Irreducible Cartesian tensors
From a geometrical point of view, a three-dimensional Cartesian tensor is an object which depends on two or more directions. This book studies such objects and their rotational properties, in particular those which are irreducible under the three-dimensional rotation group. The monograph is based on papers written by J. A. R. Coope and the author, together with some generalisations and developments. It is written primarily with the needs and interests of physicists, chemists and other physical scientists in mind. Concepts are clearly explained and the presentation draws mainly on physical concepts, rather than the definition, theorem and proof approach used in mathematical presentations. The book emphasises almost exclusively the coordinate-free representation of vectors and tensors. It is essentially self-contained and includes an appendix on some formalities of linear algebra, which covers both the mathematical and physical way of defining certain terms. It contains a comprehensive introduction, followed by ten chapters. The final chapter gives some applications to quantum mechanics. The book contains a bibliography and although it does not have an index, a useful index of symbols is provided. Chapter 2 gives an introduction to vectors and tensors from a physical viewpoint. The unit second-order tensor and the Levi-Civita tensor are introduced. Properties of rotations in two dimensions are then studied. It is shown how a rotation can be described in terms of a tensor. The diagrammatic representation of a rotation tensor is introduced and this method of representation is then used throughout the book. The chapter goes on to study rotations in three dimensions and the three-dimensional rotation group. The objective of Chapter 3 is to study how a tensor behaves under a rotation and describes how it is often possible to reduce the tensor into rotationally independent parts. If no further reduction is possible, these parts are irreducible representations of the rotation group. The chapter begins by proving that the unit second-order tensor and the Levi-Civita tensor are rotational invariants. The notion of a rotationally invariant mapping between tensor spaces then provides the background for the reduction, under the action of the rotation group, of a given tensor into its irreducible parts. This leads on to the idea of projection tensors and a discussion of various properties of the rotation group. Chapter 4 describes ``Irreducible Cartesian Tensors'' (ICTs) constructed from one or more vectors. The scalar product between two ICTs of order \(p\) is defined, and it is shown that, for tensors built from vectors, this scalar product is proportional to the Legendre polynomial of order \(p\). Integrals of products of two ICTs formed from a common vector, and then the expansion of an arbitrary tensor product of a vector with itself, are discussed. Finally, it is stressed how integrals of products of vector components can be expressed in terms of a set of tensor integrals, each of which is evaluated in terms of invariant tensors. In Chapter 5, spherical tensors are used to discuss the spherical components of an arbitrary ICT. The general connections between Cartesian and spherical tensors are then demonstrated. Chapter 6 introduces the 3-\(j\) coupling tensors together with their normalization factors and certain recursion relations satisfied by them. Products of ICTs are reduced to a linear combination of ICTs (the Clebsch-Gordan reduction) using 3-\(j\) tensors. In the following chapter, the orthogonality and expansion properties of the 3-\(j\) tensors are studied. The chapter goes on to discuss the connection between 3-\(j\) tensors and Wigner's 3-\(j\) symbols. In Chapters 8 and 9, other properties of the rotation group are given in terms of ICTs. In particular, the 6-\(j\) and 9-\(j\) symbols, together with the rotation matrices, are written in terms of ICTs. The idea of ICTs is generalised in Chapter 10 to the case of spinors, that is, to the space of tensors based on two-dimensional complex vectors with irreducibility under the action of the group \(\operatorname{SU}(2)\). This group is the Lie group of \(2\times2\) unitary matrices with unit determinant and is a natural extension of the rotation group. The presentation of this chapter is novel due to the coordinate free approach used. Finally, in Chapter 11, various properties of the quantum mechanics of angular momentum are described in terms of ICTs. The chapter ends with a discussion of some of the properties of tensors of angular momentum.
scientific article; zbMATH DE number 6806643