An Introduction to Tensors and Group Theory for Physicists

verfasst von: Nadir Jeevanjee

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics. Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations. New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students.

Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part I by introducing representation theory. Examples and exercises are provided in each chapter for good practice in applying the presented material and techniques.

Prerequisites for this text include the standard lower-division mathematics and physics courses, though extensive references are provided for the motivated student who has not yet had these. Advanced undergraduate and beginning graduate students in physics and applied mathematics will find this textbook to be a clear, concise, and engaging introduction to tensors and groups.

Reviews of the First Edition

“[P]hysicist Nadir Jeevanjee has produced a masterly book that will help other physicists understand those subjects [tensors and groups] as mathematicians understand them… From the first pages, Jeevanjee shows amazing skill in finding fresh, compelling words to bring forward the insight that animates the modern mathematical view…[W]ith compelling force and clarity, he provides many carefully worked-out examples and well-chosen specific problems… Jeevanjee’s clear and forceful writing presents familiar cases with a freshness that will draw in and reassure even a fearful student. [This] is a masterpiece of exposition and explanation that would win credit for even a seasoned author.”

—Physics Today

"Jeevanjee’s [text] is a valuable piece of work on several counts, including its express pedagogical service rendered to fledgling physicists and the fact that it does indeed give pure mathematicians a way to come to terms with what physicists are saying with the same words we use, but with an ostensibly different meaning. The book is very easy to read, very user-friendly, full of examples...and exercises, and will do the job the author wants it to do with style.”

—MAA Reviews

Inhaltsverzeichnis

Frontmatter

Linear Algebra and Tensors

Frontmatter

Chapter 1. A Quick Introduction to Tensors

Abstract

This chapter introduces the notion of a tensor as a multilinear map and explores its implications through the examples of the Levi–Civita tensor and a generic second rank tensor. This discussion sheds light on the Levi–Civita symbol and also answers 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, 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.

Nadir Jeevanjee

Chapter 2. Vector Spaces

Abstract

This chapter reviews the basic linear algebra essential for understanding tensors (linear independence, bases, linear operators, etc.), and also develops some more advanced linear algebraic notions (e.g., dual spaces and non-degenerate Hermitian forms) which are also essential but often undiscussed. This chapter also takes a more abstract point of view than is typical, which gives us the freedom to consider vector spaces made up of functions or matrices, rather than just vectors in Euclidean space. Throughout, special care is taken to distinguish the component representation of various objects (vectors, linear operators, etc.) from their existence as coordinate-free abstract objects. The machinery developed is also used to illuminate enigmatic topics such as spherical harmonics and the relationship between bras and kets and the covariant and contravariant components of a vector.

Nadir Jeevanjee

Chapter 3. Tensors

Abstract

This chapter 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 it 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. We then define the tensor product and uncover many applications of tensor products in classical and quantum physics. We close with a discussion of symmetric and anti-symmetric tensors, with examples concerning determinants and pseudovectors. The connection between anti-symmetric tensors and rotations is made, which leads naturally to the subject of Lie Groups and Lie Algebras in Part II.

Nadir Jeevanjee

Group Theory

Frontmatter

Chapter 4. Groups, Lie Groups, and Lie Algebras

Abstract

This chapter introduces abstract groups and Lie groups, which are a formalization of the notion of a physical transformation. The chapter begins with a heuristic introduction that motivates the definition of a group and gives an intuitive sense for what an “infinitesimal generator” is. Then we give the definition of an abstract group along with examples, followed by 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.

Nadir Jeevanjee

Chapter 5. Basic Representation Theory

Abstract

This chapter discusses representation theory, which formalizes the notion of an object that transforms in a certain way under a given transformation. We begin with a heuristic introduction that shows how representations naturally arise in quantum mechanics, and how the panoply of such representations calls for organizing principles, which we develop in the rest of the chapter. We then define 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.

Nadir Jeevanjee

Chapter 6. The Representation Operator and Its Applications

Abstract

This chapter 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 use these to give a quick overview of the Wigner–Eckart theorem and selection rules. These latter subjects are then made precise using the notion of a representation operator. 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.

Nadir Jeevanjee

Backmatter

Titel: An Introduction to Tensors and Group Theory for Physicists
verfasst von: Nadir Jeevanjee
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-14794-9
Print ISBN: 978-3-319-14793-2
DOI: https://doi.org/10.1007/978-3-319-14794-9