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Physics Today

Theory of dynamic critical phenomena

Bertrand I. Halperin

Published Feb 1, 2019• 23 citations
Publisher

Abstract

New mathematical approaches have extended physicists’ understanding of magnets, superfluids, and other complex systems.
DOI
10.1063/pt.3.4137

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Introducing a unified framework for describing and understanding complex interacting systems common in physics, chemistry, biology, ecology, and the social sciences, this comprehensive overview of dynamic critical phenomena covers the description of systems at thermal equilibrium, quantum systems, and non-equilibrium systems. Powerful mathematical techniques for dealing with complex dynamic systems are carefully introduced, including field-theoretic tools and the perturbative dynamical renormalization group approach, rapidly building up a mathematical toolbox of relevant skills. Heuristic and qualitative arguments outlining the essential theory behind each type of system are introduced at the start of each chapter, alongside real-world numerical and experimental data, firmly linking new mathematical techniques to their practical applications. Each chapter is supported by carefully tailored problems for solution, and comprehensive suggestions for further reading, making this an excellent introduction to critical dynamics for graduate students and researchers across many disciplines within physical and life sciences.
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Social Quantum Field Theory Toward a Mathematical Theory of Social Fields First Edition (2026) Author: Chou I-Hsien Dedication To those who believe that mathematics is not merely a collection of formulas, but a language through which the hidden structures of reality may gradually be revealed. To every mathematician who has sought unity beneath diversity, to every physicist who has transformed geometry into dynamics, to every computer scientist who has discovered structure within information, to every philosopher who has pursued clarity through logic, and to every student who has dared to ask why seemingly unrelated phenomena often obey remarkably similar mathematical principles — this work is respectfully dedicated. The development of mathematics has always depended upon imagination disciplined by rigor. Every new mathematical language begins with a simple question: can apparently different systems be understood through the same underlying structure? This question inspired analytic geometry, calculus, group theory, differential geometry, Hilbert spaces, information theory, and category theory. It is the same question that motivates the present work. Social Quantum Field Theory is offered not as the final answer, but as one possible step toward a broader mathematical synthesis. If future generations discover that many of its ideas are incomplete, they should improve them. If some ideas are incorrect, they should replace them. If entirely better formulations emerge, they should be embraced without hesitation. Such is the natural evolution of mathematics. No mathematical framework belongs permanently to its first author. Every worthwhile theory eventually becomes part of a larger intellectual tradition, refined through criticism, generalization, and collaboration across generations. The greatest hope for this work is therefore not that it remains unchanged, but that it becomes unnecessary — because future mathematicians will have constructed something even more elegant upon its foundations. May this book encourage its readers to value rigor over rhetoric, structure over appearance, proof over assertion, curiosity over certainty, and understanding over complexity. For mathematics advances not by defending established ideas, but by continually discovering deeper patterns that unite what once appeared separate. "The pursuit of structure is the pursuit of understanding." Dedicated to all researchers who seek unity through mathematics, and all future scholars who will carry that search further.