by Roberto Lalli
Gerardus 't Hooft
Nobel Prize in Physics 1999
Gerardus ‘t Hooft is an outstanding Dutch theoretical physicist who is currently one of the nine Distinguished Professors of Utrecht University in the Netherlands. When he was still working on his PhD dissertation, ‘t Hooft made fundamental contributions to the theoretical development of the Standard Model of particle physics. Under the supervision of Martinus Veltman, in 1971 ‘t Hooft provided a proof that a class of theories—namely, the Yang-Mills non-Abelian gauge field theories—was renormalizable. This proof was soon regarded as a key advancement because it made the abovementioned set of theories acceptable as a realistic description of particle behaviours. Following ‘t Hooft’s work, the research on quantum field theory was revitalised. After further theoretical developments and important empirical confirmations, the SU(2)xU(1) Yang-Mills theory of electroweak interactions as well as the SU(3) colour Yang-Mills theory of strong interactions became the Standard Model of particle physics by the end of the 1970s. For their achievements, both ‘t Hooft and his supervisor M. Veltman were awarded the Nobel Prize in Physics in 1999. After his early contribution to the renormalization problem of gauge field theory, ‘t Hooft continued to collaborate with M. Veltman on the renormalization of quantum gravity models until 1981, when Veltman moved to the USA. ‘t Hooft also made major contributions to the resolution of the quark confinement problem in quantum chromodynamics. Apart from gauge theories in elementary particle physics, his fields of interest have been including the description of black holes in quantum gravity framework, space-time lattice models, fundamental aspects of quantum theory, and the hierarchy problem; namely, the enormous discrepancy between the intensity of the weak force and that of gravity. His reflections on the foundations of physics have led him to embrace an interpretation of quantum mechanics based on hidden variables—an interpretation very different from that accepted by the majority of physicists. ‘t Hooft’s idiosyncratic approach resulted in a series of original theories, which are currently being investigated by a limited number of theoretical physicists who are not convinced by the main lines of approach of their community.
Training in Theoretical Particle Physics under Veltman
Gerardus ‘t Hooft was born on 5 July 1946, in Den Helder, in the northernmost point of the North Holland peninsula, but spent his boyhood at The Hague, which is the seat of the Netherlands’ government. His mother’s family had an impressive commitment to scientific research. When ‘t Hooft was a boy, the brother of his grandmother—Frits Zernike—won the 1953 Nobel Prize in Physics for the invention of the phase contrast microscope. Moreover, his uncle Nicolaas ‘Nico’ Godfried van Kampen was Professor of Theoretical Physics at Utrecht University. Also his father had received a good scientific education and worked as a naval engineer. Very early in his life, ‘t Hooft proved to have a natural talent for mathematics and physics—a predisposition that his relatives strongly encouraged.
Although Leyden University was closer to his hometown, in 1964 he enrolled at Utrecht University because he wished to attend his uncle’s lectures. ‘t Hooft developed a keen interest for particle physics, but his uncle strongly disliked this topic and preferred statistical mechanics—by large the prevailing field of theoretical physics in the Netherlands. A new professor of theoretical physics had been recently appointed at Utrecht University. His name was Martinus Veltman and he was the only faculty member who was doing research on theoretical particle physics. Thus, in 1969 ‘t Hooft was assigned to Veltman for working on his predoctoral thesis, the Dutch equivalent of a Master’s degree thesis.
Since 1968, Veltman was developing a programme aimed at the renormalization of the massive Yang-Mills gauge field theory—a research stream that at the time was rather idiosyncratic because most physicists believed that the Yang-Mills theory was unrenormalizable. Veltman, instead, had reached the conviction that the massive Yang-Mills theory could be renormalized by means of a modification of the Feynman rules. Employing this approach, in 1968 Veltman had achieved an initial success by demonstrating that the Yang-Mills theory with an explicit mass term could be made one-loop renormalizable. At the same time, other theorists presented convincing arguments that the massless Yang-Mills theory was also renormalizable. These encouraging results were, however, followed by some failures when Veltman tried to extend the renormalization procedures for massive Yang-Mills fields to higher orders. Particularly disturbing was Veltman’s discovery that the limit to zero of the massive Yang-Mills theory did not correspond to the massless case. Veltman was trying to overcome these difficulties when ‘t Hooft began also working on the renormalization of Yang-Mills fields for his PhD dissertation.
