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 . .
Gabor Kunstatter is a theoretical physicist who has worked on general relativity, gauge theory quantization, finite temperature quantum field theory, quantum computing and quantum gravity. His current research focuses on the quantum mechanics of black holes, quantum information and effective theories for non-singular black hole evaporation and evaporation. Dr. Kunstatter is Professor Emeritus at the University of Winnipeg and Adjunct Professor at the University of Victoria, Simon Fraser University and the University of Manitoba. He has been a visiting scientist at a variety of institutions, including M.I.T., Université de Paris (Orsay), UNAM (Mexico), University of Nottingham and CECS (Chile). Dr. Kunstatter has also served as the President of the Canadian Association of Physicists and as Dean of Science at the University of Winnipeg.
Saurya Das is a theoretical physicist whose research areas include quantum gravity theory and phenomenology and cosmology. He has worked on problems in black hole physics, testing signatures of quantum gravity in the laboratory and on dark matter and dark energy, on which he has published more than 80 papers. After doing postdoctoral research at the Pennsylvania State University and the Universities of Winnipeg and New Brunswick, Dr. Das joined the faculty the University of Lethbridge, Canada in 2003, where he is now a full professor.
