The fundamental forces of nature can all be described in terms of gauge theories. Not long after the advent of quantum theory, physicists realised that electromagnetic interactions arise from demanding invariance of quantum wave equations under local changes of phase. This idea was later extended by Yang and Mills, who showed how to construct theories based on more complicated, non-commutative Lie groups. This is the basis for the standard model of the electroweak and strong interactions. Around this time physicists also turned their attention to gravitation, and discovered that general relativity could also be formulated as a gauge theory. But this time there was a price to pay. The existence of spinor fields means that the simple geometric structure of general relativity has to be modified by the inclusion of a torsion field, leading to an Einstein–Cartan theory. For clarity, we use the term general relativity to refer to the theory defined by Einstein, with zero torsion and the connection given by the Christoffel symbol. The extended theory, with torsion present, is referred to as Einstein–Cartan theory.

While gauge theory is the dominant method in particle physics, it is less popular as a means of analysing gravitational interactions. This is, in part, due to the perception that the gauge theory equations are more complicated than their geometric counterparts.

The Lagrangian formulation of mechanics is popular in practically all modern treatments of the subject. The ideas date back to the pioneering work of Euler, Lagrange and Hamilton, who showed how the equations of Newtonian dynamics could be derived from variational principles. In these, the evolution of a system is viewed as a path in some parameter space. The path the system follows is one which extremises a quantity called the action, which is the integral of the Lagrangian with respect to the evolution parameter (usually time). The mathematics behind this approach was clear from the outset, but a thorough physical understanding had to wait until the arrival of quantum theory. In the path-integral formulation of quantum mechanics a particle is viewed as simultaneously following all possible paths. By assigning a phase factor to the action for each path and summing these, one obtains the amplitude for a quantum process. The classical limit can then be understood as resulting from trajectories that reinforce the amplitude. In this manner classical trajectories emerge as those which make the action stationary.

A closely related idea is the Hamiltonian formulation of dynamics. The advantage of this approach is that it produces a set of first-order equations, making it well suited to numerical methods. The Hamiltonian approach also exposes the appropriate geometry for classical dynamical systems, which is a symplectic manifold.

Geometric algebra was introduced in the nineteenth century by the English mathematician William Kingdon Clifford (figure 2.1). Clifford appears to have been one of the small number of mathematicians at the time to be significantly influenced by Grassmann's work. Clifford introduced his geometric algebra by uniting the inner and outer products into a single geometric product. This is associative, like Grassmann's product, but has the crucial extra feature of being invertible, like Hamilton's quaternion algebra. Indeed, Clifford's original motivation was to unite Grassmann's and Hamilton's work into a single structure. In the mathematical literature one often sees this subject referred to as Clifford algebra. We have chosen to follow the example of David Hestenes, and many other modern researchers, by returning to Clifford's original choice of name – geometric algebra. One reason for this is that the first published definition of the geometric product was due to Grassmann, who introduced it in the second Ausdehnungslehre. It was Clifford, however, who realised the great potential of this product and who was responsible for advancing the subject.

In this chapter we introduce the basics of geometric algebra in two and three dimensions in a way that is intended to appear natural and geometric, if somewhat informal. A more formal, axiomatic approach is delayed until chapter 4, where geometric algebra is defined in arbitrary dimensions.

Geometric algebra offers a number of new techniques for studying problems in electromagnetism and electrodynamics. These are described in this chapter. We will not attempt a thorough development of electrodynamics, which is a vast subject with numerous specialist areas. Instead we concentrate on a number of selected applications which highlight the advantages that geometric algebra can bring. There are two particularly significant new features that geometric algebra adds to traditional formulations of electrodynamics. The first is that, through employing the spacetime algebra, all equations can be studied in the appropriate spacetime setting. This is much more transparent than the more traditional approach based on a 3 + 1 formulation involving retarded times. The spacetime algebra simplifies the study of how electromagnetic fields appear to different observers, and is particularly powerful for handling accelerated charges and radiation. These results build on the applications of spacetime algebra described in section 5.5.3.

The second major advantage of the geometric algebra treatment is a new, compact formulation of Maxwell's equations. The spacetime vector derivative and the geometric product enable us to unite all four of Maxwell's equations into a single equation. This is one of the most impressive results in geometric algebra. And, as we showed in chapter 6, this is more than merely a cosmetic exercise. The vector derivative is invertible directly, without having to pass via intermediate, second-order equations. This has many implications for scattering and propagator theory.

Geometric algebra provides us with an invertible product for vectors. In this chapter we investigate the new insights this provides for the subject of vector calculus. The familiar gradient, divergence and curl operations all result from the action of the vector operator, ∇. Since this operator is vector-valued, we can now form its geometric product with other multivectors. We call this the vector derivative. Unlike the separate divergence and curl operations, the vector derivative has the important property of being invertible. That is to say, Green's functions exist for ∇ which enable initial conditions to be propagated off a surface.

The synthesis of vector differentiation and geometric algebra described in this chapter is called ‘geometric calculus’. We will see that geometric calculus provides new insights into the subject of complex analysis and enables the concept of an analytic function to be extended to arbitrary dimensions. In three dimensions this generalisation gives rise to the angular eigenstates of the Pauli theory, and the spacetime generalisation of an analytic function defines the wavefunction for a massless spin-1/2 particle. Clearly there are many insights to be gained from a unified treatment of calculus based around the geometric product.

The early sections of this chapter discuss the vector derivative, and its associated Green's functions, in flat spaces. This way we can quickly assemble a number of results of central importance in later chapters.

