# What is the simplest way to realize or visualize SU(3)?

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SU(3) is an important group in physics. Is there a simple system from daily life that has this symmetry?

Or is there some pretty object that has the symmetry?

The question is inspired by the buckle at the end of a belt, which behaves like SU(2): rotations around x y and z behave like the three generators. This is nicely shown by Dirac's string trick. Another example is given in  Visualizing Quaternion Rotation by Hart, Francis, L. Kauffman, , https://dl.acm.org/citation.cfm?id=197480 (free pdf via scholar.google.com). They explain how the rotations of the *palm of a hand* are exactly like SU(2), including the double cover. So it is possible to visualize SU(2).

Is there something similar for SU(3)?

edited Oct 31, 2017

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The reason why you can think of $SU(2)$ as rotations in 3 dimensions is because it is closely related with the actual group of rotations in 3 dimensions, that is $SO(3)$. The former is the double cover of the latter, that means there exists a 2:1 group homomorphism $\rho : SU(2) \to SO(3)$. Even those groups are very hard to imagine. First of all note that as a group manifold $SO(3)$ is isomorphic to $\mathbb{RP}^3$. Of course it is impossible to actually visualise this. At least I cannot. Then, $SO(3) \cong SU(2)/\mathbb{Z}_2$ where $\mathbb{Z}_2$ is the center of $SU(2)$. This is even harder to imagine or visualise despite $SU(2)$ being simply connected, $\pi_1(SU(2))=1$. Although both groups can be thought of as spheres topologically, you lose much information about their actual structure if you think in such a naive way. Now going to $SU(3)$. This group does not have a similar to the previous one group isomorphism. Therefore it is not easy to give it some interpretation as the rotation group in $3+1$ dimensions (since $SU(2)$ would correspond to rotations in 2+1). The best way to understand unitary (and not only) groups is through their representation theory, i.e. how they act on some vector space. There you can think of rotations quite nicely. Then $SU(3)$ is the group whose elements are unitary matrices with determinant one and rotate basis of a 3 dimensional complex vector space.

answered Oct 31, 2017 by (3,605 points)

Added some details above to clarify.

You don't actually visualize it as a group manifold. You visualize the action that it has on something like a spinor (i.e. your palm). And what you actually visualize is the double covering. You don't have such a map for SU(3) that will link it to some space rotation.

True, you visualize the action, not the group. And the original question was whether there is a similar visualization (of the action) for SU(3).

No.

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