Is there a simple system that has $\text{SU}(3)$ symmetry?

Question

The buckle at the end of a belt has $\text{SU}(2)$ symmetry, if the rotations around the three coordinate axes are taken as generators. See, for example, the paper by Hart, Francis and Kauffman, https://www.researchgate.net/publication/2389584_Visualizing_Quaternion_Rotation . You can also use the hand at the end of your arm to visualize SU(2), including its double cover of SO(3).

Is there something similar for $\text{SU}(3)$ symmetry? Is there a simple system, maybe from everyday life, that realizes this symmetry?

Alternatively: how do you picture $\text{SU}(3)$ in your head?

Answer 1

Outside of quantum physics, both SU(2) and SU(3) appear in the context of polarization optics. For a plane wave the rotation of the two components of the polarization (perpendicular to the propagation direction) are represented by an element of SU(2). To describe the more general case of an arbitrary wave field (without a fixed propagation direction), three components need to be rotated and SU(3) is needed. See, for example, SU(3) representation for the polarisation of light (1980) and Parameters characterizing electromagnetic wave polarization (2000).

Answer 2

One of the simplest mechanical systems having an SU(3) dynamical symmetry is the three dimensional isotropic harmonic oscillator. Its phase space is $\mathbb{R}^6$ parametrized by 3 position coordinates $x_i$ and 3 momenta $p_i$, $i=1,2,3$. A fixed energy hypersurface in phase space is given by the constraint (in units where the natural frequency is 1): $$\sum_i x_i^2 + \sum_i p_i^2 = E $$ This a the $5$-dimensional sphere $S^5$, and the set of classical trajectories (the reduced phase space corresponding to the Hamiltonian action) on this hypersurface is the complex projective space $CP^2$ , obtained from $S^5$ by a Hopf fibration and from the original phase space by Marsden-Weinstein symplectic reduction with respect to the action of the total energy function: $$CP^2 = S^5/U(1) = \mathbb{C}^3//U(1)$$ Being a $SU(3)$ homogeneous space $CP^2$ has a natural $SU(3)$action: The functions representing $SU(3)$ in the $CP^2$ Poisson algebra have simple geometric expressions:

1. First we have the three angular momentum components: $$L_{ij} = x_ip_j-x_jp_i$$
2. In addition we have 5 quadrupole moments built from the (traceless)symmetric combinations of the coordinates and momenta: $$Q_{ij} = \frac{3}{2} (x_ip_j+x_jp_i) - \delta_{ij} \sum_k x_kp_k$$ Together they generate the $SU(3)$ Lie algebra. The quadrupole moment space is closed under the $SU(2)$ action generated by the angular momenta. They transform according to the spin-2 representation.

Physically, the quadrupole moments interact with gradients of an external magnetic field. If the harmonic oscillator is charged, then its interaction energy with an external magnetic field has the form: $$E_{int} = \sum_{ijk} \epsilon_{ijk}L_{ij}B_k+ \sum_{ij} Q_{ij} \partial_{i}B_j$$ Please, see the following Melih Sener lecture note by for a detailed exposition.