Symmetry in Physics

Symmetry plays an essential role in science - not only in crystallography and quantum theory, where its role has long been explicitly recognized, but also in condensed-matter physics, thermodynamics, chemistry, biology, and others. This text discusses the concept of symmetry and its application to many areas of science. While it includes a detailed introduction to the theory of groups, which forms the mathematical apparatus for describing symmetries, it also includes a much more general discussion of the nature of symmetry and its role in science. Many problems serve to sharpen the reader's understanding, and an extensive bibliography concludes the book.

"And what is the use," thought Alice, "of a book without pictures or conversations in it?" -Lewis Carroll This book is written for modem undergraduate students - not the ideal stu­ dents that mathematics professors wish for (and who occasionally grace our campuses), but the students like many the author has taught: talented but ap­ preciating review and reinforcement of past course work; willing to work hard, but demanding context and motivation for the mathematics they are learning. To suit this audience, the author eschews density of topics and efficiency of presentation in favor of a gentler tone, a coherent story, digressions on mathe­ maticians, physicists and their notations, simple examples worked out in detail, and reinforcement of the basics. Dense and efficient texts play a crucial role in the education of budding (and budded) mathematicians and physicists.

This book is the second edition, whose original mission was to offer a new approach for students wishing to better understand the mathematical tenets that underlie the study of physics. This mission is retained in this book.

The Second International Conference “Symmetry in Nonlinear Mathematical Physics” was organized mainly due to efforts of Professor Wilhelm Fushchych. It happened, however, that this out- standing scientist and kind person passed away. The Conference was dedicated to his memory. Among 80 participants of the Conference there were representatives of 14 countries from 5 continents. The main part of the lectures was devoted to investigation of symmetries and construction of exact solutions of nonlinear differential equations.

This book consists of the articles published in the special issues of this Symmetry journal based on two-by-two matrices and harmonic oscillators. The book also contains additional articles published by the guest editor in this Symmetry journal. They are of course based on harmonic oscillators and/or two-by-two matrices. The subject of symmetry is based on exactly soluble problems in physics, and the physical theory is not soluble unless it is based on oscillators and/or two-by-two matrices. The authors of those two special issues were aware of this environment when they submitted their articles. This book could therefore serve as an example to illustrate this important aspect of symmetry problems in physics.