Physicists have used Feynman diagrams as a tool for calculating scattering amplitudes that describe particle interactions for more than six decades. Their broad utility was due initially in large part to the seminal work of Freeman Dyson, Professor Emeritus in the School of Natural Sciences. Feynman diagrams provide a way of calculating scattering amplitudes in a manner that is consistent with quantum mechanics and special relativity and more recently they have been used for increasingly complex calculations related to the physics being probed at high-energy particle accelerators, such as the Large Hadron Collider (LHC).
The many graphs depicted in Figure 1 represent a small sample of the Feynman diagrams necessary to compute the amplitude for producing four outgoing gluons from the collision of two incoming ones. This process will occur several hundred times a second at the LHC. Each Feynman diagram pictorially represents a specific way in which this process can happen, and is associated with a complicated mathematical expression. The amplitude is obtained by adding up a total of 220 diagrams.
Because the number of Feynman diagrams needed to calculate an amplitude can climb into the thousands, increasingly clever tricks have been developed to bypass their direct computation. Figure 2 represents the complete set of BCFW diagrams required to calculate the same process. These powerful new diagrams were developed at the Institute, and were made possible by the realization that the amplitudes possess remarkable properties when the gluons are associated with points in a geometric setting known as “twistor space,” rather than ordinary spacetime. For example, this six-gluon process is only not zero when the points in twistor space representing the six gluons lie along two intersecting lines (Figure 3). Recent work has revealed a direct relationship between BCFW terms and “twistor diagrams” (Figure 4), which reduce the calculation of amplitudes to simple multiplication rules. Scattering processes involving very strongly interacting gluons are impossible to calculate using Feynman diagrams. Instead, a “holographically dual” formulation of the problem has been developed (Figure 5), where the gluon scattering in a four-dimensional conformal field theory (CFT) is related to a tractable string theoretic calculation in a higher-dimensional anti–de Sitter (AdS) space.
For sixty years, Feynman diagrams have been an essential calculational and conceptual tool for theoretical physicists striving to deepen our understanding of the fundamental forces and particles of nature. Members of the Institute have played leading roles in the development of their use, from Freeman Dyson in the late 1940s and early 1950s to the current generation of theoretical physicists in the School of Natural Sciences. Most recently, clues provided by Feynman diagrams have led to powerful new methods that are revolutionizing our ability to understand the fundamental particle collisions that will occur at the Large Hadron Collider (LHC). At the same time, these clues have motivated Institute theorists to pursue a radical transcription of our ordinary physics formulated in space and time in terms of a theory without explicit reference to spacetime. The story of these developments connects one of the most pressing practical issues in theoretical particle physics with perhaps the most profound insight in string theory in the last decade––and at the same time provides a window into the history of physics at the Institute.
Surprising as it now seems, when Richard Feynman first introduced his diagrams at a meeting at a hotel in the Pocono Mountains in the spring of 1948, they were not immediately embraced by the physicists present, who included J. Robert Oppenheimer, then Director of the Institute and organizer of the meeting, and a number of then Members of the Institute, including Niels Bohr and Paul Dirac. The main event of the meeting, whose topic was how to calculate observable quantities in quantum electrodynamics, was an eight-hour talk by Julian Schwinger of Harvard, whose well-received analysis used calculations founded in an orthodox understanding of quantum mechanics. On the other hand, Feynman struggled to explain the rules and the origins of his diagrams, which used simple pictures instead of complicated equations to describe particle interactions, also known as scattering amplitudes.
