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“The deeper I look, the more elegance I glimpse.”
– Max Tegmark, in Our Mathematical Universe: My Quest for the Ultimate Nature of Reality |
Origins
This short book is essentially motivated by personal considerations. I studied Pure Mathematics many years ago; specifically I focussed on two branches: Group Theory and Number Theory. I have made reference to each of these disciplines elsewhere on my this site ^{[1]} (though some of these references may have been somewhat oblique). What I wanted to do in this book is to drill down on the former area in more detail; though, perhaps unsurprisingly, the latter will still make several guest appearances. At the time that I wrote the first draft of this Foreword, I was reading What We Cannot Know by Marcus Du Sautoy who holds the Simonyi Professorship for the Public Understanding of Science at Oxford University. Marcus’s academic interests include (once again) Group Theory and Number Theory, so we have some things in common, albeit that his work in these fields is obviously several levels above my own understanding.
In Chapter 4 of What we cannot know, Marcus references the development of Quantum Chromodynamics by (in broad chronological order) Eugene Wigner, Werner Heisenberg, Murray Gell-Mann, Kazuhiko Nishijima, Yuval Ne’eman and George Zweig (Gell-Mann being the common thread through much of the later work) ^{[2]}. This theory has helped us to understand how more familiar particles such as protons and neutrons are created from smaller components. What drew Du Sautoy’s attention (and had drawn mine on other occasions) was the role played in this theory by something that Pure Mathematicians would be more familiar with, the Special Unitary Group of Degree 3, or SU(3).
It struck me that the connection between sub-atomic Physics and Group Theory is often cited, but that the latter area is probably a closed book to the non-expert ^{[3]}. Partly in an attempt to make some important elements of Group Theory more accessible to a general audience and partly (here comes the personal element) to see if I could still grapple with Pure Mathematical concepts some 27 years after laying down my pencil, I decided to try to publish a lay-person’s guide to what Marcus describes as “a very particular symmetrical object”. Whether or not I have succeeded in this objective is something that I will leave for my readers to decide.
As to the title Glimpses of Symmetry, Symmetry is a theme that we will come back to time and time again. However I am fully aware that, especially in a relatively short piece of prose, glimpses of this important and complex area of Mathematics and Physics are the best that I can hope to achieve. However, if by providing some fleeting and imperfect images, I manage to catch the attention of at least a few readers and if some of these subsequently decide to explore the subject further, then I believe that my time will have been well-spent.
In Chapter 1, I begin this process by providing a somewhat longer introduction to the importance of Group Theory in sub-atomic Physics.
— Peter James Thomas, July 2016
A Note on the Level of Mathematics Contained in this Book The Mathematics that is included in Chapters up until Chapter 8 is mostly of no greater difficulty than what was included in the UK Further Mathematics A Level sylabus in the mid-1980s ^{[4]}; a lot of it is much simpler than that. UK A-levels are broadly the equivalent of US High School, but perhaps with some Advanced Placement classes thrown in. Curricula have no doubt changed since back when the author was studying at this level, so the time at which students are introduced to the subjects covered in this book may have changed accordingly. Chapter 6 has some more complicated elements, but not too many. From Chapter 8 to Chapter 16, some of the work introduced might sit in the first term of an undergraduate Mathematics course. Chapter 17 and onwards would most likely be covered in the second or maybe third years; though I have focussed on the simpler elements where possible. However, I would actually argue that a lot of the Mathematics is probably simpler than might be expected; instead it is the concepts that require a bit more thought to absorb. I have tried to be clear in my explanations and to omit details that are not pertinent to the journey I hope to take reader on. My assertion is that anyone with a firm grip of arithmetic, some idea about basic geometry and – more importantly – the desire to master the subject matter, will be able to work their way through the text unaided. I would however like to offer a caveat. Most of the Mathematics contained in this book I have dredged up from my memory, or worked through from first principles based on sometimes hazy recollections. I have reached for the crutch of Wikipedia when necessary, but tried to make this more the exception that the rule. I have certainly not transcribed any proofs word for word, regardless of their source. A downside of this approach is that I will undoubtedly have omitted some salient facts that a more comprehensive text booked would have included. It is also very likely that either my memory has misled me, or my logic has been faulty in some places. Equally good old fashioned mistakes may well have crept in to the text here and there. For all these reasons, these pages should not be viewed as authoritative, but rather as an impressionist’s painting of some elements of Group Theory; conveying the essential meaning while some of the details are lost in the width of the paint brush. The book is not intended to be a study aid! |
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Foreword – Notes
^{ [1]} | Below are links to couple of articles with explicit references to Group and Number Theory: |
^{ [2]} | Rather than providing a link to information on each of these esteemed scientists, the QCD link provided above will let you explore their biographies if you so desire. |
^{ [3]} | Good luck with the Wikipedia page on Special Unitary Groups if you have no background in the area! |
^{ [4]} | Chapter 7 has some concepts which are most likely introduced in the first year of a university course in Mathematics. |
Text: © Peter James Thomas 2016-18. |