Mathematical Structure of Quantum Theory Lecture 4: Lie Groups and Lie Algebras

In this lecture, I discuss the relevance of Lie groups and Lie algebras in quantum mechanics. First, I define Lie groups and Lie algebras. Then, I use homomorphism to show the connection between the two, and I use a geometric representation to show that the Lie group represents the manifold and the Lie algebras are the tangent space. Afterwards, I list important Lie groups/algebras in quantum mechanics, illustrating examples like Heisenberd’s uncertainty principle, Yang-Mills theory, quantum entanglement, and spherical harmonics. Finally, as a bonus content, I discuss how Lie groups/algebras are relevant to general relativity and how they give rise to Killing vectors and conserved quantities. Additionally, I discuss how we can use the Killing vectors to determine surface gravity and black hole temperature. Minor announcement: Since I am taking two graduate-level courses on computational physics and astrophysics, I decided that maybe in a couple of days, I will start a new series titled: “Computational & Theoretical Methods in Astrophysics“. Chapters: 0:00:00 - Introduction 0:00:59 - Outline of lecture 0:01:26 - Summary of the last lecture 0:03:26 - Definition of a differentiable manifold 0:04:09 - Definition of a Lie group 0:04:49 - GL(n) as a Lie group 0:06:24 - Definition of a Lie algebra 0:07:57 - Angular momenta/Poisson brackets as Lie algebras 0:11:26 - Correspondence between Lie groups and Lie algebras 0:13:01 - Exponential maps 0:15:03 - Deriving the Hadamard formula by induction 0:18:07 - Deriving the BCH formula using differential equations 0:21:15 - Lie group-Lie algebra homomorphism 0:22:15 - Geometric picture of Lie groups/algebras 0:23:36 - Connection to Noether’s theorem and symmetry 0:25:43 - Importance of Lie groups/algebras in quantum mechanics 0:26:59 - Heisenberg group/algebras 0:28:42 - Connection to uncertainty 0:30:25 - U(1) group and Yang-Mills theory 0:31:58 - U(1) gauge invariance 0:34:11 - Let there be light! 0:35:46 - SU(2) and angular momenta (cool) 0:36:24 - SU(2) and entanglement (cooler) 0:36:42 - Tensor product representation of groups 0:38:36 - Decomposability 0:39:00 - Connection to entangled states 0:39:32 - Clebsch-Gordon decomposition theorem 0:40:04 - Connection to Clebsch-Gordon coefficients 0:40:52 - Bell states and decomposability 0:42:13 - Discussion on basis dependence 0:42:31 - Coordinate free representations 0:43:09 - SO(3) group and angular momenta 0:44:49 - Spherical harmonics as integral representations of SO(3) 0:47:30 - Vector transformations 0:49:25 - Tensor transformations 0:50:17 - Lie derivatives and parallel transports 0:51:38 - Killing’s equation and Killing vectors 0:52:40 - Killing vectors in Minkowski spacetime 0:55:24 - Killing vectors associated with rotations 0:55:56 - Killing vectors associated with boosts 0:55:56 - Killing vectors as generators of the Lorentz algebra 0:58:24 - Isomorphism between SU(2) and SO(1,3) 0:58:38 - Killing vectors in Schwarzschild spacetime 0:59:14 - Rotational Killing vectors in Schwarzschild spacetime 0:59:33 - Visualizing the generators associated with rotations 1:01:36 - Conservation laws in Schwarzschild spacetime 1:02:02 - Poincaré group/algebras 1:02:29 - Surface gravity of black holes 1:04:06 - Discussion on black hole temperature 1:04:32 - Content of next lecture