Traveling on a Greyhound bus from San Francisco to Princeton at the end of the summer of 1948 to take up his appointment as a Member of the Institute, the twenty-four-year-old Dyson had an epiphany that would turn Feynman diagrams into the working language of particle physicists all over the world. Earlier, in June, Dyson had embarked on a four-day road trip to Albuquerque with Feynman, whom he had met at Cornell the previous year. Then he spent five weeks at a summer school at the University of Michigan in Ann Arbor where Schwinger presented detailed lectures about his theory. Dyson had taken these opportunities to speak at length with both Feynman and Schwinger and, as the bus was making its way across Nebraska, Dyson began to fit Feynman’s pictures and Schwinger’s equations together. “Feynman and Schwinger were just looking at the same set of ideas from two different sides,” Dyson explains in his autobiographical book, Disturbing the Universe. “Putting their methods together, you would have a theory of quantum electrodynamics that combined the mathematical precision of Schwinger with the practical flexibility of Feynman.” Dyson combined these ideas with those of a Japanese physicist, Shinichiro Tomonaga, whose paper Hans Bethe had passed on to him at Cornell, to map out the seminal paper, “The Radiation Theories of Tomonaga, Schwinger and Feynman,” as the bus sped on through the Midwest. Published in the Physical Review in 1949, this work marked an epoch in physics.
While Feynman, Schwinger, and Tomonaga were awarded the Nobel Prize in Physics in 1965 for their contributions to developing an improved theory of quantum electrodynamics, it was Dyson who derived the rules and provided instructions about how the Feynman diagrams should be drawn and how they should be translated into their associated mathematical expressions. Moreover, he trained his peers to use the diagrams during the late 1940s and 1950s, turning the Institute into a hotbed of activity in this area. According to David Kaiser of the Massachusetts Institute of Technology, author of Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, “Feynman diagrams spread throughout the U.S. by means of a postdoc cascade emanating from the Institute for Advanced Study.”
Feynman diagrams are powerful tools because they provide a transparent picture for particle interactions in spacetime and a set of rules for calculating the scattering amplitudes describing these interactions that are completely consistent with the laws of quantum mechanics and special relativity. These rules allow any process involving particle scattering to be converted into a collection of diagrams representing all the ways the collision can take place. Each of these diagrams corresponds to a particular mathematical expression that can be evaluated. The exact description of the scattering process involves summing an infinite number of diagrams. But in quantum electrodynamics, a simplification occurs: because the electric charge is a small number, the more interactions a diagram involves the smaller the contribution it makes to the sum. Thus, to describe a process to a given accuracy, one only has to sum up a finite number of diagrams.
“Freeman was the person who realized that once you force the quantum mechanical answer to look like it is consistent with the laws of special relativity, then it is very natural to do the calculations in terms of Feynman diagrams,” says Nima Arkani-Hamed, Professor in the School of Natural Sciences. “Almost nobody thinks about Feynman diagrams the way Feynman originally arrived at them. You open up any textbook and the derivation of these things uses this very beautiful, profound set of insights that Freeman came up with.”
By the 1980s and 1990s, Feynman diagrams were being used to calculate increasingly complicated processes. These included not only collisions of the familiar electrons and photons, governed by quantum electrodynamics, but also the interaction of particles such as quarks and gluons, which are the basic constituents of protons and neutrons, and are governed by the theory known as quantum chromodynamics. These calculations are essential to understanding and interpreting the physics probed at modern high-energy particle accelerators. However, theorists found that using Feynman diagrams in this wider context led to an explosion in their number and complexity.
Increasingly clever tricks were developed in the late 1980s to calculate these more complicated processes without actually calculating the Feynman diagrams directly. This led to a surprising realization. While each step in the calculation was very complicated, involving a huge number of terms, cancellations between them led to a final answer that was often stunningly simple. “The answer seems to have all sorts of incredible properties in it that we are learning more and more about, which are not obvious when you draw Feynman diagrams. In fact, keeping spacetime manifest is forcing the introduction of so much redundancy in how we talk about things that computing processes involving only a few gluons can require thousands of diagrams, while the final answer turns out to be given by a few simple terms,” says Arkani-Hamed. “A big, overarching goal is to figure out some way of getting to that answer directly without going through this intermediary spacetime description.”
