Incoming links: Looking Glass Universe
For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.
Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.
As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.
The most beautiful idea in physics - Noether's Theorem by Looking Glass Universe (2015)
Source. One sentence stands out: the generated quantities are called the generators of the transforms.The Biggest Ideas in the Universe | 15. Gauge Theory by Sean Carroll (2020)
Source. This attempts a one hour hand wave explanation of it. It is a noble attempt and gives some key ideas, but it falls a bit short of Ciro's desires (as would anything that fit into one hour?)The Symmetries of the universe by ScienceClic English (2021)
Source. youtu.be/hF_uHfSoOGA?t=144 explains intuitively why symmetry implies consevation!Quantum superposition is really weird because it is fundamentally different than "either definite state but I don't know which", because the superposition state leads to different measurements than the non-superposition state.
Examples:
- www.youtube.com/watch?v=tt8gVXDsh7Q "Interference in quantum mechanics" by Looking Glass Universe (2015) shows how a left-right spin measurement has a defined value for a superposed half up half down state, but not for a pure up state.TODO can this be conducted? As mentioned in the video, this is closely linked to the fact that you can describe the wave function in multiple different bases (up/down or left/right), which is also at the root of the uncertainty principle.
- Video "Quantum Mechanics 9b - Photon Spin and Schrodinger's Cat II by ViaScience (2013)" gives a similar photon version
- it seems that the single particle double slit experiment can also be thought of as in terms of a superposition of "the particle goes through the right" and "the particle goes through the right", although it is a bit harder to thing about as it is not a discrete process