge.” It was a passionate defense of the value of the freely roaming, creative spirit, and a sharp denunciation of American universities at the time, which Flexner considered to have become large-scale education factories that placed too much emphasis on the practical side of knowledge. Columbia University, for example, offered courses on “practical poultry raising.” Flexner was convinced that the less researchers needed to concern themselves with direct applications, the more they could ultimately contribute to the good of society.

By John Wheeler 

The paper by Niels Bohr and John Wheeler on the mechanism of nuclear fission appeared in the Physical Review of September 1, 1939, the same day the war began.

The following excerpt is from remarks given by John Archibald Wheeler on March 27, 2000, in connection with the play Copenhagen by Michael Frayn. Wheeler was a Professor of Physics at Princeton University from 1938 until his retirement in 1976 and a Member of the Institute’s School of Mathematics (prior to the founding of the School of Natural Sciences) in the spring of 1937, when it was still temporarily housed in Fine Hall (now Jones Hall) at Princeton University. Niels Bohr, who had a twenty-year association with the Institute, first visited in the academic year 1938–39, when the Institute completed Fuld Hall. For more about Bohr and his relationship with Albert Einstein, one of the Institute’s first Professors, see the Spring 2009 Institute Letter.

If two such great thinkers as Bohr and Einstein, who had such a high regard for each other, could be brought together for a prolonged period, would not something emerge of great value to all of us? This thought and this hope animated the guiding spirits of the Princeton Institute for Advanced Study to invite Niels Bohr to come as a guest of the Institute for the entire spring semester of 1939. However, four days before Bohr boarded his America-bound ship, he learned from Otto Robert Frisch that Frisch and his aunt Lisa Meitner had solid evidence that a neutron splits the nucleus of uranium. As he crossed the Atlantic, Bohr’s vision turned more and more from the problem of quantum mechanics to the problems of nuclear physics. So January and February, March and April of 1939 saw him working, discussing, calculating, and writing, day after day, not with Einstein on quantum physics as intended, but with me on the nuclear physics of fission. Yes, of course, there were meetings Bohr had with Einstein but they were occasional and did not lead to the big push it takes to formulate a solid well-argued position. No. Fission, and what it meant and how it differed from one nucleus to another, and what those differences offered in the way of using the nucleus for a chain reaction stood at the center of our attention. . . .

By David H. Weinberg 

An SDSS-III plugplate, which admits light from preselected galaxies, stars, and quasars, superposed on an SDSS sky image.

Why is the expansion of the universe speeding up, instead of being slowed by the gravitational attraction of galaxies and dark matter? What is the history of the Milky Way galaxy and of the chemical elements in its stars? Why are the planetary systems discovered around other stars so different from our own solar system? These questions are the themes of SDSS-III, a six-year program of four giant astronomical surveys, and the focal point of my research at the Institute during the last year.

In fact, the Sloan Digital Sky Survey (SDSS) has been a running theme through all four of my stays at the Institute, which now span nearly two decades. As a long-term postdoctoral Member in the early 1990s, I joined in the effort to design the survey strategy and software system for the SDSS, a project that was then still in the early stages of fundraising, collaboration building, and hardware development. When I returned as a sabbatical visitor in 2001–02, SDSS observations were—finally—well underway. My concentration during that year was developing theoretical modeling and statistical analysis techniques, which we later applied to SDSS maps of cosmic structure to infer the clustering of invisible dark matter from the observable clustering of galaxies. By the time I returned for a one-term visit in 2006, the project had entered a new phase known as SDSS-II, and I had become the spokesperson of a collaboration that encompassed more than three hundred scientists at twenty-five institutions around the globe. With SDSS-II scheduled to complete its observations in mid-2008, I joined a seven-person committee that spent countless hours on the telephone that fall, sorting through many ideas suggested by the collaboration and putting together the program that became SDSS-III.

In his conjectures, now collectively known as the Langlands program, Robert Langlands drew on the work of Hermann Weyl (above), André Weil, and Harish-Chandra, among others with extensive ties to the Institute.

It has been said that the goals of modern mathematics are recon­struction and development.¹ The unifying conjectures between number theory and representation theory that Robert Langlands, Professor Emeritus in the School of Mathematics, articulated in a letter to André Weil in 1967, continue a tradition at the Institute of advancing mathematical knowledge through the identification of problems central to the understanding of active areas or likely to become central in the future.

“Two striking qualities of mathematical concepts regarded as central are that they are simultaneously pregnant with possibilities for their own development and, so far as we can judge from a history of two and a half millennia, of permanent validity,” says Langlands. “In comparison with biology, above all with the theory of evolution, a fusion of biology and history, or with physics and its two enigmas, quantum theory and relativity theory, mathematics contributes only modestly to the intellectual architecture of mankind, but its central contributions have been lasting, one does not supersede another, it enlarges it.”²

In his conjectures, now collectively known as the Langlands program, Langlands drew on the work of Harish-Chandra, Atle Selberg, Goro Shimura, André Weil, and Hermann Weyl, among others with extensive ties to the Institute. 

