I have several questions about symmetry in quantum mechanics.

- It is often said that the degeneracy is the dimension of irreducible representation. I can understand that if the Hamiltonian has a symmetric group $G$, then the state space with the same energy eigenvalue will carry a representation of $G$. However, why this representation is usually irreducible?
- Is it true that the representation of continuous symmetric group must be unitary and cannot be anti-unitary?
- What is the difference between geometric symmetry and dynamical symmetry? By dynamical symmetry I mean for example the $\mathrm{SO}(4)$ symmetry of hydrogen. Some text refers dynamical symmetry to "internal" symmetry. How to state the definition of dynamical symmetry strictly?

This post imported from StackExchange Physics at 2014-08-11 14:59 (UCT), posted by SE-user Andrew