Notes on representation theory and quantum mechanics by Noah Miller in 2018
1 Group representations
A representation of a group \[G\] on a vector space \[V\] is a group homomorphism \[ \pi \colon G \rightarrow GL(V)\]
2 Schur’s lemma
3 Spherical Harmonics as Representations of SO(3) progress indicators
4 Unitary Representations
5 All Representations Break Up Into a Direct Sum of Irreducible Representations
6 A Brief Safari of The Hydrogen Atom
7 Lie Algebras
8 The Lie Algebras of U(n), SL(n,C), and SU(n)
9 Lie Algebra Representations
10 Classifying the Irreducible Representations of U(1)
11 Lie Algebra Complexifications
12 Classifying the Irreducible Representations of SU(2)
13 The Spin 1 Representation of SU(2)
14 An Explicit Construction of Unitary SU(2) Representations
15 The Odd Couple: SU(2) and SO(3)
16 What’s The Deal With Spin?
17 The Adjoint Representation And The Power Of Magical Thinking
18 Operators That Generate Representations of U(1) Have Quantized Eigenvalues
19 Functions on Phase Space Comprise a Lie Algebra
20 The Moment Map: Lie Algebra → Conserved Quantities
21 Quantization is a Lie Algebra Representation
22 Quantizing a Group Action
23 Symplectomorphisms and Degree 2 Polynomials