Notably local symmetries appear to map to forces, and local means "around the identity", notably: local symmetries of the Lagrangian imply conserved currents.
More precisely: local symmetries of the Lagrangian imply conserved currents.
TODO Ciro Santilli really wants to understand what all the fuss is about:
Oh, there is a low dimensional classification! Ciro is a sucker for classification theorems! https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras
The fact that there are elements arbitrarily close to the identity, which is only possible due to the group being continuous, is the key factor that simplifies the treatment of Lie groups, and follows the philosophy of continuous problems are simpler than discrete ones.
- https://youtu.be/kpeP3ioiHcw?t=2655 "Particle Physics Topic 6: Lie Groups and Lie Algebras" by Alex Flournoy (2016). Good SO(3) explicit exponential expansion example. Then next lecture shows why SU(2) is the representation of SO(3). Next ones appear to eventually get to the physical usefulness of the thing, but I lost patience. Not too far out though.
- https://www.youtube.com/playlist?list=PLRlVmXqzHjURZO0fviJuyikvKlGS6rXrb "Lie Groups and Lie Algebras" playlist by XylyXylyX (2018). Tutorial with infinitely many hours
- Applications of Lie groups to differential equations
- Baker-Campbell-Hausdorff formula
- Continuous problems are simpler than discrete ones
- Continuous symmetry
- Derivation of the Schrodinger equation
- Exponential map
- Group of Lie type
- Lagrangian mechanics
- Lie algebra
- Lie algebra of
- Lie bracket of a matrix Lie group
- Lie group-Lie algebra correspondence
- Lie Groups, Physics, and Geometry by Robert Gilmore (2008)
- Matrix exponential
- Noether's theorem
- One parameter subgroup
- Representation theory
- What does it mean that photons are force carriers for electromagnetism?