An Introduction to Tensors and Group Theory for Physicists

The Introduction gives a brief overview of the modern component-free definition of tensors as multilinear maps, and then uses this definition to answer many of the questions students often have when seeing tensors for the first time. In particular, we discuss the meaning of components and the origin of the tensor transformation law (which is taken as the definition of a tensor in the old-fashioned formulation), as well as the difference between a second rank tensor and a matrix. We also demonstrate how second rank tensors are related to linear operators. We then make these considerations concrete by applying them to the moment of inertia tensor from classical mechanics. The discussion is neither totally complete nor precise but is meant to introduce the main ideas quickly, to give the reader a sense of where the next two chapters are heading.

Chapter 2 reviews the basic linear algebra essential for understanding tensors, and also develops some more advanced linear algebraic notions (such as dual spaces and non-degenerate Hermitian forms) which are also essential but usually are omitted in the standard ‘linear algebra for scientists and engineers’ course. This chapter also takes a more abstract point of view than that usually taken in lower-division linear algebra courses, in that it begins by discussing abstract vector spaces, which are defined axiomatically. This gives us the freedom to consider vector spaces made up of functions or matrices, rather than just vectors in Euclidean space. The utility of this is illustrated immediately through numerous physical examples. Following this, the elementary notions of span, linear independence, bases, components, and linear operators are discussed, and special care is taken to distinguish the component representation of vectors and linear operators from their existence as coordinate-free abstract objects. From here we move on to more advanced material, introducing dual spaces as well as non-degenerate Hermitian forms; the latter are the appropriate framework for the various scalar products that occur in physics. We conclude by showing how a non-degenerate Hermitian form allows us to turn vectors into dual vectors, which explains the relationship between bras and kets, as well as that between the covariant and contravariant components of a vector.

Chapter 3 begins with the abstract, coordinate-free definition of a tensor. This definition is standard in the math literature and in texts on General Relativity, but is otherwise not accessible in the physics literature. A major feature of this book is that is provides a relatively quick route to this definition, without the full machinery of differential geometry and tensor analysis. After the definition and some examples we thoroughly discuss change of bases and make contact with the usual coordinate-dependent definition of tensors. Matrix equations for a change of basis are also given. This is followed by a discussion of active and passive transformations, a subtle topic that is rarely fleshed out fully in other texts. We then define the tensor product and uncover many applications of tensor products in classical and quantum physics; in particular, we discuss the unwritten rule that adding degrees of freedom in Quantum Mechanics means taking the tensor product of the corresponding Hilbert spaces, and we give several examples. We then close with a discussion of symmetric and antisymmetric tensors. Important machinery such as the wedge product is introduced, along with examples concerning determinants and pseudovectors. The connection between antisymmetric tensors and rotations is made, which leads naturally to the subject of Lie Groups and Lie Algebras in Part II.

Chapter 4 introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with the definition of an abstract group along with examples, then specializes to a discussion of the groups that arise most often in physics, particularly the rotation group O(3) and the Lorentz group SO(3,1) _o . These groups are discussed in coordinates and in great detail, so that the reader gets a sense of what they look like in action. Then we discuss homomorphisms of groups, which allows us to make precise the relationship between the rotation group O(3) and its quantum-mechanical ‘double-cover’ SU(2). We then define matrix Lie groups and demonstrate how the so-called ‘infinitesimal’ elements of the group give rise to a Lie algebra, whose properties we then explore. We discuss many examples of Lie algebras in physics, and then show how homomorphisms of matrix Lie groups induce homomorphisms of their associated Lie algebras.

Chapter 5 discusses representation theory, which formalizes the notion of an object that transforms in a certain way under a given transformation (e.g. vectors under rotations, or antisymmetric tensors under boosts). We begin by defining a representation of a group as a vector space on which that group acts, and we give many examples, using the vector spaces we met in Chap. 2 and the groups we met in Chap. 4. We then discuss how to take tensor products of representations, and we see how this reproduces the additivity of quantum numbers in Quantum Mechanics. We then define irreducible representations, which are in a sense the ‘smallest’ ones we can work with, and we compute these representations for SU(2). These just end up being the familiar spin j representations, where j is a half-integer. We then use these results to compute the irreducible representations of the Lorentz group as well.

Chapter 6 applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and then unify them using the notion of a representation operator. We then use this framework to derive a generalized selection rule, from which the various quantum-mechanical selection rules can be derived, and we also discuss the Wigner–Eckart theorem. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the ‘Dirac bilinears’ of QED.