Editing Eightfold way (physics) (section)

===SU(3)===

There is an abstract three-dimensional vector space:
<math display="block">
 \text{up quark} \to \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad
 \text{down quark} \to \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad
 \text{strange quark} \to \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix},
</math>
and the laws of physics are ''approximately'' invariant under a determinant-1 [[unitary transformation]] to this space (sometimes called a ''flavour rotation''):
<math display="block">
 \begin{pmatrix} x \\ y \\ z \end{pmatrix} \mapsto
 A \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad
 \text{where}\ A\ \text{is in}\ SU(3).</math>

Here, [[SU(3)]] refers to the [[Lie group]] of 3×3 unitary matrices with determinant 1 ([[special unitary group]]). For example, the flavour rotation
<math display="block">
 A = \begin{pmatrix}
 \phantom- 0 & 1 & 0 \\
          -1 & 0 & 0 \\
 \phantom- 0 & 0 & 1
\end{pmatrix}</math>
is a transformation that simultaneously turns all the up quarks in the universe into down quarks and conversely. More specifically, these flavour rotations are exact symmetries if ''only'' [[strong force]] interactions are looked at, but they are not truly exact symmetries of the universe because the three quarks have different masses and different electroweak interactions.

This approximate symmetry is called ''[[flavour symmetry]]'', or more specifically ''flavour SU(3) symmetry''.

{{see also|Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group}}