QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS by Johannes C. Familton A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D Columbia University 2015 Approved by ______________________________________________________________________ Chairperson of Supervisory Committee _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Program Authorized to Offer Degree ___________________________________________________________________ Date _______________________________________________________________________________ COLUMBIA UNIVERSITY QUATERNIONS: A HISTORY OF COMPLEX NONCOMMUTATIVE ROTATION GROUPS IN THEORETICAL PHYSICS By Johannes C. Familton Chairperson of the Supervisory Committee: Dr. Bruce Vogeli and Dr Henry O. Pollak Department of Mathematics Education 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 Algebraic approach to Quaternion Geometry ...16 D. Gibbs and Heaviside: Combining Hamilton and Grassmann............... 21 i. Gibbs: a well-educated American ............................................. 22 ii. Heaviside: a self-taught English electrician ............................. 24 E. The Joining of Quaternions with Grassmann algebras: William Kingdon Clifford................................................................................... 27 F. Sophus Lie ........................................................................................ 30 Chapter III: Mathematical Development ........................................................... 34 A. Algebra of Quaternions ......................................................................... 35 B. Vector analysis and Quaternions ........................................................... 42 C. Geometry of Quaternions ...................................................................... 47 i. Euler’s Rotation Theorem ....................................................... 49 ii. Quaternion Rotations in 3-Dimensions .................................... 50 iii. Rodrigues’ Rotation Formula………………………………..54 D. Octonians ............................................................................................... 59 E. Grassmann algebras ............................................................................... 62 F. Clifford algebras .................................................................................... 68 G. Lie Groups, Lie Algebras ...................................................................... 72 H. Quaternions, Rotation Groups and their associated Lie groups ........... 76 i. The rotation group SU(2) and its associated Lie group and algebra........................................................................................ 77 ii. The rotation group SO(3) and its associated Lie group and algebra ...................................................................................... 81 iii. The rotation group SU(3) and its associated Lie group and algebra........................................................................................ 85 iv. The rotation group SO(4) and its associated Lie group and algebra........................................................................................ 87 i Chapter IV: Applications of Quaternions in Physics ........................................ 88 A. Tait and the Elementary Treatise of Quaternions ................................ 89 i. Kinematics and Rigid body motion .......................................... 91 ii. Electricity and magnetism ......................................................... 96 iii. The operator “del”, and the Laplacian ...................................... 97 B. Maxwell and Quaternions ..................................................................99 C. Quaternions and Special Relativity .....................................................105 i. Quaternions and Lorentz transformations ..............................108 ii. Clifford algebras and Lorentz transformations ......................110 D. Quaternions and Quantum Mechanics ................................................111 i. Spinors ...........................................................................................112 ii. The Heisenberg Uncertainty Principal ..........................................112 E. Quaternions in Particle Physics…………………………….………115 i. Isospin ......................................................................................115 ii. Higher Symmetries ..................................................................117 iii. Electroweak isospin.................................................................117 F. Octonions and String Theory ..............................................................118 Chapter V: Jury Evaluation of the Source Book .............................................121 Chapter VI: Summery, Conclusions and Recommendations .........................128 i. Summery ........................................................................................129 ii. Conclusions ....................................................................................131 iii. Recommendations .........................................................................133 Bibliography .....................................................................................................137 ii iii LIST OF FIGURES Number Page 1. A “2-blade” ........................................................................................... 18 2. The Fino plane for quaternions ............................................................. 41 3. Quaternion rotations through the complex plane .................................. 48 4. Graphical representation of quaternion units product as 90ᵒ rotation in 4-dimentional space ............................................................................... 48 5. A vector diagram for the derivation for the rotation formula ............... 56 6. The Fino plane for Octinons .................................................................. 62 7. Geometric interpretation of ba = -ab ............................................... 64 8. abc ..................................................................................................... 65 9. Vectors rotated by complex phase rotations ......................................... 71 10. Three 2-blades ........................................................................................ 71 11. Rotation on a plane ................................................................................ 73 12. SO(2) .................................................................................................... 74 13. Double cover of SO(2) ......................................................................... 75 14. The 3-Sphere SU(2) projected from R4 ................................................ 78 15. SO(3) .................................................................................................... 82 16. Torus illustrating 2 types of closed loops ............................................ 82 17. SO(3) Double Cover ............................................................................. 84 18. Maxwell’s Diagrams ............................................................................104 19. Macfarlane’s Figure 7…….…. ...........................................................107 20. Two ways a string can wrap around a torus ........................................119 iv v LIST OF TABLES Number Page 1. Multiplication Table for the Quaternion Group. ................................... 40 2. Multiplication Table for Octonions ....................................................... 60 3. Maxwell’s Equations ...........................................................................100 4. Hyperbolic Quaternion Table ..............................................................106 vi ACKNOWLEDGMENTS This thesis is dedicated to Dr. Richard Friedberg Professor emeritus of Barnard College, Columbia University who first person to introduce me to quaternions as an undergraduate in an independent study course on complex analysis. The author wishes to express sincere appreciation to Dr. Bruce Vogeli who was there for me through thick and thin, and there was a lot of thin, throughout the Teacher’s College program. I would also like to thank Dr. Henry Pollak for his advice and concern along the way.