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Now the gloves come off.

In the simplest theory of oscillating ellipsoids, all 6 numbers describing an ellipsoid are treated equally, so we get a harmonic oscillator in 6 dimensions. This has SO(6) symmetry due to rotations in 6d space. But the phase space of this harmonic oscillator is not ℝ⁶ but ℂ⁶, with complex coordinates describing both the position *and momentum* of the oscillator! In fact the oscillator has U(6) symmetry, due to unitary transformations of ℂ⁶. The group U(6) is a lot bigger than SO(6). Yum!

In this simplest possible theory we want a Hamiltonian that's invariant under U(6). The usual Hamiltonian for a quantum harmonic oscillator has this property. Indeed, U(6) acts on the Hilbert space of the 6d quantum harmonic oscillator, so the Casimir of U(6) gives an invariant operator on this Hilbert space... and this works out to be the harmonic oscillator Hamiltonian!

But suppose we want a more realistic yet still tractable theory. One way is to say our theory only needs to have SO(6) symmetry. Then we get another possible term in the Hamiltonian: the Casimir for SO(6). Or we can admit our theory only needs to have SO(3) symmetry, due to rotations in physical 3d space. This suggests another possible term: the Casimir for SO(3).

In the SO(6) SD model, the Hamiltonian is a linear combination of 3 terms:

• the Casimir for SU(6)
• the Casimir for SO(5)
• the Casimir for SO(3)

For more details, check out the paper where I got the picture:

• J.E. Garcia-Ramos, A. Leviatan, P. Van Isacker, Partial dynamical symmetry in quantum Hamiltonians with higher-order terms, https://arxiv.org/abs/0812.4109

I'm still puzzled by the geometry involved... these guys ignore that.

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