Professor Martin Liebeck of Imperial College London, who taught me most of what I have since forgotten about Group Theory (no fault implied on his side). As well as lecturing me as an undergraduate, [then] Dr Liebeck was my MSc. supervisor and I rather wish that I had spent more time listening to him than in other pursuits.
Professor Robert Vaughan previously of Imperial College London, now at Penn State, who fulfilled a similar role with respect to Number Theory. Professor Vaughan lectured me as both an undergraduate and post graduate. He was guilty of awarding me a mark of 120% for my MSc. Number Theory exam, something which had both compelling logic and a sense of the surreal. It’s probably fair to say that I have never worked my brain harder than in preparation for my MSc. Number Theory exam.
As explicitly acknowledged in the Foreword, reading Marcus Du Sautoy’s latest book, What we Cannot Know could be described as the distal cause of this book. I have read a few of Marcus’s tomes and – as I mention – he is maybe a shadowy reflection of what I might have been had I continued to pursue a career in academic Mathematics . While I enjoyed reading Marcus’s work immensely (and would not want this comment to detract from my assessment of its value), a final unexplained reference to the mysterious SU(3) × SU(2) × U(1) was enough to push me into writing this book.
I have been an avid reader of popular science and Mathematics books since learning to read. Going back in time, two titles that had a profound impact on me were Riddles in Mathematics (which was awarded to my father as a school prize and opened a new world of Mathematical possibilities to me) and Mathematics and the Imagination (which broadened this view to include a larger universe).
About half way through writing this book, I decided to re-read one of Professor Ian Stewart’s books on Group Theory, Why Beauty Is Truth: The History of Symmetry. The clue is in the title and this work is more a mathematical history than a maths book (not that I am questioning Professor Stewart’s ability, he is an esteemed Mathematician). While – at the author’s express intent – the Mathematics in Why Beauty is Truth is very much behind the scenes, revisiting some of the key events and players in the development of what is now called Group Theory helped me to structure elements of this book and to recall some interesting areas and results, which I might otherwise have overlooked.
At the beginning of this book, I stated that I am a Mathematician, not a Physicist. I have a background that enables me to remember or rederive most of the Mathematics that appears in these pages; this is not the case where it comes to Physics. Given this, I wanted some reliable guide to the terra incognita I was planning to visit. My able companion in this respect has been Group Theory in a Nutshell for Physicists by Theoretical Physicist Anthony Zee from the University of California at Santa Barbara. While I turned to Anthony’s book to educate me on the application of Group Theory to Physics, he ended up teaching me a few things about Group Theory itself that I either never knew or had forgotten. At the very least he provided a more tangible perspective on a number of topics, particularly Lie Theory. I would recommend both this book and the ones he has written for a more general audience.
In the same vein, Scott Aaronson‘s book, Quantum Computing since Democritus, helped me to develop a different perspective on Quantum Mechanics and thereby break through my writers’ block. While I remain a Pure Mathematician, I understand Statistics a whole lot better than I do the use of systems of partial differential equations in Physics, so it was a relief to have a more tangible (for me) way to grapple with the mysteries of quantum states and their transitions. Scott is Professor of Computer Science at The University of Texas at Austin, and director of its Quantum Information Center.
Carey G. Butler, CEO of Heurist GmbH, know each other from community question and answer site Quora. Carey was kind and patient enough to review several chapters of this book, making helpful suggestions and pointing out errors. The quality of the end product has benefited greatly from his assiduous input.
To be completed
Except where listed below, all images that appear in this book are drawn by the author and therefore © Peter James Thomas 2016-2018. Published under a Creative Commons Attribution 4.0 International License.
||Drawn by the author, but based on numerous similar diagrams from multiple sources
Chapter 1 – Introduction
||Sourced from: Wikimedia Commons – Fermilab, Office of Science, United States Department of Energy, Particle Data Group, Creative Commons BY 3.0
Chapter 2 – What is a Group?
Chapter 3 – Shifting Shapes
Chapter 4 – Rationality & Reality
Chapter 5 – Tabular Amasser
||© 1999 Warner Bros Entertainment & Village Roadshow Films Limited. Sourced from: youtube.com
||Ownership unknown. Sourced from: dillonmarcus.com
Chapter 6 – Matrix Revolutions
Chapter 7 – Imaginary Battleships
Chapter 8 – Simplicity
||© Lord Bullgod. Sourced from: Flicka
||© Eric W. Weisstein 1996-9. Sourced from: msu.edu
Chapter 9 – Normality
Chapter 10 – Profundity
Chapter 11 – Root of the Problem
Chapter 12 – Mont Évariste
Chapter 13 – First Contact – U(1)
Chapter 14 – Determination – U(2) & SU(2)
Chapter 15 – It’s Space Jim…
Chapter 16 – …But not as we know it
Chapter 17 – MatricesRedux
Chapter 18 – The Lie of the Land
Chapter 19 – Making Connections
Chapter 20 – Power to Truth
Chapter 21 – SU(3) and the Meaning of Lie
Chapter 22 – The Final Frontier
Chapter 23 – Probable Cause
Chapter 24 – Emmy
Acknowledgements – Notes
With apologies to both Lawrence Kasdan and Marcus himself.