Geometric Algebra For Physicists, Notas de estudo de Física

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GEOMETRIC ALGEBRA

FOR PHYSICISTS

CHRIS DORAN

and

ANTHONY LASENBY

University of Cambridge

Contents

Preface ix

• 1 Introduction Notation xiii
• 1.1 Vector (linear) spaces
• 1.2 The scalar product
• 1.3 Complex numbers
• 1.4 Quaternions
• 1.5 The cross product
• 1.6 The outer product
• 1.7 Notes
• 1.8 Exercises
• 2 Geometric algebra in two and three dimensions
• 2.1 A new product for vectors
• 2.2 An outline of geometric algebra
• 2.3 Geometric algebra of the plane
• 2.4 The geometric algebra of space
• 2.5 Conventions
• 2.6 Reflections
• 2.7 Rotations
• 2.8 Notes
• 2.9 Exercises
• 3 Classical mechanics
• 3.1 Elementary principles
• 3.2 Two-body central force interactions
• 3.3 Celestial mechanics and perturbations
• 3.4 Rotating systems and rigid-body motion CONTENTS
• 3.5 Notes
• 3.6 Exercises
• 4 Foundations of geometric algebra
• 4.1 Axiomatic development
• 4.2 Rotations and reflections
• 4.3 Bases, frames and components
• 4.4 Linear algebra
• 4.5 Tensors and components
• 4.6 Notes
• 4.7 Exercises
• 5 Relativity and spacetime
• 5.1 An algebra for spacetime
• 5.2 Observers, trajectories and frames
• 5.3 Lorentz transformations
• 5.4 The Lorentz group
• 5.5 Spacetime dynamics
• 5.6 Notes
• 5.7 Exercises
• 6 Geometric calculus
• 6.1 The vector derivative
• 6.2 Curvilinear coordinates
• 6.3 Analytic functions
• 6.4 Directed integration theory
• 6.5 Embedded surfaces and vector manifolds
• 6.6 Elasticity
• 6.7 Notes
• 6.8 Exercises
• 7 Classical electrodynamics
• 7.1 Maxwell’s equations
• 7.2 Integral and conservation theorems
• 7.3 The electromagnetic field of a point charge
• 7.4 Electromagnetic waves
• 7.5 Scattering and diffraction
• 7.6 Scattering
• 7.7 Notes
• 7.8 Exercises
• 8 Quantum theory and spinors CONTENTS
• 8.1 Non-relativistic quantum spin
• 8.2 Relativistic quantum states
• 8.3 The Dirac equation
• 8.4 Central potentials
• 8.5 Scattering theory
• 8.6 Notes
• 8.7 Exercises
• 9 Multiparticle states and quantum entanglement
• 9.1 Many-body quantum theory
• 9.2 Multiparticle spacetime algebra
• 9.3 Systems of two particles
• 9.4 Relativistic states and operators
• 9.5 Two-spinor calculus
• 9.6 Notes
• 9.7 Exercises
• 10 Geometry
• 10.1 Projective geometry
• 10.2 Conformal geometry
• 10.3 Conformal transformations
• 10.4 Geometric primitives in conformal space
• 10.5 Intersection and reflection in conformal space
• 10.6 Non-Euclidean geometry
• 10.7 Spacetime conformal geometry
• 10.8 Notes
• 10.9 Exercises
• 11 Further topics in calculus and group theory
• 11.1 Multivector calculus
• 11.2 Grassmann calculus
• 11.3 Lie groups
• 11.4 Complex structures and unitary groups
• 11.5 The general linear group
• 11.6 Notes
• 11.7 Exercises
• 12 Lagrangian and Hamiltonian techniques
• 12.1 The Euler–Lagrange equations
• 12.2 Classical models for spin-1/2 particles
• 12.3 Hamiltonian techniques
• 12.4 Lagrangian field theory CONTENTS
• 12.5 Notes
• 12.6 Exercises
• 13 Symmetry and gauge theory
• 13.1 Conservation laws in field theory
• 13.2 Electromagnetism
• 13.3 Dirac theory
• 13.4 Gauge principles for gravitation
• 13.5 The gravitational field equations
• 13.6 The structure of the Riemann tensor
• 13.7 Notes
• 13.8 Exercises
• 14 Gravitation
• 14.1 Solving the field equations
• 14.2 Spherically-symmetric systems
• 14.3 Schwarzschild black holes
• 14.4 Quantum mechanics in a black hole background
• 14.5 Cosmology
• 14.6 Cylindrical systems
• 14.7 Axially-symmetric systems
• 14.8 Notes
• 14.9 Exercises
• Bibliography
• Index

Preface

The ideas and concepts of physics are best expressed in the language of mathe- matics. 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 mathe- matical 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 applica- tions, 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 de- velopment 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 ob- jects. 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 book we aim to dispel this belief by giving a straightforward introduction to this new and fundamentally different approach to vectors and vector multiplication. In this language much of the standard subject matter taught to physicists can be formulated in an elegant and highly condensed fashion. And the portability of the techniques we discuss enables us to reach a range of advanced topics with little extra work. This book is intended to be of interest to both students and researchers in physics. The early chapters grew out of an undergraduate lecture course that we have run for a number of years in the Physics Department at Cambridge Uni-

ix

PREFACE

not complaining about the lost evenings as I worked on this book. I promise to finish the decorating now it is complete. AL thanks Joan and his children Robert and Alison for their constant enthu- siasm and support, and their patience in the face of many explanations of topics from this book.

