RETROSPECTIVE RETROSPECTIVE Ken Wilson: A scientific appreciation

Frank A. Wilczek1 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

This brief retrospective is an overview and die. We will have colossal domains, per- celebration of Kenneth Wilson’sprofound meated by huge subdomains, themselves contributions to science, meant to be acces- permeated by very large subdomains. By sible to a broad scientific audience. Wilson’s considering the differently sized domains lecture on receipt of the 1982 Nobel Prize (1) as material objects in their own right, we gives a more technical and detailed descrip- discover a plausible explanation for why tion of his central work, including historical the emergent structure is both self-similar perspective and extensive references. and self-determining (i.e., scale invariant and Phase transitions, such as the boiling of universal). August de Morgan’s couplet (6) water, are a familiar phenomenon of every- captures the essence of this picture: day life. J. W. Gibbs (2), J. van der Waals (3), fl fl and others, in now-classic work, illumi- Great eas have little eas upon their backs to bite ’em, nated many aspects of phase transitions. fl fl And little eas have lesser eas, and so Ken Wilson. However, one feature, known as critical be- ad infinitum. havior, stubbornly eluded theoretical under- standing even as it revealed astonishing Kadanoff ’s picture is insightful and com- In experiments at ever higher energies we regularities. pelling, but does not in itself provide a quan- probe ever shorter distances. To describe such By way of illustration, it will be easiest to titative tool. Wilson’s breakthrough work (7, experiments, we now gradually “integrate in” discuss a concrete special case. As a ferro- 8) in 1971 did just that. The mathematical short-distance fluctuations. In 1969 Wilson magnet is heated, its magnetism diminishes, technique is too complicated for meaningful sketched (10) how one might apply this ver- until at a critical temperature (the Curie tem- presentation here. It is both possible and ap- sion of renormalization group ideas, given perature) it vanishes altogether. Closer study propriate, however, to mention the techni- a theory of the strong interaction. At Prince- reveals surprising regularities, first clearly que’sleadingideas. ton in the spring of 1972, Wilson gave a series codified by B. Widom (4). Just below the We have fluctuations on many different of seminars on his recent work, which both critical temperature, the strength of the length scales that overlap and affect one David Gross and I attended. Those inspiring remaining magnetic field vanishes as a frac- another. Because we are most interested in lectures informed our subsequent research tional power, called the “critical exponent,” behavior at very large scales, we will be sat- and helped to lure me from mathematics of the temperature difference. Amazingly, fi fi is ed to nd an effective theory that averages to physics. Building on Wilson’sindications, the same value of the critical exponent is “ ” fl fl or integrates out the in uence of uctua- andworkingbackward,weusedobserved observed in many different materials with tions smaller than some cut-off. Wilson was phenomena to infer the required theory vastly different Curie temperatures, a prop- able to formulate equations that describe how (11). [H. D. Politzer (12) performed the erty we call “universality.” Thus, critical the effective theory changes as one changes key calculation independently.] That the- exponents are nontrivial numbers, char- the cut-off slightly. This concept, that one ory,quantumchromodynamicsorQCD,is acteristic of macroscopic matter, that can derive equivalent effective theories with emerge “from nothing.” Why do they exist, now accepted as the fundamental theory of different cut-offs, is called the “renormaliza- why are they universal, and how can we the strong force. tion group.” At the critical point, where the compute them? Wilson was quick to appreciate the prom- dynamics is scale-invariant, the form of the Leo Kadanoff (5) supplied a physical pic- ise of QCD, but he realized that to calculate effective theory must be independent of the ’ ture that answers the first two of these ques- the theory s consequences at (relatively) low ’ cut-off. That requirement, formulated as an energies or long distances would be a de- tions. (Please note, however, that Kadanoff s ’ discussion is considerably more sophisticated equation, determines the theory. In collabo- manding task. Wilson sapproach(13)was than the following caricature.) We can think ration with Michael Fisher, Wilson found revolutionary in its directness: He formulated of the magnet as having domains, within a simple way to get approximate solutions. the theory in computer-friendly form, essen- fi which the underlying spins share a common Their paper (9), with the wonderful title tially as a complicated de nite integral in “ ” alignment. Below the critical temperature Critical phenomena in 3.99 dimensions, a space of enormously large dimension, and asingledomainpermeatestheentiresam- launched an avalanche of extensions and then set out to perform the integral nu- ple, and so there is a nonzero net magne- applications. merically. Many years passed before com- tization. Above the critical temperature no We also must deal with fluctuations in the puters and algorithms were up to the job, fi single domain permeates the sample, and quantum-field theories relevant to high- but lattice gauge theory amply ful lled the magnetization averages to zero. Near the energy physics: they are quantum fluctua- critical temperature there are domains that tions, rather than thermal fluctuations, but Author contributions: F.A.W. wrote the paper. nearly permeate the sample but eventually they pose related mathematical challenges. 1E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1312463110 PNAS | August 6, 2013 | vol. 110 | no. 32 | 12855–12856 Downloaded by guest on October 1, 2021 Wilson’svision.Thehighestawardinthe students encounter in learning, and to over- Ken Wilson radiated joy in his work. As I field, awarded annually at the major interna- come them. It would be unwise to bet against visualize him delivering a seminar, invariably tional conference in Lattice Field Theory, Wilson’s prophetic visions. he is smiling. bears Wilson’sname. Wilson loved to work with computers— preferably supercomputers—and was a 1 Wilson KG (1993) Nobel Lecture: “The renormalization group and critical 7 Wilson KG (1971) Renormalization group and critical phenomena. phenomena.” Nobel Lectures 1981–1990, ed Ekspöng G (World Scientific, I. Renormalization group and the Kadanoff scaling picture. Phys Rev B prophet on their behalf. It is no accident Singapore). Available at http://nobelprize.org/nobel_prizes/physics/laureates/ 4:3174–3183. that Paul Ginsparg, the primary architect 1982/wilson-lecture.pdf#search=’wilson‘. Accessed July 15, 2013. 8 Wilson KG (1971) Renormalization group and critical phenomena. II. of arXiv.org, was his student. 2 Gibbs JW (1874–1878) On the Equilibrium of Heterogeneous Phase-space cell analysis of critical behavior. Phys Rev B 4:3184–3205. ’ Substances. Transactions of the Connecticut Academy of Arts and 9 Wilson KG, Fisher ME (1972) Critical exponents in 3.99 In later years Wilson s interests turned to- Sciences 3:108–248, 343–524. dimensions. Phys Rev Lett 28:240–243. ward other challenges. He strove to bring the 3 Van der Waals J (1910) The Equation of State for Gases and Liquids 10 Wilson KG (1969) Non-lagrangian models in current algebra. (Nobel Lecture). Available at http://nobelprize.org/nobel_prizes/physics/ Phys Rev 179:1499–1512. power of supercomputing and the lessons of laureates/1910/waals-lecture.pdf. Accessed July 15, 2012. 11 Gross D, Wilczek F (1973) Ultraviolet behavior of non-Abelian critical behavior and QCD to bear on quan- 4 Widom B (1965) Equation of state in the neighborhood of the gauge theories. Phys Rev Lett 30:1343–1346. tum chemistry. In a more radical departure, critical point J. Chem Phys 43:3898–3905. 12 Politzer HD (1973) Reliable perturbative results for strong 5 Kadanoff L (1966) Scaling laws for Ising models near Tc. Physics 2:263. interactions? Phys Rev Lett 30:1346–1349. Wilson proposed to use computers to diag- 6 De Morgan A (1872) A Budget of Paradoxes # 377 (Longmans 13 Wilson KG (1974) Confinement of quarks. Phys Rev D Part Fields nose, scientifically, the specificdifficulties that Green, London). 10(8):2445–2459.

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