In 2003, Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, came up with a proposal along these lines. He found a remarkable rewriting of the leading approximation to the interactions between gluons that led directly to the simple form of their scattering amplitudes, without using Feynman diagrams. This work immediately led to a major innovation: a new diagrammatic representation of amplitudes, called “CSW diagrams” (after Freddy Cachazo (then a Member), Witten’s student Peter Svrcek, and Witten). This led to a number of new insights into amplitudes that, via a circuitous path, led to a second, apparently unrelated representation of the amplitudes known as “BCFW diagrams” (after former Members Ruth Britto and Bo Feng, as well as Cachazo and Witten). These powerful new diagrams highlight and exploit properties of the physics that are invisible in Feynman diagrams, and they provide a much more efficient route to the final answer.
These new methods have triggered a breakthrough in a critical area relevant to describing physics at the LHC. This enormous machine will experimentally probe our understanding of nature at extremely short distances, and could reveal major new physical principles, such as an extended quantum notion of spacetime known as supersymmetry. In order to establish the discovery of new particles and forces, it is necessary to accurately understand the predictions from current theories. But these calculations had been hampered by the complexity of the relevant Feynman graphs. Many processes experimental physicists were interested in were considered to be impossible to calculate theoretically in practice. Now, this is no longer the case, and already computer code exploiting the BCFW technique is being developed for application to the data the LHC will produce.
In addition to their practical value, these new ideas have opened up a number of new frontiers of purely theoretical research, both in exploring further the inner workings of scattering amplitudes and in investigating their relationship with deeper theories of space and time. About a year and a half ago, Arkani-Hamed became intrigued by the BCFW formalism, and with his student Jared Kaplan he found a simple physical argument for why it is applicable to calculating scattering amplitudes for gluons and gravitons, not just in four dimensions of spacetime as originally formulated, but in any number of dimensions. “This idea of BCFW is somehow a powerful and general fact about physics in any number of dimensions,” says Arkani-Hamed. Their work also suggested that the amplitudes for the scattering of gravitons might be especially simple. “Even the simplest processes for gravity involve ludicrously complicated Feynman diagrams, and yet not only are the amplitudes just as simple in this new language, there is even some indication that they might be simpler,” says Arkani-Hamed. “Perhaps this is because the things that are the most complicated from the Feynman diagram perspective are the simplest from this other perspective that we are searching for.”
Expressing amplitudes in terms of variables that encode directly only the physical properties of particles, such as their momentum and spin, is a key step in this program. This was achieved in the 1980s for four-dimensional theories. But if a general rewriting of basic interactions for particles like gluons and gravitons is sought, it must be possible to extend these successes to higher dimensions, especially since extra dimensions appear naturally in string theory. This year, Institute Member Donal O’Connell and one of Arkani-Hamed’s students, Clifford Cheung, showed that this could also be achieved in six dimensions. Using their formalism, and BCFW diagrams in six dimensions, O’Connell and Cheung were able to discover very compact expressions for scattering gauge bosons and gravitons in higher dimensions, which also unify a multitude of four-dimensional expressions.
Witten’s 2003 proposal used a set of ideas that Roger Penrose first suggested in the 1960s called twistor theory, which posits that instead of keeping track of points in spacetime, physicists should look at the light rays that come out of those points and follow them out to infinity. Witten’s method of calculating the scattering amplitudes suggested a new string theory that lives in twistor space rather than in ordinary spacetime; the structure of this string theory is directly related to the CSW diagrammatic construction.
The BCFW diagrams arose from studying general properties of relativistic quantum theory formulated in spacetime. However, in a recent collaboration, Arkani-Hamed, Cachazo, Cheung, and Kaplan, found to their surprise that the BCFW formalism is also most naturally expressed in twistor space. Their reasoning leads to a direct mapping of ordinary physics in spacetime to a simpler description in twistor space. “What is extremely cool about this business is that we are trying to come up with an explanation for marvelous patterns found in theories describing our world,” says Arkani-Hamed. “We have a lot of clues now, and I think there is a path towards a complete theory that will rewrite physics in a language that won’t have spacetime in it but will explain these patterns.”
