Geometric Algebra and its Application to Mathematical Physics Chris J. L. Doran Sidney Sussex College A dissertation submitted for the degree of Doctor of Philosophy in the University of Cambridge. February 1994 Preface This dissertation is the result of work carried out in the Department of Applied Mathematics and Theoretical Physics between October 1990 and October 1993. Sections of the dissertation have appeared in a series of collaborative papers [1] —[10]. Except where explicit reference is made to the work of others, the work contained in this dissertation is my own. Acknowledgements Many people have given help and support over the last three years and I am grateful to them all. I owe a great debt to my supervisor, Nick Manton, for allowing me the freedom to pursue my own interests, and to my two principle collaborators, Anthony Lasenby and Stephen Gull, whose ideas and inspiration were essential in shaping my research. I also thank David Hestenes for his encouragement and his company on an arduous journey to Poland. Above all, I thank Julie Cooke for the love and encouragement that sustained me through to the completion of this work. Finally, I thank Stuart Rankin and Margaret James for many happy hours in the Mill, Mike and Rachael, Tim and Imogen, Paul, Alan and my other colleagues in DAMTP and MRAO. I gratefully acknowledge financial support from the SERC, DAMTP and Sidney Sussex College. To my parents Contents 1 Introduction 1 1.1 Some History and Recent Developments . 4 1.2 Axioms and Definitions . 9 1.2.1 The Geometric Product . 14 1.2.2 The Geometric Algebra of the Plane . 16 1.2.3 The Geometric Algebra of Space . 19 1.2.4 Reflections and Rotations . 22 1.2.5 The Geometric Algebra of Spacetime . 27 1.3 Linear Algebra . 30 1.3.1 Linear Functions and the Outermorphism . 30 1.3.2 Non-Orthonormal Frames . 34 2 Grassmann Algebra and Berezin Calculus 37 2.1 Grassmann Algebra versus Clifford Algebra . 37 2.2 The Geometrisation of Berezin Calculus . 39 2.2.1 Example I. The “Grauss” Integral . 42 2.2.2 Example II. The Grassmann Fourier Transform . 44 2.3 Some Further Developments . 48 3 Lie Groups and Spin Groups 50 3.1 Spin Groups and their Generators . 51 3.2 The Unitary Group as a Spin Group . 57 3.3 The General Linear Group as a Spin Group . 62 3.3.1 Endomorphisms of <n ..................... 70 3.4 The Remaining Classical Groups . 76 3.4.1 Complexification — so(n,C) . 76 3.4.2 Quaternionic Structures — sp(n) and so∗(2n) . 78 i 3.4.3 The Complex and Quaternionic General Linear Groups . 82 3.4.4 The symplectic Groups Sp(n,R) and Sp(n,C) . 83 3.5 Summary . 85 4 Spinor Algebra 87 4.1 Pauli Spinors . 88 4.1.1 Pauli Operators . 94 4.2 Multiparticle Pauli States . 95 4.2.1 The Non-Relativistic Singlet State . 99 4.2.2 Non-Relativistic Multiparticle Observables . 100 4.3 Dirac Spinors . 102 4.3.1 Changes of Representation — Weyl Spinors . 107 4.4 The Multiparticle Spacetime Algebra . 109 4.4.1 The Lorentz Singlet State . 115 4.5 2-Spinor Calculus . 118 4.5.1 2-Spinor Observables . 121 4.5.2 The 2-spinor Inner Product . 123 4.5.3 The Null Tetrad . 124 4.5.4 The ∇A0A Operator . 127 4.5.5 Applications . 129 5 Point-particle Lagrangians 132 5.1 The Multivector Derivative . 132 5.2 Scalar and Multivector Lagrangians . 136 5.2.1 Noether’s Theorem . 137 5.2.2 Scalar Parameterised Transformations . 137 5.2.3 Multivector Parameterised Transformations . 139 5.3 Applications — Models for Spinning Point Particles . 141 6 Field Theory 157 6.1 The Field Equations and Noether’s Theorem . 158 6.2 Spacetime Transformations and their Conjugate Tensors . 161 6.3 Applications . 166 6.4 Multivector Techniques for Functional Differentiation . 175 ii 7 Gravity as a Gauge Theory 179 7.1 Gauge Theories and Gravity . 180 7.1.1 Local Poincaré Invariance . 183 7.1.2 Gravitational Action and the Field Equations . 187 7.1.3 The Matter-Field Equations . 193 7.1.4 Comparison with Other Approaches . 199 7.2 Point Source Solutions . 203 7.2.1 Radially-Symmetric Static Solutions . 204 7.2.2 Kerr-Type Solutions . 217 7.3 Extended Matter Distributions . 221 7.4 Conclusions . 227 iii List of Tables 1.1 Some algebraic systems employed in modern physics . 8 3.1 Bivector Basis for so(p,q)........................ 57 3.2 Bivector Basis for u(p,q)........................ 61 3.3 Bivector Basis for su(p,q)........................ 61 3.4 Bivector Basis for gl(n,R) . 65 3.5 Bivector Basis for sl(n,R) . 66 3.6 Bivector Basis for so(n,C) . 77 3.7 Bivector Basis for sp(n) ........................ 79 3.8 Bivector Basis for so∗(n) ........................ 82 3.9 Bivector Basis for gl(n,C) . 83 3.10 Bivector Basis for sl(n,C) . 83 3.11 Bivector Basis for sp(n,R) . 84 3.12 The Classical Bilinear Forms and their Invariance Groups . 