Book contents
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Geometric algebra in two and three dimensions
- 3 Classical mechanics
- 4 Foundations of geometric algebra
- 5 Relativity and spacetime
- 6 Geometric calculus
- 7 Classical electrodynamics
- 8 Quantum theory and spinors
- 9 Multiparticle states and quantum entanglement
- 10 Geometry
- 11 Further topics in calculus and group theory
- 12 Lagrangian and Hamiltonian techniques
- 13 Symmetry and gauge theory
- 14 Gravitation
- Bibliography
- Index
1 - Introduction
Published online by Cambridge University Press: 05 January 2013
- Frontmatter
- Contents
- Preface
- Notation
- 1 Introduction
- 2 Geometric algebra in two and three dimensions
- 3 Classical mechanics
- 4 Foundations of geometric algebra
- 5 Relativity and spacetime
- 6 Geometric calculus
- 7 Classical electrodynamics
- 8 Quantum theory and spinors
- 9 Multiparticle states and quantum entanglement
- 10 Geometry
- 11 Further topics in calculus and group theory
- 12 Lagrangian and Hamiltonian techniques
- 13 Symmetry and gauge theory
- 14 Gravitation
- Bibliography
- Index
Summary
The goal of expressing geometrical relationships through algebraic equations has dominated much of the development of mathematics. This line of thinking goes back to the ancient Greeks, who constructed a set of geometric laws to describe the world as they saw it. Their view of geometry was largely unchallenged until the eighteenth century, when mathematicians discovered new geometries with different properties from the Greeks' Euclidean geometry. Each of these new geometries had distinct algebraic properties, and a major preoccupation of nineteenth century mathematicians was to place these geometries within a unified algebraic framework. One of the key insights in this process was made by W.K. Clifford, and this book is concerned with the implications of his discovery.
Before we describe Clifford's discovery (in chapter 2) we have gathered together some introductory material of use throughout this book. This chapter revises basic notions of vector spaces, emphasising pictorial representations of the underlying algebraic rules – a theme which dominates this book. The material is presented in a way which sets the scene for the introduction of Clifford's product, in part by reflecting the state of play when Clifford conducted his research. To this end, much of this chapter is devoted to studying the various products that can be defined between vectors. These include the scalar and vector products familiar from three-dimensional geometry, and the complex and quaternion products. We also introduce the outer or exterior product, though this is covered in greater depth in later chapters.
- Type
- Chapter
- Information
- Geometric Algebra for Physicists , pp. 1 - 19Publisher: Cambridge University PressPrint publication year: 2003