1 Introduction 9
1.1 The goal of physics . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 The connection between physics and mathematics . . . . . . . 10
1.3 Paradigm shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . 16
2 Symmetry and Physics 17
2.1 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 What is Symmetry? . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Role of Symmetry in Physics . . . . . . . . . . . . . . . . . . . 18
2.3.1 Symmetry as a guiding principle . . . . . . . . . . . . . 18
2.3.2 Symmetry and Conserved Quantities: Noether''s Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.3 Symmetry as a tool for simplifying problems . . . . . . 19
2.4 Symmetries were made to be broken . . . . . . . . . . . . . . 20
2.4.1 Spacetime symmetries . . . . . . . . . . . . . . . . . . 20
2.4.2 Parity violation . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Spontaneously broken symmetries . . . . . . . . . . . . 24
2.4.4 Variational calculations: Lifeguards and light rays . . . 27
3 Formal Aspects of Symmetry 30
3.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Symmetries and Operations . . . . . . . . . . . . . . . . . . . 30
3.2.1 Denition of a symmetry operation . . . . . . . . . . . 30
3.2.2 Rules obeyed by symmetry operations . . . . . . . . . 32
3.2.3 Multiplication tables . . . . . . . . . . . . . . . . . . . 35
3.2.4 Symmetry and group theory . . . . . . . . . . . . . . . 36
3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 The identity operation . . . . . . . . . . . . . . . . . . 37
3.3.2 Permutations of two identical objects . . . . . . . . . . 37
3.3.3 Permutations of three identical objects . . . . . . . . . 38
3.3.4 Rotations of regular polygons . . . . . . . . . . . . . . 39
3.4 Continuous vs discrete symmetries . . . . . . . . . . . . . . . 40
3.5 Symmetries and Conserved Quantities:
Noether''s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Supplementary: Variational Mechanics and the Proof of Noether''s
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.1 Variational Mechanics: Principle of Least Action . . . . 42
3.6.2 Euler-Lagrange Equations . . . . . . . . . . . . . . . . 47
3.6.3 Proof of Noether''s Theorem . . . . . . . . . . . . . . . 48
4 Symmetries and Linear Transformations 52
4.1 Learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Review of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Coordinate free denitions . . . . . . . . . . . . . . . . 53
4.2.2 Cartesian Coordinates . . . . . . . . . . . . . . . . . . 58
4.2.3 Vector operations in component form . . . . . . . . . . 59
4.2.4 Position vector . . . . . . . . . . . . . . . . . . . . . . 60
4.2.5 Dierentiation of vectors: velocity and acceleration . . 62
4.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . 63
4.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2 Translations . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.4 Re
ections . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Linear Transformations and matrices . . . . . . . . . . . . . . 68
4.4.1 Linear transformations as matrices . . . . . . . . . . . 68
4.4.2 Identity Transformation and Inverses . .
