Linear algebra and group theory for physicists
Linear algebra and group theory belong to those mathematical disciplines that also find extensive use in theoretical physics. Therefore a solid theoretical foundation in these areas of mathematics, together with their concrete applications to physics, is an indispensible prerequisite for any budding scholar in modern natural science. The book under review, taking this requirement into account, is especially directed to undergraduate and graduate students in physics, who wish to acquire a profound basic knowledge in (multi-)linear algebra and group theory as used (and specifically handled) in this realm. The author, highly experienced and acknowledged teacher in mathematical physics over decades, and well-known by a number of other textbooks in this field, provides here a comprehensive, tailor-made introduction to those aspects of linear algebra and group theory that are of particular importance in particle physics. The first edition of this textbook was published in 1996, and has gained a remarkable popularity since then, mainly due to its exceptionally clear and detailed exposition. As to the contents, the book contains eleven chapters, each of which is subdivided into numerous, sometimes up to twenty sections. Whilst the first four chapters on elementary group theory and linear algebra cover most of the very standard material, though in a physicist's understanding, terminology, and notation, and are basically included to make the book entirely self-contained, the remaining seven chapters form the heart of the text, offering partly highly special topics and applications with a view toward physics. Chapter 1 introduces the elements of group theory up to the homomorphism theorem, with a special emphasis on symmetric groups and other concrete examples. Chapter 2 very briefly discusses other algebraic structures such as rings, division rings, fields, vector spaces, associative algebras, quaternions, Lie rings, and Lie algebras. Chapter 3 is devoted to the basics of linear algebra, inclusive of bases in vector spaces, subspaces, linear maps, matrix calculus, eigenvalues and eigenvectors, canonical forms of matrices, Euclidean spaces, tensor spaces, the exponential of a matrix, Hermitean and unitary matrices, and -- with a view to quantum mechanics -- the Dirac bracket notation. Chapter 4 concludes the purely introductory mathematical part of the book with explaining the elements of representation theory in general. Chapter 5 goes into greater depth by discussing the representation theory of finite groups in a more detailed manner. Chapter 6 turns then to a special feature of this book, namely to a finer analysis of the representations of linear associative algebras, with a particular emphasis on the physically relevant Dirac and Kemmer algebras. Chapter 7 deals with representations of symmetric groups and their description via Young tableaux, including applications to the construction of the symmetry classes of certain tensors occuring in particle physics. Chapter 8 explains in greater detail the rotation group and its various representations, culminating in a discussion of the Clebsch-Gordan theorem and the Wigner-Eckart theorem in their physical context. This is followed, in Chapter 9, by a systematic derivation of the 32 crystallographic point-groups of first and second type. Chapter 10 gives a very elaborated exposition of the Lorentz group and its representations, and is therefore of particular physical relevance. Finally, Chapter 11 touches upon the classification of Lie groups, which may serve as a brief introduction to Dykin diagrams for physicists. Written as a mathematical textbook for students in physics, thereby using both the way of mathematical reasoning and the characteristic terminology generally established in this field of science, this book certainly serves its purpose in a masterly manner. On the other hand, as theoretical physics and abstract mathematics become increasingly interrelated, the present textbook also offers a wealth of concrete, inspiring applications and examples valuable for mathematicians who wish to widen their scope.
scientific article; zbMATH DE number 5009677