Kenneth Wilson's Nobel Prize winning breakthrough in the renormalization group theory of phase transition and critical phenomena almost overlapped with the violent "cultural revolution" years (1966–1976) in China. An unexpected chance in 1972 brought the author of these lines close to the Wilson–Fisher ϵ-expansion of critical exponents and eventually led to a joint paper with Lu Yu published entirely in Chinese without any English title and abstract. Even the original acknowledgment was deleted because of mentioning foreign names like Kenneth Wilson and Kerson Huang. In this article I will tell the 40-year old story as a much belated tribute to Kenneth Wilson and to reproduce the essence of our work in English. At the end, I give an elementary derivation of the Callan–Symanzik equation without referring to field theory.
Kenneth Wilson's Nobel Prize winning breakthrough in the renormalization group theory of phase transition and critical phenomena almost overlapped with the violent "cultural revolution" years (1966–1976) in China. An unexpected chance in 1972 brought the author of these lines close to the Wilson–Fisher ϵ-expansion of critical exponents and eventually led to a joint paper with Lu Yu published entirely in Chinese without any English title and abstract. Even the original acknowledgment was deleted because of mentioning foreign names like Kenneth Wilson and Kerson Huang. In this article I will tell the 40-year old story as a much belated tribute to Kenneth Wilson and to reproduce the essence of our work in English. At the end, I give an elementary derivation of the Callan–Symanzik equation without referring to field theory.
The 1982 Nobel prize in physics has been awarded to Kenneth G. Wilson of Cornell University “for his theory for critical phenomena in connection with phase transitions.” In its announcement the Royal Swedish Academy of Sciences said that Wilson's renormalization-group approach to critical phenomena, particularly two Physical Review papers in 1971, “gave a complete theoretical description of the behavior close to the critical point and gave also methods to calculate numerically the crucial quantities.”
The tasks posed for renormalization group theory (RGT) within statistical physics by critical phenomena theory in the 1960's are set out briefly in contradistinction to quantum field theory (QFT), which was the origin for Ken Wilson's concerns. Kadanoff's 1966 block spin scaling picture and its difficulties are presented; Wilson's early vision of flows is described from the author's perspective. How Wilson's subsequent breakthrough ideas, published in 1971, led to the epsilon expansion and the resulting clarity is related. Concluding sections complete the general picture of flows in a space of Hamiltonians, universality and scaling. The article represents a 40% condensation (but with added items) of an earlier account: Rev. Mod. Phys.70, 653–681 (1998).
This article offers a historical development review of the theory of phase transitions and critical phenomena, the branch of physics whose development from Thomas Andrews's 1869 discovery of the critical point through Kenneth Wilson's renormalization group transformed the discipline's understanding of how collective behavior emerges from microscopic parts. The review reconstructs the development across six phases: the continuity surprise, Andrews's critical point and Johannes Diderik van der Waals's equation of state, whose 1873 dissertation unified the liquid and the gas and whose corresponding states law organized the fluids; the mean field era, Pierre Curie's magnetic measurements, Paul Weiss's molecular field hypothesis, and Lev Landau's order parameter, whose symmetry-based framework dominated the discipline for four decades; the exact solution, Wilhelm Lenz and Ernst Ising's model, Rudolf Peierls's argument, Lars Onsager's 1944 solution of the two-dimensional model, and Chen Ning Yang's spontaneous magnetization, whose eighth-root exponent destroyed the mean field's claims; the scaling synthesis, Edward Guggenheim's empirical corresponding states, Benjamin Widom's equation of state, and Leo Kadanoff's block spins, whose scaling laws connected the exponents; the renormalization group, Kenneth Wilson's iterative integration of fluctuations and Michael Fisher's epsilon expansion, whose fixed points explained universality itself; and the precision era, whose microgravity calorimetry and conformal bootstrap computations pushed the exponents to accuracies beyond experiment. Three synthetic claims are advanced. First, the theory's history is the defeat of mean field reasoning by exact mathematics and careful experiment, followed by the renormalization group's explanation of both the failure and the rare successes. Second, the universality discovery established that microscopic detail is irrelevant at criticality, the deepest expression of Philip Anderson's principle that more is different. Third, the renormalization group redefined physical theory itself, replacing single equations with methods that connect scales. An agenda is proposed spanning bootstrap precision, nonequilibrium criticality, and quantum matter.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].