# Is this really $SO(4)$ algebra?

The commutation relations involving the components of Runge-Lenz vector of the Hydrogen atom problem, $${\vec A}$$ and the angular momentum $${\vec L}$$ are given by $$[L_i,L_j]=i\hslash\varepsilon_{ijk}L_k,\\ [A_i,A_j] = -i\hslash\varepsilon_{ijk} \frac{2H}{m} L_k,\\ [L_i,A_j]=i\hslash\varepsilon_{ijk}A_k$$ I am doubtful about the commutation relation in the middle the right-hand side of which contains the Hamiltonian operator. The righthand side does not seem to be a simple linear combination of $$L_1,L_2$$ and $$L_3$$. If $$H$$ were a constant, then this is the same as $$SO(4)$$ algebra. But how can this be $$SO(4)$$ where $${H}$$ is an operator? Any help?

• Will "an operator commuting with all your generators behaves like a constant for all practical purposes" mean anything to you? Commented Feb 2, 2022 at 16:23
• You should treat $H$ as a constant w.r.t. the $SO(4)$ generators - this is just the same thing as saying that $H$ is invariant under $SO(4)$ transformations. Commented Feb 2, 2022 at 18:14

Since both the $$L_i$$ and the $$A_i$$ are constants of motion, their Poisson bracket with the Hamiltonian vanishes and so it is "as good as a constant" for purposes of the algebraic structure - like a constant, it commutes with everything. Essentially you get one differently scaled $$\mathfrak{so}(4)$$ algebra for each subspace defined by constant energy $$H(q,p) = E$$.
• Just to be really precise, for $E<0$. Commented Feb 2, 2022 at 16:37
• @ACuriousMind How about quantum mechanics? In that case, how do you scale the $A_i$'s to get the standard SO(4) algebra structure? Surely, we cannot divide by the square root of the hamiltonian operator. Commented Feb 2, 2022 at 16:40
• @Solidification I don't know what you mean - you still get one $\mathfrak{so}(4)$ on each eigenspace of $H$. Commented Feb 2, 2022 at 16:42
• Ah! I think I got it. You are saying that this set of commutation relations when applied to a subspace of the Hilbert space corresponding to a fixed energy $E$, is a SO(4) algebra. Is that right? Commented Feb 2, 2022 at 16:46