Is there a relationship between the degeneracy of the Hamiltonian and the set of non-commuting operators that commute with the Hamiltonian?

Asked Aug 29 '22 at 15:19

Active Aug 31 '22 at 16:17

Viewed 41 times

I'd like to know if there is any relationship between the number of the degeneracy of the Hamiltonian and the set of non-commuting operators that commute with the Hamiltonian.

If I know a set of $n$ independent non-commuting operators, how do I know if that set is complete to characterise the whole ket space of the Hamiltonian?

Knowing that the set is complete, is there any relationship between $n$ and the number of the degeneracy?

edited Aug 31 '22 at 16:17

asked Aug 29 '22 at 15:19

Matteo Melloni

How did you define d=3 degeneracy? – Cosmas Zachos Aug 29 '22 at 16:25
@CosmasZachos It's an example. I'm asking if there is a general rule. – Matteo Melloni Aug 29 '22 at 16:29
Do these operators close into a Lie algebra? The ground state is non-degenerate for linearly realized symmetries. Linked and also. – Cosmas Zachos Aug 31 '22 at 16:24
Suppose the non-commuting operators form some sort of closed algebra under multiplication, then the ground state subspace must give a representation of this algebra. – Meng Cheng Aug 31 '22 at 16:29

Is there a relationship between the degeneracy of the Hamiltonian and the set of non-commuting operators that commute with the Hamiltonian?

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