# Glimpses of Symmetry

 < ρℝεν | ℂσητεητs | ℕεχτ >

 Glimpses of Symmetry Reflections on the Mathematics underpinning the Standard Model of Particle Physics by Peter James Thomas
 For Jennifer, Charlotte & Beatrice with whom I form my own Special Unitary Group.
 Author’s Note This book is currently a work-in-progress. In case you stumble across it, please note that Chapters whose titles are italicised and appear in square brackets below (for example [Chapter 12 Mont Évariste]) are either incomplete or, in some cases, not yet started.

 PART I – IN THE BEGINNING Foreword Chapter 1 Introduction – The Symmetry of Reality PART II – FIRST STEPS Chapter 2 What is a Group? – A Collective Noun – Setting a Good Example – Smooth Operators – The Formal Answer to “What is a Group?” – In Addition it May be Noted… – Modular Arithmetic – The Symmetry Angle Chapter 3 Shifting Shapes – Let’s Get Physical – Turning Triangles – On Further Reflection… – Movers and Shakers – Turtles all the Way Down – Cavorting Cubes PART III – EXTENDING THE CONCEPT OF NUMBER Chapter 4 Rationality and Reality – Multiplying the Multitude – Divide and Conquer – A Magic Mirror – Expanding our Horizons Chapter 5 Tabular Amasser – The Matrix is Everywhere – What have Matrices ever done for us? – Laying our Cards on the Table – The Perennial Question Chapter 6 Matrix Revolutions – Turning the Tables – Direction of Travel – Generic Gyrations – From Dihedral to Orthogonal – Moveable Mirrors Chapter 7 Imaginary Battleships – Mare Complexionis, The Sea of Complexity – Fleet Manoeuvres – ‘Fessing Up – The Complex Numbers as a Group – amo, amas, amat… PART IV – GROUP DECOMPOSITION Chapter 8 Simplicity – Subsets and Subgroups – Exceptions to the Rule – Primed for Action Chapter 9 Normality – What Passes for Normal Round Here – The Deciding Factor – Subgroups and Cosets – Using Cosets to Create Quotient Groups Chapter 10 Profundity – A Simple Algorithm – The Quotient Group of a Maximal Normal Subgroup – Multiplication Redux PART V – SOLUTIONS OF POLYNOMIAL EQUATIONS Chapter 11 Root of the Problem – Many Names, Many Numbers – Roots of Unity – From Algebra to Geometry… – From Geometry to Trigonometry… – From Trigonometry to Group Theory… [Chapter 12 Mont Évariste] – TBC PART VI – UNITARY & SPECIAL UNITARY GROUPS Chapter 13 First Contact – U(1) – To Infinity and Beyond… – Grandes Complications – 1 × 1 is Complex… – U(1), SO(2) and Isomorphism Chapter 14 Determination – U(2) & SU(2) – 2 × 2 is more Complex… – The Shape of Things U(2) Come – U(2) Can be a Group – Singular Determination – SU(2) a Worked Example PART VII – VECTOR SPACES & VECTORS Chapter 15 It’s Space Jim… – The Red Arrows – How do we Group Vectors? – Tipping the Scales – Being Productive Chapter 16 …But not as we know it – Pastures New – The Nature of Space (not Time) – Back to Bases – Exempli Locis Chapter 17 Matrices Redux – More Marvellous Matrix Multiplications – Establishing Ownership – A Good Characteristic – Eigenlob für Eigenvalues PART VIII – LIE ALGEBRAS & LIE GROUPS Chapter 18 The Lie of the Land – Getting Crotchety – Crossing Space – Skews me! – A Singularly Uncommon Algebra? – Without a Trace Chapter 19 Making Connections – su(2) – su(3) – u(1) – As Smooth as Silk – Going off on a Tangent Chapter 20 Power to Truth – There and Back Again – What difference does it make? – In Summary – Euling the Wheels [Chapter 21 SU(3) and the Meaning of Lie] – SU(3) Unmasked – Putting a New Spin on Things – The Journey of a Thousand Miles… – TBC PART IX – TBD [Chapter 22 The Final Frontier] – TBC [Chapter 23 Placeholder] – TBC PART X – CLOSING THOUGHTS [Epilogue] Acknowledgements About the Author
 < ρℝεν | ℂσητεητs | ℕεχτ >
 Text: © Peter James Thomas 2016-17. Images: © Peter James Thomas 2016-17, unless stated otherwise. Published under a Creative Commons Attribution 4.0 International License.