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

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?

• Regarding how to picture $SU(3)$, I guess since $SU(3)/SU(2) \simeq S^5$ one could "picture" $SU(3)$ as a 5-sphere's worth of $SU(2)$s (the non-trivial one, as $\pi_4(S^3) = Z/2$). – Oliver Nash Nov 1 '17 at 12:44
• Quantum chromodynamics is somehow related to SU(3), I believe – user74900 Nov 2 '17 at 6:25

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).

• Could you recommend a text to read in more detail about rotations of an internal degree of freedom from a mathematical point of view? – მამუკა ჯიბლაძე Oct 31 '17 at 7:38
• The belt trick (or plate trick) that Francesco refers to is a trick to visualize SU(2), not just SO(3), so goes beyond the local isomorphism of those groups: en.wikipedia.org/wiki/Plate_trick. It is hard to state precisely how the trick enables us to visualize SU(2), but that is certainly the purpose of the trick. – Ben McKay Oct 31 '17 at 11:49
• Carlo, I added a reference to Francesco's question. – Motion Mountain Oct 31 '17 at 12:37
• @მამუკა ჯიბლაძე: Feynman Lectures on Physics, vol. 3, has a whole chapter titled "Spin 1 particles". – Alexandre Eremenko Oct 31 '17 at 13:19
• somehow I interpreted the "every day life" part of the OP that this should be an application of SU(3) to a classical, as opposed to quantum system; but indeed, in the quantum context SU(2s+1) appears for any spin-s particle. In the classical context I only know of the application to light polarisation. – Carlo Beenakker Oct 31 '17 at 13:26

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.

• This is not special to three dimensions, right? I.e., the set of classical trajectories of an $n$-dimensional isotropic harmonic oscillator is $CP(n-1)$ and carries an $SU(n)$ action etc? – Oliver Nash Nov 8 '17 at 15:05
• Yes, of course, you are correct – David Bar Moshe Nov 8 '17 at 15:08