Weyl, whose appointment to the Institute’s Faculty in 1933 followed those of Albert Einstein and Oswald Veblen, was a strong believer in the overall unity of mathematics, across disciplines and generations. Weyl had a major impact on the progress of the entire field of mathematics, as well as physics, where he was equally comfortable. His work spanned topology, differential geometry, Lie groups, representation theory, harmonic analysis, and analytic number theory, and extended into physics, including relativity, electromagnetism, and quantum mechanics. “For [Weyl] the best of the past was not forgotten,” notes Michael Atiyah, a former Institute Professor and Member, “but was subsumed and refined by the mathematics of the present.”³

By Jeremy Bernstein 

Debates at the fifth Solvay Conference in Brussels in 1927 helped shape the modern interpretation of quantum mechanics. Participants included Niels Bohr (second row, far right) and Albert Einstein (first row, fifth from left).

In the two years I spent at the Institute, 1957–59, I had the opportunity of meeting two of the founders of the quantum theory—Niels Bohr and Paul Dirac. In the case of Bohr, perhaps “meeting” overstates the case. He was a Mem­ber in the spring of 1958 and Oppenheimer, who had known him since the 1920s and who had a feeling of adulation for him, decided that a fitting thing to do was to have a sort of seminar in which the physicists would trot out their wares with Bohr looking on and possibly commenting. As it happened, I had had a brief collaboration with T. D. Lee and C. N. Yang, who had won the Nobel Prize that fall. They had better things to tell Bohr than our modest work, so I was the designated spokesman. I was given ten minutes and took about three. After which Bohr commented, “Very interesting,” which meant he did not think so. If he had had any real interest, he would have engaged in a Socratic dialogue, which would have proceeded until he was satisfied. There is a famous story concerning Erwin Schrödinger—with whom I later spent an afternoon in Vienna—arriving in Copenhagen after having created his version of the quantum theory. Bohr disagreed with some of what Schrödinger was saying and pursued him into his bedroom where the now sick Schrödinger had taken refuge.

On a visit to the Institute ten years earlier, Bohr had written his wonderful account of his discussions with Einstein about the theory. Bohr found writing incredibly difficult and always had an amanuensis who acted as a sounding board. In this case, it was Abraham Pais who told the following story. Einstein had given Bohr his office for the visit and was in the adjoining smaller office of his assistant. Where the assistant had gone is not recorded. Bohr was facing away from the door and saying, “Einstein, Einstein” several times. As if summoned by a genie, Einstein stealthy came into the office. Before Bohr could turn around, Einstein helped himself to some of Bohr’s pipe tobacco. When Bohr did turn around, Einstein explained that his doctor had ordered him not to “buy” any more tobacco, but there was no injunction against his “stealing” some.

Robbert Dijkgraaf will become the ninth Director of the Institute, as of July 1, 2012.

On November 14, the Institute for Advanced Study announced the appointment of Robbert Dijkgraaf as its ninth Director, succeeding, as of July 1, 2012, Peter Goddard, who has served as Director since January 2004.

Below, Dijkgraaf speaks about his enthusiasm for the Institute and for using knowledge, creativity, and collaboration to further our understanding of a world of diverse facts, structures, ideas, and cultures.

––––––––

I am delighted to come to the Institute for Advanced Study, one of the intellectual centers of the world. The position of Director is highly distinguished, and the list of former Directors is quite intimidating. But I am particularly looking forward to combining at the highest level three elements that have been important in my professional life: the opportunity to collaborate with the very best scientists and scholars; to organize a stimulating environment for great talent from around the world; and to play an active role in science education, advocacy, and diplomacy to engage future generations.

Taking up my appointment as Director of the Institute will feel a bit like coming home. My family and I have only the best recollections of our stays in Princeton. I also expect that in many ways my life will become more focused. My present position as President of the Royal Netherlands Academy of Arts and Sciences requires giving attention to many different areas, from elementary school programs to industrial affairs, from government policy to international relations. The Institute is remarkably effective as a place for concentration and inspiration.

By Matthew Kahle 

Matthew Kahle, Member (2010-11) in the School of Mathematics, writes about his interest in thinking about what it might be like inside a black hole. This illustration (Figure 1.), from Kip Thorne's Black Holes and Time Warps: Einstein's Outrageous Legacy (W. W. Norton & Company, Inc., 1994), suggests a few probabilities.

I sometimes like to think about what it might be like inside a black hole. What does that even mean? Is it really “like” anything inside a black hole? Nature keeps us from ever knowing. (Well, what we know for sure is that nature keeps us from knowing and coming back to tell anyone about it.) But mathematics and physics make some predictions.

John Wheeler suggested in the 1960s that inside a black hole the fabric of spacetime might be reduced to a kind of quantum foam. Kip Thorne described the idea in his book Black Holes & Time Warps as follows (see Figure 1).

“This random, probabilistic froth is the thing of which the singularity is made, and the froth is governed by the laws of quantum gravity. In the froth, space does not have any definite shape (that is, any definite curvature, or even any definite topology). Instead, space has various probabilities for this, that, or another curvature and topology. For example, inside the singularity there might be a 0.1 percent probability for the curvature and topology of space to have the form shown in (a), and a 0.4 percent probability for the form in (b), and a 0.02 percent probability for the form in (c), and so on.”

In other words, perhaps we cannot say exactly what the properties of spacetime are in the immediate vicinity of a singularity, but perhaps we could characterize their distribution. By way of analogy, if we know that we are going to flip a fair coin a thousand times, we have no idea whether any particular flip will turn up heads or tails. But we can say that on average, we should expect about five hundred heads. Moreover, if we did the experiment many times we should expect a bell-curve shape (i.e., a normal distribution), so it is very unlikely, for example, that we would see more than six hundred heads.