# Why we only consider ''group'' symmetry but not general symmetry?

tom.stoer
Again my question: is there a simple example for a dynamical symmetry where one can see explicitly [H, gi] = f(gi) ≠ 0?

In the cases SU(N) for the N-dim. harmonic oscillator and SO(4) for the hydrogen atom H commutes with all generators.

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tom.stoer
thanks; I found a hint in another paper

in case of the 1/r potential one can either multiply H by r which cancels 1/r or one can find a canonical transformation which does exactly this; then it's possible to rescale by E and find a combination of SO(4,2) generators which is equal to rH

not very systematic; as far as I can see it's better to know the result before one starts with the calculation ;-)

strangerep
Again my question: is there a simple example for a dynamical symmetry where one can see explicitly [H, gi] = f(gi) ≠ 0?
A simple Heisenberg oscillator algebra with $a,a^*$ satisfying CCRs, and a Hamiltonian such as $H=a^*a$.
(This example is more revealing if considered in terms of ordinary (Glauber) coherent states.)

For something less trivial, you could try looking up material on "generalized coherent states". There's a old textbook by Perelomov and a newer one by Gazeau. Also this classic review paper:

WM Zhang, DH Feng, R Gilmore,
Coherent states: Theory and some applications,
Rev. Mod. Phys. 62, 867-927 (1990).

A. Neumaier
2019 Award
Chapter 12 of Barut and Raczka's text on group theory has a rigorous treatment, while the chapter in Wybourne on the H-atom is more algebraic than topological, but it is an extended treatment. Actually in the H-atom you have several dynamical groups and corresponding dynamical algebras, but, AFAIK, SO(4,2) is the largest one.
Another relevant reference is

R. Gilmore,
Lie Groups, Lie Algebras & Some of Their Applications,
Wiley 1974, Dover, 2002

who treats the SO(4,2) description of the hydrogen atom in Exercises on pp. 427-430, both in a nonrelativistic and relativistic setting, with and without spin.