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How symmetry is related to the degeneracy?

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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

asked Mar 25, 2014 in Theoretical Physics by Andrew (135 points) [ no revision ]

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