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Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics

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Quaternions: a History of Complex Noncommutative Rotation Groups in Theoretical Physics QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS Johannes C. Familton Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2015 © 2015 Johannes C. Familton All rights reserved ABSTRACT Quaternions: a history of complex Noncommutative rotation groups in theoretical physics Johannes C. Familton The purpose of this dissertation is to clarify the emergence of quaternions in order to make the history of quaternions less opaque to teachers and students in mathematics and physics. ‘Quaternion type Rotation Groups’ are important in modern physics. They are usually encountered by students in the form of: Pauli matrices, and SU(2) & SO(4) rotation groups. These objects did not originally appear in the neat form presented to students in modern mathematics or physics courses. What is presented to students by instructors is usually polished and complete due to many years of reworking. Often neither students of physics, mathematics or their instructors have an understanding about how these objects came into existence, or became incorporated into their respected subject in the first place. This study was done to bridge the gaps between the history of quaternions and their associated rotation groups, and the subject matter that students encounter in their course work. TABLE OF CONTENTS List of Figures ..................................................................................................... iv List of Tables ...................................................................................................... vi Acknowledgements ........................................................................................... vii Chapter I: Introduction......................................................................................... 1 A. Need for Study ......................................................................................... 2 B. Purpose of Study ..................................................................................... 2 C. Procedure of Study .................................................................................. 4 Chapter II: Historical Development .................................................................... 6 A. What are Quaternions? ............................................................................ 6 B. William Rowan Hamilton and the discovery of Quaternions .............. 8 C. Grassmann: A more Geometric approach to Algebra........................15 D. Gibbs and Heaviside: Combining Hamilton and Grassmann .............. 18 i. Gibbs: a well-educated American ............................................ 19 ii. Heaviside: a self-taught English electrician ............................. 20 E. The Joining of Quaternions with Grassmann algebras: William Kingdon Clifford .................................................................... 24 F. Sophus Lie ........................................................................................ 26 Chapter III: Mathematical Development .......................................................... 30 A. Algebra of Quaternions ......................................................................... 31 B. Gibbs Vector analysis and Quaternions ................................................ 36 C. Geometry of Quaternions ...................................................................... 40 i. Euler’s Rotation Theorem ...................................................... 41 ii. Quaternion Rotations in 3-Dimensions .................................... 42 iii. Rodrigues’ Rotation Formula………………………………..45 D. Octonians ............................................................................................... 49 E. Grassmann algebras ............................................................................... 53 F. Clifford algebras .................................................................................... 58 G. Lie Groups, Lie Algebras ...................................................................... 60 H. Quaternions, Rotation Groups and their associated Lie groups ........... 63 i. The rotation group SU(2) and its associated Lie group and algebra ....................................................................................... 64 ii. The rotation group SO(3) and its associated Lie group and algebra ...................................................................................... 67 iii. The rotation group SU(3) and its associated Lie group and algebra ....................................................................................... 71 iv. The rotation group SO(4) and its associated Lie group and algebra ....................................................................................... 72 i Chapter IV: Applications of Quaternions in Physics ........................................ 73 A. Tait and the Elementary Treatise of Quaternions ................................ 74 i. Kinematics and Rigid body motion .......................................... 76 ii. Electricity and magnetism ........................................................ 81 iii. The operator “del”, and the Laplacian ..................................... 81 B. Maxwell and Quaternions ..................................................................83 C. Quaternions and Special Relativity ....................................................... 89 i. Quaternions and Lorentz transformations ................................ 91 ii. Clifford algebras and Lorentz transformations ........................ 93 D. Quaternions and Quantum Mechanics .................................................. 94 i. Spinors ............................................................................................. 95 ii. The Heisenberg Uncertainty Principal ........................................... 95 E. Quaternions in Particle Physics…………………………….………..97 i. Isospin ....................................................................................... 97 ii. Higher Symmetries ................................................................... 99 iii. Electroweak isospin .................................................................. 99 F. Octonions and String Theory .............................................................. 101 Chapter V: Jury Evaluation of the Source Book ............................................. 103 Chapter VI: Summery, Conclusions and Recommendations ........................ 110 i. Summery ....................................................................................... 110 ii. Conclusions ................................................................................... 112 iii. Recommendations ......................................................................... 114 Bibliography ..................................................................................................... 117 ii LIST OF FIGURES Number Page 1. The Fino plane for quaternions ............................................................. 34 2. Quaternion Angle .................................................................................. 35 3. Geometric interpretation for the proof of Theorem 1 ........................... 35 4. Quaternion rotations through the complex plane ................................. 40 5. Graphical representation of quaternion units product as 90ᵒ rotation in 4-dimentional space ............................................................ 41 6. A vector diagram for the derivation for the rotation formula .............. 46 7. The Fino plane for Octinons ................................................................. 52 8. A “2-blade” ............................................................................................ 53 9. Geometric interpretation of ba = -ab ............................................... 55 10. abc ..................................................................................................... 55 11. Rotation on a plane ................................................................................ 61 12. SO(2) .................................................................................................... 62 13. Double cover of SO(2) ......................................................................... 63 14. The 3-Sphere SU(2) projected from R4 ................................................ 65 15. SO(3) .................................................................................................... 68 16. Torus illustrating 2 types of closed loops ............................................ 69 17. SO(3) Double Cover ............................................................................. 69 18. Maxwell’s Diagrams ............................................................................. 89 19. Macfarlane’s Figure 7…….…. ............................................................. 90 20. Two ways a string can wrap around a torus ....................................... 102 iv LIST OF TABLES Number Page 1. Multiplication Table for the Quaternion Group. .................................. 34 2. Multiplication Table for Octonions ...................................................... 51 3. Maxwell’s Equations ............................................................................. 84 4. Hyperbolic Quaternion Table................................................................ 89 vi ACKNOWLEDGMENTS This thesis is dedicated to Dr. Richard Friedberg Professor emeritus of Barnard
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