Glimpses of Symmetry

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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…
The Sign of the Four
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
A Study in Quartet
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
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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.