An Idiosyncratic Path Towards the Renormalization of the Electroweak Theory
The majority of physicists considered Veltman’s research to be on the fringe of theoretical particle physics. The Yang-Mills theory had been formulated in 1954 in order to extend the gauge principle—on which quantum electrodynamics (QED) was based—to include weak and strong interactions. After an initial enthusiasm towards the Yang-Mills non-Abelian gauge fields, hopes waned that such fields were a realistic description of nature. The main problem was related to the mass of the Yang-Mills gauge bosons. By the early 1960s, it was understood that the Yang-Mills theory should have massless gauge bosons in order to be renormalizable. On the other hand, the only massless particle that had until then been detected was the mediator of the electromagnetic field; namely, the photon. Following the success of renormalized QED, many theorists came to believe that renormalizability was an indispensable factor for a theory to be a good candidate for the explanation of phenomena. Since they were supposed to be unrealistic, in the 1960s few theoretical physicists were working on Yang-Mills field theories.
In the meanwhile, a number of theoretical physicists were developing concepts and theories that led to the formulation of the Standard Model unified theory of electromagnetic and weak interactions by the end of the 1960s. Around 1964, various researchers employed the concept of spontaneous symmetry breaking of the original gauge symmetry to derive the so-called Higgs mechanism. This mechanism allowed for the generation of mass in field quanta that were massless in the unbroken Lagrangian. Between 1967 and 1968 Weinberg and, independently, Salam developed a theory of electroweak unification based on the SU(2)xU(1) gauge scheme initially proposed by S. Glashow in 1961. Explaining the origin of gauge bosons’ mass through the Higgs mechanism, the Weinberg-Salam theory overcame some of the theoretical difficulties affecting Glashow’s early proposal. However, one important element was still missing. While the authors believed the theory to be renormalizable, no formal proof was given. This theoretical shortcoming—along with the fact that the theory predicted the existence of weak neutral currents, which had never been observed—made the theory uninteresting. Nobody, not even the author, quoted Weinberg’s paper before 1971. Developing Veltman’s project and in relative isolation with respect to the aforementioned achievements, ‘t Hooft provided such a proof in 1971 as the main result of his PhD dissertation.
‘t Hooft’s Rediscovery of the Electroweak Theory and the Proof of Its Renormalizability
When ‘t Hooft was assigned to Veltman, the latter immediately gave the former the paper written by Yang and Mills in 1954 and exposed his views of the problem. However, ‘t Hooft did not begin working on this theory as his first research project. Veltman was also interested in the sigma-model—a field theory first put forward by Schwinger and employed by M. Gell-Mann and M. Levy as a heuristic tool for calculation in partially conserved axial current (PCAC) framework. The sigma-model was not a gauge theory, but had interesting proprieties such as spontaneous symmetry breaking, in which pions were interpreted as the massless Goldstone bosons. In 1969, S. Adler and, independently, J. Bell and R. Jackiw had discovered an anomalous non-conservation of chiral currents in the sigma-model. This anomaly led to the empirical prediction that massless neutral pions can decay into photons while the formal PCAC theory forbade this decay. ‘t Hooft became interested in the sigma-model and in the issues of spontaneous symmetry breaking and renormalization. After having completed his predoctoral thesis, he continued to work under the supervision of Veltman to a PhD dissertation aimed at the renormalization of the Yang-Mills theory.
Two events were fundamental in ‘t Hooft’s path towards his demonstration that the Yang-Mills theory was renormalizable. First, Veltman proposed a course on path integrals to be given at Utrecht University, and ‘t Hooft was in charge of writing the lecture notes. In this way, ‘t Hooft became an expert of this theoretical tool on which he later relied on in his proof of the renormalizability of the Yang-Mills theory. The second event was a conference held in Summer 1970 at Cargèse in which B. Lee, J. L. Gervais, and K. Symanzik discussed the renormalization of the sigma-model. The three physicists argued that the sigma-model was renormalizable provided that the vector bosons acquired mass through spontaneous symmetry breaking. Their lectures were instrumental in convincing ‘t Hooft that it was possible to renormalize the massive Yang-Mills theory using the same approach employed for the renormalization of the sigma-model. As soon as he came back to Utrecht, ‘t Hooft successfully developed this intuition.
