The AdS/CFT Correspondence

The holy grail of 20th century physics was to unify the two grand theories describing the universe. The theory of the very large, General Relativity, proposed by Albert Einstein in 1915, and the theory of the very small, quantum field theory, the result of the work of many physicists in the early 20th century. These two pillars of modern physics, each proven beyond any doubt, have remained stubbornly antagonistic, hinting at a problem at the heart of our understanding of physics. Reliably applicable in their separate disciplines, the two theories have been instrumental in countless discoveries, yet at the few intersections where the two converge, such as the big bang or in black holes, critical inconsistencies arise. More specifically, what has remained elusive is a description of gravity on the quantum scale, or “quantum gravity”. Without it, the ultimate goal of finding one theory which describes all the fundamental forces, including gravity, a ‘theory of everything’, remains the theoretical physicist’s distant dream.

In the late 1960s a theory initially proposed to solve an unrelated issue showed itself by accident to be a possible contender in the search to find answers to the problem of quantum gravity. What became known as string theory was born and despite numerous problems on the way, it has, due to its mathematically rigorous framework, over the last forty years developed into a major branch of theoretical physics (1). Its premise was that at the ultimate microscopic level, our universe is made of infinitesimally small strings, vibrating in many different dimensions, and which, depending on the frequency of the vibration, manifest themselves as the different point particles we observe in particle colliders and which make up the Standard Model of Particles. One particularly surprising effect of the string theory framework is that one of the vibrational states of strings gives rise to the graviton, the quantum mechanical particle that carries gravitational force, making string theory the first viable candidate for a theory of quantum gravity.

Yet the strings in string theory inhabit such a minute scale (the Planck scale), that they are impossible to probe with current collider technology: to observe these strings would require colossal amounts of energy, putting an observational proof of string theory out of reach for many decades or even centuries to come. Despite its undisputed mathematical beauty and consistency, string theory’s unprovability has made it an easy target for its sceptics.

Another intriguing concept arising from the framework of string theory is the holographic principle, first proposed by the Dutch physicist Gerard t’Hooft in 1993 and extended by Leonard Susskind and others soon after. The theory predicts that one of the three dimensions of space could be a form of illusion—that in fact all the particles and fields that make up reality are moving about in a two-dimensional realm like the “Flatland” of Edwin A. Abbott. Gravity, too, would be part of the illusion: a force that is not present in the two-dimensional world but that materialises with the emergence of the illusory third dimension (2). While this was considered another promising discovery within the context of string theory, the difficulty in proving it made it impossible to escape the realm of mathematically beautiful conjecture. Susskind himself stated at the time that he did not expect the theory to be proven for many years to come (3).

Then, in late 1997, Juan Maldacena, a young Argentinian theoretical physicist working at Harvard, wrote a paper with the title The Large N Limit of Superconformal field theories and Supergravity. First published in the March 1998 issue of Advances in Theoretical and Mathematical Physics, it was to become one of the most referenced papers in the history of physics, and would revolutionise the entire discipline. In it, Maldacena demonstrated an intimate connection between Einstein’s general theory of relativity and quantum physics, and that in some cases quantum field theory and string theory are completely equivalent. In other words, that two subjects physicists have been studying for many decades might in fact be the same (4).

In his paper Maldacena considered the relationship between two seemingly different model universes. One is a cosmos similar to our own. Although it neither expands nor contracts, it has three dimensions, is filled with quantum particles and obeys Einstein’s equations of gravity. Known as ‘anti-de Sitter space’ (AdS), it is commonly referred to as ‘the bulk’ (5). The other model is also filled with elementary particles, but it has one dimension fewer and lacks gravity. Commonly known as ‘the boundary’, it is a mathematically defined membrane that lies an infinite distance from any given point in the bulk, yet completely encloses it, much like the 2D surface of a balloon enclosing a 3D volume of air. The boundary particles obey the equations of a quantum system known as conformal field theory (CFT). Maldacena discovered that the boundary and the bulk are completely equivalent. Like the 2D circuitry of a computer chip that encodes the 3D imagery of a computer game, the relatively simple, gravity-free equations that prevail on the boundary contain the same information and describe the same physics as the more complex equations that rule the bulk. Maldacena’s insight was simple, yet audacious: take any process involving particles and fields in the first universe, and it could equally well be described as a process involving gravity, black holes and strings in the second universe—and vice versa (6).

AdS/CFT thus offers a working example of quantum gravity in which everything is defined: it is only necessary to study it to find the answers to many of the paradoxes that have plagued so much of 20th century theoretical physics (7). The mathematically intricate world of strings, which exist in many extra dimensions, may be merely a hologram: the real action plays out in a simpler, flatter cosmos, in which there is no gravity (8).

The implications of Maldacena’s paper were profound: not only did it offer a way to put the still unproven theory of strings on a solid footing, but it also solved the apparent inconsistencies between quantum physics and Einstein's theory of gravity. The correspondence provides physicists with a mathematical Rosetta stone, a duality that allows them to translate back and forth between the two languages, and solve problems in one model that seem intractable in the other (9). Joseph Polchinski, one of the leading figures in string theory, described Maldacena’s correspondence as “the greatest equation ever, since it contains all the central concepts of fundamental physics: Maxwell’s equations, (…), the Dirac and Klein-Gordon equations, quantum mechanics, quantum field theory and general relativity. Moreover, in addition to these known principles of nature, it contains several more that theorists have found appealing: supersymmetry, string theory, and extra dimensions, and it ties these all together in an irreducible way. And this is just the tip of a much larger iceberg.” (10)

1.) Dennis Overbye, ‘String Theory, at 20, Explains It All (or Not)’, The New York Times, 7.12.2004

2.) Juan Maldacena, ‘The Illusion of Gravity’, Scientific American, November 2005, p.67

3.) Leonard Susskind, Black Hole Wars, New York, 2008, p. 412

4.) Jan Zaanen, ‘A black hole full of answers’, Nature, Vol. 448, No. 30, August 2007, p.1000–1001

5.) Ron Cowen, ‘The quantum source of space-time’, Nature, Vol. 527, 16 November 2015, pp. 290–293

6.) Zeeya Merali, ‘String theory finds a bench mate’, Nature, Vol. 478, 20 October 2011, p. 303

7.) Natalie Wolchover, ‘Physicists uncover geometric ‘Theory Space’, Quanta Magazine, 23 February 2017

8.) Ron Cowen, ‘Simulations back up theory that Universe is a hologram’, Nature, 23 February 2017

9.) Ibid

10.) Joseph Polchinski, ‘Introduction to Gauge/Gravity Duality’, Lecture at TASI, June 1–7, 2010, University of Colorado at Boulder