This line of thought has connections to the concepts of duality and holography, which grew out of developments in string theory in the 1990s and have dominated much of the activity in the field for the past decade. A “duality” is an exact quantum equivalence between two completely different classical theories. The first examples of this remarkable phenomenon were discovered in four-dimensional supersymmetric theories by Nathan Seiberg, Professor in the School of Natural Sciences, in collaboration with Witten. This led to the realization that all string theories are different manifestations of a single underlying theory. Holography is the most striking example of duality to date, relating a gravitational theory in a curved spacetime to a non-gravitational particle theory in a lower-dimensional flat space. The analogy is to familiar holograms, which encode three-dimensional information on a two-dimensional surface. Juan Maldacena, also a Professor in the School of Natural Sciences, found the first example of a holographic duality, in which everything that happens in the bulk of spacetime can be mapped to processes occurring on its boundary. Maldacena’s conjecture is now known as the AdS/CFT correspondence, and provides a dictionary for translating the physics of anti–de Sitter space (AdS)—a negatively curved space with an extra fifth dimension, containing gravity and strings—to a conformal field theory (CFT), a four-dimensional particle theory that lives on the boundary of the spacetime. “Things about gravity are mysterious; things about particle theories are much less mysterious. Incredibly, the AdS/CFT correspondence maps mysterious things about gravity to well-understood things about particle physics, giving us the first working example of what a theory with emergent spacetime looks like,” says Arkani-Hamed. “It encourages the thought that, even in a nearly flat spacetime like our own, there is a picture of scattering processes, which takes incoming particles and converts them to outgoing particles with some very simple rules that bypass evolution through the intermediary spacetime.”
Exploitation of the AdS/CFT correspondence has led to many remarkable new developments. A key point is that the four-dimensional CFT involved in the correspondence is a close cousin of quantum chromodynamics, which is the theory relevant at the LHC. AdS/CFT thus provides a sort of theoretical laboratory for the exploration of phenomena related to the hadron collider. While the CFT is similar to the theories that describe nature, it is different in that it is far more symmetric. In fact, the theory enjoys so much symmetry that it has a property known as integrability, which has allowed, for the first time, the exact computation of a quantity relevant to scattering amplitudes. Already there is much progress in an area where a Feynman diagram computation is hopeless: when the coupling analogous to the electric charge is large, one would have to sum all possible diagrams. But via AdS/CFT, Maldacena and Member Fernando Alday have shown that in the large coupling regime, the scattering amplitude can be computed by turning it into a tractable calculation in a string theory living in AdS space. This work led to another major surprise: the scattering amplitudes were shown to have unexpected symmetries that were later sought and found in diagrammatic calculations at weak coupling. These symmetries are related to the integrability properties, and give new hope that scattering amplitudes in the CFT can be determined exactly.
Arkani-Hamed suspects that the key to further progress will be finding the analog for the AdS/CFT correspondence for flat space. An essential problem in gravity is the inability to precisely talk about physical observables that are localized in space and time. “The general rule seems to be that we can only describe gravitational systems when we’re sitting at the boundary of spacetime at infinity, because that is where notionally we can repeat experiments infinitely many times with an infinitely big apparatus to make infinitely precise measurements. None of these things are exactly possible with finite separations,” says Arkani-Hamed. “The AdS/CFT correspondence already tells us how to formulate physics in this way for negatively curved spacetimes; we are trying to figure out if there is some analog of that picture for describing scattering amplitudes in flat space. Since a sufficiently small portion of any spacetime is flat, figuring out how to talk about the physics of flat space holographically will likely represent a real step forward in theoretical physics.” ■