The hydrogen atom with Hamiltonian obviously has $SO(3)$ symmetry since it just depends on the radius.
$$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r}$$
This is generated by angular momentum $\mathbf{L} = \mathbf{r}\times \mathbf{p} $.
In quantum mechanics class we learn there is $SO(4)$ symmetry due to the Runge-Lenz vector:
$$ \mathbf{A}= \frac{1}{2m}(\mathbf{p} \times \mathbf{L} - \mathbf{L} \times \mathbf{p}) - k \frac{\mathbf{r}}{r}$$
Even classical symmetry like gravity has this kind of symmetry.
I remember reading one time there is even greater symmetry for hydrogen atom. Possibly $SO(4,2)$ as in this article by Hagen Kleinert.
Has anyone heard of this?
How can one see that the Hydrogen atom has $SO(4)$ symmetry?
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