Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis

This book presents a comprehensive account of the renormalization-group (RG) method and its extension, the doublet scheme, in a geometrical point of view.

It extract long timescale macroscopic/mesoscopic dynamics from microscopic equations in an intuitively understandable way rather than in a mathematically rigorous manner and introduces readers to a mathematically elementary, but useful and widely applicable technique for analyzing asymptotic solutions in mathematical models of nature.

The book begins with the basic notion of the RG theory, including its connection with the separation of scales. Then it formulates the RG method as a construction method of envelopes of the naive perturbative solutions containing secular terms, and then demonstrates the formulation in various types of evolution equations. Lastly, it describes successful physical examples, such as stochastic and transport phenomena including second-order relativistic as well as nonrelativistic fluid dynamics with causality and transport phenomena in cold atoms, with extensive numerical expositions of transport coefficients and relaxation times.

Requiring only an undergraduate-level understanding of physics and mathematics, the book clearly describes the notions and mathematical techniques with a wealth of examples. It is a unique and can be enlightening resource for readers who feel mystified by renormalization theory in quantum field theory.

Sheds light on geometrical notion of renormalization group theory Offers systematic asymptotic analysis in evolution equations Describes contents a priori and clearly

Auteur

Teiji Kunihiro is Professor Emeritus at Kyoto University in Japan and specializes in research in nuclear and hadron physics theory and mathematical physics. He received his Doctor of Science in Physics from Kyoto University in 1981. After serving as Associate Professor and Professor at Ryukoku University, he was appointed as Professor at the Yukawa Institute for Theoretical Physics, Kyoto University in 2000, and was Vice Director of the institute from 2006 to 2007, before moving to the Department of Physics in 2008.

Yuta Kikuchi is a Goldhaber Fellow at Brookhaven National Laboratory in the USA at the completion of the present book, and appointed to a scientist at Cambridge Quantum Computing starting in 2022. He received his Doctor of Science in Physics from Kyoto University in 2018. He was awarded the Research Fellowship for Young Scientists by Japan Society for the Promotion of Science (JSPS) in 2015, and the Goldhaber distinguished fellowship by Brookhaven National Laboratory in 2020. He currently focuses on designing quantum algorithms for near-term quantum computing.

Kyosuke Tsumura is Primary Research Scientist at the Analysis Technology Center, Fujifilm Corporation in Japan. He joined the Analysis Technology Center as a Researcher in 2006 and was promoted to current position. He received his Doctor of Science in Physics from Kyoto University in 2013. He is Leader of a project developing a novel computational method for efficient drug design.

Texte du rabat

This book presents a comprehensive account of the renormalization-group (RG) method and its extension, the doublet scheme, in a geometrical point of view. It extract long timescale macroscopic/mesoscopic dynamics from microscopic equations in an intuitively understandable way rather than in a mathematically rigorous manner and introduces readers to a mathematically elementary, but useful and widely applicable technique for analyzing asymptotic solutions in mathematical models of nature. The book begins with the basic notion of the RG theory, including its connection with the separation of scales. Then it formulates the RG method as a construction method of envelopes of the naive perturbative solutions containing secular terms, and then demonstrates the formulation in various types of evolution equations. Lastly, it describes successful physical examples, such as stochastic and transport phenomena including second-order relativistic as well as nonrelativistic fluid dynamics with causality and transport phenomena in cold atoms, with extensive numerical expositions of transport coefficients and relaxation times. Requiring only an undergraduate-level understanding of physics and mathematics, the book clearly describes the notions and mathematical techniques with a wealth of examples. It is a unique and can be enlightening resource for readers who feel mystified by renormalization theory in quantum field theory.

Contenu

Notion of Effective Theories in Physical Sciences.- Divergence and Secular Term in the Perturbation Series of Ordinary Differential Equations.- Traditional Resummation Methods.- Elementary Introduction of the RG method in Terms of the Notion of Envelopes.- Ei-Fujii-Kunihiro Formulation and Relation to Kuramoto's reduction scheme.- Relation to the RG Theory in Quantum Field Theory.- Resummation of the Perturbation Series in Quantum Methods.- Illustrative Examples.- Slow Dynamics Around Critical Point in Bifurcation Phenomena.- Dynamical Reduction of A Generic Non-linear Evolution Equation with Semi-simple Linear Operator.- A Generic Case when the Linear Operator Has a Jordan-cell Structure.- Dynamical Reduction of Difference Equations.- Slow Dynamics in Some Partial Differential Equations.- Some Mathematical Formulae.- Dynamical Reduction of Kinetic Equations.- Relativistic First-Order Fluid Dynamic Equation.- Doublet Scheme and its Applications.- Relativistic Causal Fluid dynamic Equation.- Numerical Analysis of Transport Coefficients and Relaxation Times.- Reactive Multi-component Systems.- Non-relativistic Case and Application to Cold Atoms.- Summary and Future Prospects.