Veltman suggested ‘t Hooft to focus the investigation on the renormalization of the massless unbroken Yang-Mills theory. While by the end of the 1950s a number of theorists had already provided a demonstration of the renormalizability of the massless Yang-Mills theory, the status of affairs remained quite unclear. As ‘t Hooft stressed, there was no straightforward prescription for the subtraction of infinities. The aim of ‘t Hooft’s first 1971 publication “Renormalization of Massless Yang-Mills Fields” was to put into practice what was only theorised in the previous analyses. In this paper, ‘t Hooft learned how to manipulate the Feynman rules and devised a new cut-off—or regulator—method, working only up to one loop—a method that was later considered a precursor of dimensional regularization. ‘t Hooft’s work came to represent the first detailed analysis concerning the renormalizability of massless Yang-Mills fields. More importantly, this paper was the preparatory work for his breakthrough discovery of the renormalizability of massive Yang-Mills fields. After having read ‘t Hooft’s first paper, Veltman insisted that the only physically significant theory was that in which one could explain the existence of massive vector bosons, and ‘t Hooft accepted the challenge. According to ‘t Hooft’s recollections, the extension of his methods to include the massive case was relatively straightforward. Nevertheless, his proof had a deep impact because it changed the attitude of many theoretical physicists towards massive Yang-Mills fields. Employing spontaneous symmetry breaking, ‘t Hooft rediscovered the Higgs mechanism and the electroweak theory, while at the same time writing the Feynman rules of such a theory. Thanks to the analytical methods he had developed, in his second 1971 paper “Renormalizable Lagrangians for Massive Yang-Mills Fields” he showed that the spontaneously broken gauge theory was renormalizable to all orders of the perturbation expansion.
Other physicists were not familiar with the methods employed by ‘t Hooft, and it was not clear that his work solved one of the main problems of the Weinberg-Salam electroweak theory. The Korean-born American physicist Ben Lee was the first expert to understand the importance of ‘t Hooft’s paper. Along with J. Zinn-Justin, Lee developed a different approach to the renormalization of gauge theories employing methods more familiar to the community of theoretical physicists. Thanks to their analyses circulated in 1971 and published in 1972, physicists became convinced that those gauge theories whose intermediate vector bosons acquire mass through the Higgs mechanism were renormalizable. One of the consequences of this novel understanding was that the Weinberg-Salam model for electroweak interactions gained respectability and experiments were planned to test its predictions. Starting from 1973, several experiments precisely verified some of the more interesting predictions of the electroweak theory such as the existence of weak neutral currents and of the charm quark. In 1979, the theory was publicly recognised through the Nobel Prize in Physics awarded to Glashow, Weinberg, and Salam “for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current.”
The two papers ‘t Hooft had published in 1971 constituted his PhD dissertation leading to his graduation in 1972. After these fundamental publications, ‘t Hooft continued to work on the renormalizability of gauge field theories by devising novel regulator methods. In 1972, ‘t Hooft and Veltman developed ‘t Hooft’s technique and put forward what is now called the dimensional regularization, which was independently proposed by South American physicists Giambiagi and Bollini. While pursuing this work, by 1972 ‘t Hooft had established that massless Yang-Mills fields are asymptotically free. This means that the gauge coupling constant tends to zero when energy increases —or, equivalently, distance decreases. This was an important achievement, which probably ‘t Hooft underestimated. He did not publish his analysis, while briefly reporting the final result in a small conference. One year later, Politzer and, independently, Gross and Wilczek made the same discovery and published it. This analysis allowed for a quantum field theoretical explanation of the scaling phenomena into the nucleons. Thanks to this demonstration, the quantum field theory of strong interactions of coloured quarks and gluons—christened quantum chromodynamics—became acceptable as the second pillar of the Standard Model of particle physics, the first being the electroweak theory. In 2004, such an endeavour gained Politzer, Gross, and Wilczek the Nobel Prize in Physics “for the discovery of asymptotic freedom in the theory of the strong interaction.”
