Lie Bracket
Source lecture(s): PHY622 Lec10
Physical Meaning
Measures non-commutativity of infinitesimal generators.
Variable Definitions
| Symbol | Meaning |
|---|---|
| \(T_a\) | generators |
| \(f_{abc}\) | structure constants |
Assumptions
Smooth manifold, flows near identity.
Derivation
Consider \([X,Y]=XY-YX\); at identity the commutator of vector fields yields the Lie bracket.
Applications
Quantum mechanics (commutators), gauge theory, robotics.
Connections to Other Equations
- lie-groups
- group-representations