Cambridge C.J.L. Doran July 2002 A.N. Lasenby

xi

NOTATION

(iv) The outer (exterior) product is written with a wedge, A ∧ B. The outer product is also only employed between homogeneous multivectors. (v) Inner and outer products are always performed before geometric prod- ucts. This enables us to remove unnecessary brackets. For example, the expression a·b c is to be read as (a·b)c. (vi) Angled brackets 〈M 〉p are used to denote the result of projecting onto the terms in M of grade p. The subscript zero is dropped for the projection onto the scalar part. (vii) The reverse of the multivector M is denoted either with a dagger, M †, or with a tilde, M˜. The latter is employed for applications in spacetime. (viii) Linear functions are written in an upright font as F(a) or h(a). This helps to distinguish linear functions from multivectors. Some exceptions are encountered in chapters 13 and 14, where caligraphic symbols are used for certain tensors in gravitation. The adjoint of a linear function is denoted with a bar, ¯h(a). (ix) Lie groups are written in capital, Roman font as in SU(n). The corre- sponding Lie algebra is written in lower case, su(n). Further details concerning the conventions adopted in this book can be found in sections 2.5 and 4.1.

xiv

Introduction

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 to- gether 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 ma- terial 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 re- search. 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 material in this chapter is intended to be fairly basic, and those impatient to uncover Clifford’s insight may want to jump straight to chapter 2. Readers unfamiliar with the outer product are encouraged to read this chapter, however, as it is crucial to understanding Clifford’s discovery.

1

1.1 Vector (linear) spaces

a (^) a

b

b

a + b a

b

a + b

b + c

a + b + c

c

Figure 1.1 A geometric picture of vector addition. The result of a + b is formed by adding the tail of b to the head of a. As is shown, the resultant vector a + b is the same as b + a. This finds an algebraic expression in the statement that addition is commutative. In the right-hand diagram the vector a + b + c is constructed two different ways, as a + (b + c) and as (a + b) + c. The fact that the results are the same is a geometric expression of the associativity of vector addition.

The preceding set of rules serves to define a vector space completely. Note that the + operation connecting scalars is different from the + operation connecting the vectors. There is no ambiguity, however, in using the same symbol for both. The following two definitions will be useful later in this book:

(i) Two vector spaces are said to be isomorphic if their elements can be placed in a one-to-one correspondence which preserves sums, and there is a one-to-one correspondence between the scalars which preserves sums and products. (ii) If U and V are two vector spaces (sharing the same scalars) and all the elements of U are contained in V, then U is said to form a subspace of V.

1.1.2 Bases and dimension

The concept of dimension is intuitive for simple vector spaces — lines are one- dimensional, planes are two-dimensional, and so on. Equipped with the axioms of a vector space we can proceed to a formal definition of the dimension of a vector space. First we need to define some terms.

(i) A vector b is said to be a linear combination of the vectors a 1 ,... , an if scalars λ 1 ,... , λn can be found such that

b = λ 1 a 1 + · · · + λnan =

∑^ n

i=

λiai. (1.5)

(ii) A set of vectors {a 1 ,... , an} is said to be linearly dependent if scalars

3

INTRODUCTION

λ 1 ,... , λn (not all zero) can be found such that

λ 1 a 1 + · · · + λnan = 0. (1.6)

If such a set of scalars cannot be found, the vectors are said to be linearly independent. (iii) A set of vectors {a 1 ,... , an} is said to span a vector space V if every element of V can be expressed as a linear combination of the set. (iv) A set of vectors which are both linearly independent and span the space V are said to form a basis for V.

