Every mathematician has only a few tricks
12 days ago
- #symmetry
- #physics
- #mathematics
- Two commuting matrices are simultaneously diagonalizable, simplifying the diagonalization of one if the eigenvectors of the other are known.
- Translation invariance leads to using Fourier transforms, solving wave equations for light, acoustics, quantum electrons, and heat equations.
- Discrete translation symmetry in crystals uses Bloch-Floquet theory, explaining band structures and differences between conductors and insulators.
- Rotational invariance helps in solving the Hydrogen atom's eigenvalues/eigenvectors, linking to finite-dimensional representations of SO(3).
- SU(3) symmetry in particle physics organizes the complex zoology of particles through its representations.