Irreducible Cartesian Tensors

Robert F. Snider

• Publisher: De Gruyter
• Year: 2017
• ISBN:
  9783110563733
• Edition: 1
• Topic: Naturwissenschaften
• Subtopic: Quantentheorie

This monograph covers the concept of cartesian tensors with the needs and interests of physicists, chemists and other physical scientists in mind. After introducing elementary tensor operations and rotations, spherical tensors, combinations of tensors are introduced, also covering Clebsch-Gordan coefficients. After this, readers from the physical sciences will find generalizations of the results to spinors and applications to quantum mechanics.

1Introduction

The method of Irreducible Cartesian Tensors is a means of simultaneously accomplishing two objectives. First, since we live in a 3-dimensional world, we have an inherent sense of a Cartesian vector as having a direction (and also a magnitude). For example, we inherently point in a direction when describing where something is. A Cartesian tensor is just a generalization of that to when something depends on two or more directions, such as the pressure tensor that describes the direction of the force applied by a fluid to a wall, when the wall’s direction is described by the vector normal to its surface. Most objects that we deal with are described in terms of Cartesian directions, and this is the natural way we visualize objects in the real world. The second objective deals with the inherent isotropy of space. All directions are equivalent, or if one direction is preferred, then that preference is associated with some phenomena, e.g. gravity. But if we rotate in an arbitrary manner both the object of interest and the device (or object, e.g. the earth in the case of gravity) responsible for the preference in direction, then the physical properties remain unchanged. Such an invariance implies something about the phenomena and how its description can be quantified. Thus a classification of an object according to its rotational properties is useful. Such a classification can always be made in terms of the irreducible representations of the 3-dimensional rotation group. An irreducible Cartesian tensor belongs to an irreducible representation of the rotation group, but it is written in terms of Cartesian directions, thus providing a means of satisfying both objectives.

This book was written with the needs and interests of chemists, physicists and other physical scientists in mind. The subject is an area of mathematics or, more correctly, a combination of areas of mathematics, as described in the previous paragraph, but the method of presentation draws on physical concepts rather than the definition, theorem or proof commonly used in mathematical presentations. There has been an attempt to make the presentation self-contained, and Appendix A gives an introduction to some of the mathematical concepts used in the book, with an attempt to cover both the mathematical and physical (physicist’s, chemist’s, etc.) way of defining these terms. The book also presents a number of relations between well-known quantities that the author believes are new. A mathematician may find some of these of interest, in particular the chapter on spinors, which presents, to the author’s knowledge, a novel way of formulating some of their properties.

In the teaching of physics and chemistry, the presentation of the rotation group is usually associated with the quantum mechanics of angular momentum. It is stressed that these two notions are fundamentally two different things. The description of the orientation of an object and how this changes under a rotation has inherently nothing quantum or even mechanical about it but merely describes how its orientation is, or could be, changed by a rotation. For example, the direction vector pointing to a tree can be rotated to point to a bush. Nothing has moved, but a comparison of the sightings can be made, and the angle between them measured, possibly for the purpose of taking a picture. In contrast, angular momentum is a mechanical property associated with the physical motion of a physical object. Most of this book involves the properties of the rotation group without any reference to mechanics. However, Chap. 11 does discuss a number of aspects of the quantum mechanics of angular momentum and describes these in terms of Irreducible Cartesian Tensors. A combination of these two notions is the classification of tensors of the angular momentum operator according to the irreducible representations of the rotation group. Thus the rotation group enters both as a mechanical property (the angular momentum vector operator) and as a means of mathematically classifying the functions of this operator; see Chap. 11.

It is most common to express the irreducible representations of the 3-dimensional rotation group in terms of spherical harmonics and “spherical tensors”; see, for example, Refs [1–7]. These are sets of quantities that form irreducible representations of the rotation group but are also classified by how they rotate about a particular axis, usually chosen as the $\widehat{z}$

-axis. As such, they have an inherent bias to this axis, and if it is desired to classify an object about some other set of directions, then it is necessary to transform or take appropriate linear combinations in order to do this. Both the set of final directions and the initial axis of classification enter into such a computation. Irreducible Cartesian Tensors have an advantage here! Since there is no preferential direction under which these are given, only the desired final set of directions needs to be considered.

A further difficulty with spherical tensors is that they involve complex numbers and phase factors. When working with and combining such quantities, it is necessary to be very careful that all phase factors are entered correctly. Irreducible Cartesian Tensors are real, so there is at most a plus or minus sign that must be worried about when applying them. The formula for a particular spherical harmonic usually has to be looked up with its combinations of sines and cosines of angles. In contrast, the structure of an Irreducible Cartesian Tensor in natural form is immediate, having a definite symmetry, namely being symmetric and traceless between every pair of indices. This can be visualized geometrically. There are normalization factors for both systems, so there is no advantage to either system in that regard. But in contrast to the many phase factors in the spherical tensor formalism, even the sign that enters when combining Irreducible Cartesian Tensors is associated with some symmetry property of those tensors involved in the combination.

Manipulations involving, in particular combining, irreducible representations of the rotation group can be given diagrammatic interpretations. Some of these are very well known and have simple structures, in particular the scalar n-j symbols ( n ≥ 6). These diagrams, being rotational invariants, are the same for both the spherical and Cartesian ways of expressing the irreducible representations. But for more directionally dependent combinations of irreducible representations, the diagrammatic interpretation based on spherical tensors requires an elaboration on phases; see in particular the work by Yutsis and coworkers [7]. In contrast, the diagrammatic interpretation for combining Irreducible Cartesian Tensors needs to worry only about which sets of symmetric traceless indices are to be contracted together.

In many calculations the object of interest is a function of several vectors, some of which may be averaged (integrated) over. The object then remains only a function of those vectors that have not been integrated over. Often that dependence is linear in each of the remaining directions or at most a small power of each. In such a case, the calculation of the object is equivalent to the calculation of the set of coefficients of the various combinations of the vectors on which the object explicitly depends. Such coefficients are tensors, involving the integration over certain directions, but inherently are rotational invariants since they do not depend on any directions. The method of Irreducible Cartesian Tensors is appropriate for carrying out such calculations since the coefficients are recognized as invariant Cartesian tensors. A classification using symmetry can then reduce the number of independent integrals that need to be calculated. Chapter 4 elaborates on the possible invariants resulting from an integral over a single spatial variable, the direction of the vector r. An introduction to the analogous classification of invariants arising from the trace over a product of angular momentum operators is given in Chap. 11. It is the method of Irreducible Cartesian Tensors that suggests using rotational invariants for carrying out such calculations.

It would take a historian to do justice to the multitude of contributions to vector and tensor calculus and to the number of different terminologies and ways of presenting these quantities. The present work involves Cartesian vectors and tensors and ignores all aspects of curvilinear coordinates and their associated transformations. Even within this limited perspective, the final result has arisen as a combination of the work of researchers with many different points of view. Such developments occurred in parallel, so there is no simple order to the story. I comment here on my connections with the subject and how I remember those influences that brought me to my present understanding of Irreducible Cartesian Tensors.

The author started using a diagrammatic method for the depiction of Cartesian tensors when he was involved in integrating several complicated functions in gas kinetic theory [8]. It was already well known in fluid dynamics and kinetic theory that a second order tensor is conveniently separated into its trace, antisymmetric and s...