These definitions all carry an obvious, intuitive picture if one thinks of vectors in a plane or in three-dimensional space. For example, it is clear that two independent vectors in a plane provide a basis for all vectors in that plane, whereas any three vectors in the plane are linearly dependent. These axioms and definitions are sufficient to prove the basis theorem, which states that all bases of a vector space have the same number of elements. This number is called the dimension of the space. Proofs of this statement can be found in any textbook on linear algebra, and a sample proof is left to work through as an exercise. Note that any two vector spaces of the same dimension and over the same field are isomorphic. The axioms for a vector space define an abstract mathematical entity which is already well equipped for studying problems in geometry. In so doing we are not compelled to interpret the elements of the vector space as displacements. Often different interpretations can be attached to isomorphic spaces, leading to different types of geometry (affine, projective, finite, etc.). For most problems in physics, however, we need to be able to do more than just add the elements of a vector space; we need to multiply them in various ways as well. This is necessary to formalise concepts such as angles and lengths and to construct higher-dimensional surfaces from simple vectors. Constructing suitable products was a major concern of nineteenth century mathematicians, and the concepts they introduced are integral to modern math- ematical physics. In the following sections we study some of the basic concepts that were successfully formulated in this period. The culmination of this work, Clifford’s geometric product, is introduced separately in chapter 2. At various points in this book we will see how the products defined in this section can all be viewed as special cases of Clifford’s geometric product.

1.2 The scalar product

Euclidean geometry deals with concepts such as lines, circles and perpendicular- ity. In order to arrive at Euclidean geometry we need to add two new concepts

4

INTRODUCTION

Here the δij is the Kronecker delta function, defined by

δij =

1 if i = j, 0 if i = j.

We can expand any vector a in this basis as

a =

∑^ n

i=

aiei = aiei, (1.13)

where we have started to employ the Einstein summation convention that pairs of indices in any expression are summed over. This convention will be assumed throughout this book. The {ai} are the components of the vector a in the {ei} basis. These are found simply by

ai = ei ·a. (1.14)

The scalar product of two vectors a = aiei and b = biei can now written simply as

a·b = (aiei)·(bj ej ) = aibj ei ·ej = aibj δij = aibi. (1.15)

In spaces where the inner product is not positive-definite, such as Minkowski spacetime, there is no equivalent version of the Schwarz inequality. In such cases it is often only possible to define an ‘angle’ between vectors by replacing the cosine function with a cosh function. In these cases we can still introduce ortho- normal frames and use these to compute scalar products. The main modification is that the Kronecker delta is replaced by ηij which again is zero if i = j, but can take values ±1 if i = j.

1.3 Complex numbers

The scalar product is the simplest product one can define between vectors, and once such a product is defined one can formulate many of the key concepts of Euclidean geometry. But this is by no means the only product that can be defined between vectors. In two dimensions a new product can be defined via complex arithmetic. A complex number can be viewed as an ordered pair of real numbers which represents a direction in the complex plane, as was realised by Wessel in

1797. Their product enables complex numbers to perform geometric operations, such as rotations and dilations. But suppose that we take the complex number z = x + iy and square it, forming

z^2 = (x + iy)^2 = x^2 − y^2 + 2xyi. (1.16)

In terms of vector arithmetic, neither the real nor imaginary parts of this ex- pression have any geometric significance. A more geometrically useful product

6

1.4 Quaternions

is defined instead by

zz∗^ = (x + iy)(x − iy) = x^2 + y^2 , (1.17)

which returns the square of the length of the vector. A product of two vectors in a plane, z and w = u + vi, can therefore be constructed as

zw∗^ = (x + iy)(u − iv) = xu + vy + i(uy − vx). (1.18)

The real part of the right-hand side recovers the scalar product. To understand the imaginary term consider the polar representation

z = |z|eiθ, w = |w|eiφ^ (1.19)

so that

zw∗^ = |z||w|ei(θ^ −^ φ). (1.20)

The imaginary term has magnitude |z||w| sin(θ − φ), where θ − φ is the angle between the two vectors. The magnitude of this term is therefore the area of the parallelogram defined by z and w. The sign of the term conveys information about the handedness of the area element swept out by the two vectors. This will be defined more carefully in section 1.6. We thus have a satisfactory interpretation for both the real and imaginary parts of the product zw∗. The surprising feature is that these are still both parts of a complex number. We thus have a second interpretation for complex addition, as a sum between scalar objects and objects representing plane segments. The advantages of adding these together are precisely the advantages of working with complex numbers as opposed to pairs of real numbers. This is a theme to which we shall return regularly in following chapters.

1.4 Quaternions

The fact that complex arithmetic can be viewed as representing a product for vectors in a plane carries with it a further advantage — it allows us to divide by a vector. Generalising this to three dimensions was a major preoccupation of the physicist W.R. Hamilton (see figure 1.2). Since a complex number x + iy can be represented by two rectangular axes on a plane it seemed reasonable to represent directions in space by a triplet consisting of one real and two complex numbers. These can be written as x + iy + jz, where the third term jz represents a third axis perpendicular to the other two. The complex numbers i and j have the properties that i^2 = j^2 = −1. The norm for such a triplet would then be

(x + iy + jz)(x − iy − jz) = (x^2 + y^2 + z^2 ) − yz(ij + ji). (1.21)

The final term is problematic, as one would like to recover the scalar product here. The obvious solution to this problem is to set ij = −ji so that the last term vanishes.

7