After several corroborations have been found, the Standard Model is currently the orthodoxy in theoretical physics. ‘t Hooft’s paper on the renormalizability of massive Yang-Mills fields stands out as a turning point that changed the attitude of physicists towards non-Abelian gauge fields, previously regarded mainly as a mathematical curiosity. During the 1980s and 1990s, the consensus around the Standard Model continued to grow thanks to the combination of theoretical developments and empirical confirmations. In 1999, almost thirty years after his breakthrough discovery, ‘t Hooft shared with Veltman the Nobel Prize in Physics, “for elucidating the quantum structure of electroweak interactions in physics.”
A Scientific Life on the Fringe
After ‘t Hooft made his explosive entering in the world of theoretical physics, he worked on fundamental theories developing an uncommon interest for foundational issues. In the 1970s, Veltman and ‘t Hooft continued to collaborate in several investigations concerning renormalization procedures, mainly aimed at finding promising lines of attack to the renormalization of quantum gravity models. In 1981, Veltman decided to accept a position in the USA interrupting their long and productive collaboration.
At the same time, ‘t Hooft pursued important research projects on the gauge theory of quarks and tried to solve the quark confinement problem. While the quark model and, successively, quantum chromodynamics successfully explained some features of the strong interaction, it was not clear why no experiment had ever detected any isolated quark, characterised by its fractional charge. It was hypothesised that the quarks were perpetually confined, but no formal proof existed. In the 1970s, ‘t Hooft was one of the leading theorists working on quark confinement trying to understand the mathematical structure and its physically testable consequences. This work led ‘t Hooft to the discovery of instantons as well as of magnetic monopole solutions of the Higgs theory—a prediction that is still under investigation. By the 1980s, the physical mechanism of the quark confinement seemed to have been solved based in the framework of lattice chromodynamics – a non-perturbative approach to the gauge theory of quarks and gluons. The evolution of the lattice chromodynamics approach has led ‘t Hooft to believe that the confinement problem has become rather a mathematical than a physical one. As a consequence, he turned his interest towards more physical topics.
In 1984, ‘t Hooft was not impressed by the mathematical coherence of the superstring theory and continued to work in relative isolation with respect to the new trends in theoretical physics. By pursuing a somewhat personal research project, he aimed at replacing general relativity and the Standard Model with a novel paradigm starting from the idea of quantum black hole. In this context, ‘t Hooft devised a feature of the quantum gravitation degrees of freedom, that was later christened “Holographic principle” in discussion with L. Susskind.
‘t Hooft has also held idiosyncratic views about the interpretation of quantum mechanics favouring a deterministic interpretation based on hidden variables—an approach that has further distanced him from the mainstream of theoretical physics. ‘t Hooft’s interest in foundational questions also led him to become the editor-in-chief of the monthly journal Foundations of Physics in 2007. The journal is devoted to the conceptual bases of modern physics. The approach of this journal was patently different from that of other physics publications in that it favours methodological, philosophical, and logical discussions on fundamental theories. As he emphasised in his Nobel Prize autobiographical sketch, some progresses in string theories have led to a renewed interest in quantum black holes and the holographic principle. As a consequence, ‘t Hooft has found himself “to be nearly back in the ‘mainstream’ of physics.”
Hoddeson, L., Brown, L. M., Riordan, M. & Dresden, M. (1995) The Rise of the Standard Model: Particle Physics in the 1960s and the 1970s. Cambridge University Press, Cambridge.
Pais, A. (1986) Inward Bound: Of Matter and Forces in the Physical World. Clarendon Press, Oxford.
Pickering, A. (1999). Constructing quarks: A sociological history of particle physics. University of Chicago Press, Chicago.
‘t Hooft, G. (2000) Gerardus ‘t Hooft – Biographical. Nobelprize.org. Nobel Media AB 2013. Accessed 2 Mar 2014. http://www.nobelprize.org/nobel_prizes/physics/laureates/1999/thooft-bio.html
‘t Hooft, G. (2000) A confrontation with infinity. International Journal of Modern Physics A, 15(28), pp. 4395-4406.
Veltman, M. J. (2000). From weak interactions to gravitation. International Journal of Modern Physics, A 15 (29), pp. 4557-4573.
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