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Symmetry Methods in Physics
·
Module 5
Continuous Symmetries: Lie Groups and Lie Algebras
1
Introduction to Lie Groups and Their Physical Examples
Define a Lie group and identify physical examples such as SO(2), SO(3), SU(2), and U(1).
2
Generators of a Lie Group from One-Parameter Subgroups
Derive the generators of a Lie group from its one-parameter subgroups.
From Group to Lie Algebra: Commutation Relations
Compute the Lie algebra commutation relations from the group multiplication law near the identity.
4
Structure Constants and Jacobi Identity in Lie Algebras
Define structure constants and verify the Jacobi identity for a given Lie algebra.
5
Exponential Map: Connecting Lie Algebras to Lie Groups
Apply the exponential map to relate Lie algebra elements to Lie group elements.
Lie Algebra of SO(3) and Angular Momentum
Derive the Lie algebra of SO(3) and identify its generators with angular momentum operators.
7
SU(2) and SO(3) Homomorphism
Derive the Lie algebra of SU(2) and establish the 2-to-1 homomorphism from SU(2) to SO(3).
8
Baker-Campbell-Hausdorff Formula in su(2)
State the Baker-Campbell-Hausdorff formula and apply it to combine exponentials of generators in the su(2) algebra.
Previous module
Symmetry in Quantum Mechanics
Next module
Representations of SU(2) and Physical Applications