The ideas and concepts of physics are best expressed in the language of mathematics. But this language is far from unique. Many different algebraic systems exist and are in use today, all with their own advantages and disadvantages. In this book we describe what we believe to be the most powerful available mathematical system developed to date. This is geometric algebra, which is presented as a new mathematical tool to add to your existing set as either a theoretician or experimentalist. Our aim is to introduce the new techniques via their applications, rather than as purely formal mathematics. These applications are diverse, and throughout we emphasise the unity of the mathematics underpinning each of these topics.

The history of geometric algebra is one of the more unusual tales in the development of mathematical physics. William Kingdon Clifford introduced his geometric algebra in the 1870s, building on the earlier work of Hamilton and Grassmann. It is clear from his writing that Clifford intended his algebra to describe the geometric properties of vectors, planes and higher-dimensional objects. But most physicists first encounter the algebra in the guise of the Pauli and Dirac matrix algebras of quantum theory. Few then contemplate using these unwieldy matrices for practical geometric computing. Indeed, some physicists come away from a study of Dirac theory with the view that Clifford's algebra is inherently quantum-mechanical.

In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators.

Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question.

In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.

In chapter 2 we introduced geometric algebra in two and three dimensions. We now turn to a discussion of the full, axiomatic framework for geometric algebra in arbitrary dimensions, with arbitrary signature. This will involve some duplication of material from chapter 2, but we hope that this will help reinforce some of the key concepts. Much of the material in this chapter is of primary relevance to those interested in the full range of applications of geometric algebra. Those interested solely in applications to space and spacetime may want to skip some of the material below, as both of these algebras are treated in a self-contained manner in chapters 2 and 5 respectively. The material on frames and linear algebra is important, however, and a knowledge of this is assumed for applications in gravitation.

The fact that geometric algebra can be applied in spaces of arbitrary dimensions is crucial to the claim that it is a mathematical tool of universal applicability. The framework developed here will enable us to extend geometric algebra to the study of relativistic dynamics, phase space, single and multiparticle quantum theory, Lie groups and manifolds. This chapter also highlights some of the new algebraic techniques we now have at our disposal. Many derivations can be simplified through judicious use of the geometric product at various intermediate steps. This is true even if the initial and final expressions contain only inner and outer products.

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.

The geometric algebra of spacetime is called the spacetime algebra. Historically, the spacetime algebra was the first modern implementation of geometric algebra to gain widespread attention amongst the physics community. This is because it provides a synthetic framework for studying spacetime physics. There are two main approaches to the study of geometry, which can be loosely referred to as the algebraic and synthetic traditions. In the algebraic approach one works entirely with the components of a vector and manipulates these directly. Such an approach leads naturally to the subject of tensors, and places considerable emphasis on how coordinates transform under changes of frame. The synthetic approach, on the other hand, treats vectors as single, abstract entities x or a, and manipulates these directly. Geometric algebra follows in this tradition.

For much of modern physics the synthetic approach has come to dominate. The most obvious examples of this are classical mechanics and electromagnetism, both of which helped shape the development of abstract vector calculus. For these subjects, presentations typically perform all of the required calculations with the three-dimensional scalar and cross products. We have argued that geometric algebra provides extra efficiency and clarity, though it is not essential to a synthetic treatment of three-dimensional physics. But for spacetime calculations the cross product cannot be defined. Despite the obvious advantages of synthetic treatments, most relativity texts revert to a more basic, algebraic approach involving the components of 4-vectors and Lorentz-transform matrices.

In the preceding chapters of this book we have dealt entirely with a single geometric interpretation of the elements of a geometric algebra. But the relationship between algebra and geometry is seldom unique. Geometric problems can be studied using a variety of algebraic techniques, and the same algebraic result can typically be pictured in a variety of different ways. In this chapter, we explore a range of alternative geometric systems, and discover how geometric algebra can be applied to each of them. We will find that there is no unique interpretation forced on the multivectors of a given grade. For example, to date we have viewed bivectors solely as directed plane segments. But in projective geometry a bivector represents a line, and in conformal geometry a bivector can represent a pair of points.

Ideas from geometry have always been a prime motivating factor in the development of mathematics. By the nineteenth century mathematicians were familiar with affine, Euclidean, spherical, hyperbolic, projective and inversive geometries. The unifying framework for studying these geometries was provided by the Kleinian viewpoint. Under this view a geometry consists of a space of points, together with a group of transformations mapping the points onto themselves. Any property of a particular geometry must be invariant under the action of the associated symmetry group.

In this chapter we study the use of geometric algebra in classical mechanics. We will assume that readers already have a basic understanding of the subject, as a complete presentation of classical mechanics with geometric algebra would require an entire book. Such a book has been written, New Foundations for Classical Mechanics by David Hestenes (1999), which looks in detail at many of the topics discussed here. Our main focus in this chapter is to areas where geometric algebra offers some immediate benefits over traditional methods. These include motion in a central force and rigid-body rotations, both of which are dealt with in some detail. More advanced topics in Lagrangian and Hamiltonian dynamics are covered in chapter 12, and relativistic dynamics is covered in chapter 5.

Classical mechanics was one of the areas of physics that prompted the development of many of the mathematical techniques routinely used today. This is particularly true of vector analysis, and it is now common to see classical mechanics described using an abstract vector notation. Many of the formulae in this chapter should be completely familiar from such treatments. A key difference comes in adopting the outer product of vectors in place of the cross product. This means, for example, that angular momentum and torque both become bivectors.