86 3.13 The General Linear Groups . 86 4.1 Spin Currents for 2-Particle Pauli States . 101 4.2 Two-Particle Relativistic Invariants . 118 4.3 2-Spinor Manipulations . 123 iv Chapter 1 Introduction This thesis is an investigation into the properties and applications of Clifford’s geometric algebra. That there is much new to say on the subject of Clifford algebra may be a surprise to some. After all, mathematicians have known how to associate a Clifford algebra with a given quadratic form for many years [11] and, by the end of the sixties, their algebraic properties had been thoroughly explored. The result of this work was the classification of all Clifford algebras as matrix algebras over one of the three associative division algebras (the real, complex and quaternion algebras) [12]–[16]. But there is much more to geometric algebra than merely Clifford algebra. To paraphrase from the introduction to “Clifford Algebra to Geometric Calculus”[24], Clifford algebra provides the grammar from which geometric algebra is constructed, but it is only when this grammar is augmented with a number of secondary definitions and concepts that one arrives at a true geometric algebra. In fact, the algebraic properties of a geometric algebra are very simple to understand, they are those of Euclidean vectors, planes and higher-dimensional (hyper)surfaces. It is the computational power brought to the manipulation of these objects that makes geometric algebra interesting and worthy of study. This computational power does not rest on the construction of explicit matrix representations, and very little attention is given to the matrix representations of the algebras used. Hence there is little common ground between the work in this thesis and earlier work on the classification and study of Clifford algebras. There are two themes running through this thesis: that geometric algebra is the natural language in which to formulate a wide range of subjects in modern mathematical physics, and that the reformulation of known mathematics and physics in terms of geometric algebra leads to new ideas and possibilities. The development 1 of new mathematical formulations has played an important role in the progress of physics. One need only consider the benefits of Lagrange’s and Hamilton’s reformulations of classical mechanics, or Feynman’s path integral (re)formulation of quantum mechanics, to see how important the process of reformulation can be. Reformulations are often interesting simply for the novel and unusual insights they can provide. In other cases, a new mathematical approach can lead to significant computational advantages, as with the use of quaternions for combining rotations in three dimensions. At the back of any programme of reformulation, however, lies the hope that it will lead to new mathematics or physics. If this turns out to be the case, then the new formalism will usually be adopted and employed by the wider community. The new results and ideas contained in this thesis should support the claim that geometric algebra offers distinct advantages over more conventional techniques, and so deserves to be taught and used widely. The work in this thesis falls broadly into the categories of formalism, refor- mulation and results. Whilst the foundations of geometric algebra were laid over a hundred years ago, gaps in the formalism still remain. To fill some of these gaps, a number of new algebraic techniques are developed within the framework of geometric algebra. The process of reformulation concentrates on the subjects of Grassmann calculus, Lie algebra theory, spinor algebra and Lagrangian field theory. In each case it is argued that the geometric algebra formulation is computationally more efficient than standard approaches, and that it provides many novel insights. The new results obtained include a real approach to relativistic multiparticle quan- tum mechanics, a new classical model for quantum spin-1/2 and an approach to gravity based on gauge fields acting in a flat spacetime. Throughout, consistent use of geometric algebra is maintained and the benefits arising from this approach are emphasised. This thesis begins with a brief history of the development of geometric algebra and a review of its present state. This leads, inevitably, to a discussion of the work of David Hestenes [17]–[34], who has done much to shape the modern form of the subject. A number of the central themes running through his research are described, with particular emphasis given to his ideas on mathematical design. Geometric algebra is then introduced, closely following Hestenes’ own approach to the subject. The central axioms and definitions are presented, and a notation is introduced which is employed consistently throughout this work.