Editing Eightfold way (physics) (section)

==Flavor symmetry==
{{main|Flavour (particle physics)}}

===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}}

===Connection to representation theory===
{{main|Particle physics and representation theory}}
{{see also|Compact group#Representation theory of a connected compact Lie group}}Assume we have a certain particle—for example, a proton—in a quantum state <math>|\psi\rangle</math>. If we apply one of the flavour rotations ''A'' to our particle, it enters a new quantum state which we can call <math>A|\psi\rangle</math>. Depending on ''A'', this new state might be a proton, or a neutron, or a superposition of a proton and a neutron, or various other possibilities. The set of all possible quantum states spans a vector space.

[[Representation theory]] is a mathematical theory that describes the situation where elements of a group (here, the flavour rotations ''A'' in the group SU(3)) are [[automorphism]]s of a vector space (here, the set of all possible quantum states that you get from flavour-rotating a proton). Therefore, by studying the representation theory of SU(3), we can learn the possibilities for what the vector space is and how it is affected by flavour symmetry.

Since the flavour rotations ''A'' are approximate, not exact, symmetries, each orthogonal state in the vector space corresponds to a different particle species. In the example above, when a proton is transformed by every possible flavour rotation ''A'', it turns out that it moves around an 8&nbsp;dimensional vector space. Those 8&nbsp;dimensions correspond to the 8&nbsp;particles in the so-called "baryon octet" (proton, neutron, [[Sigma baryon|{{SubatomicParticle|Sigma+}}, {{SubatomicParticle|Sigma0}}, {{SubatomicParticle|Sigma-}}]], [[Xi baryon|{{SubatomicParticle|Xi-}}, {{SubatomicParticle|Xi0}}]], [[Lambda baryon|{{SubatomicParticle|Lambda}}]]). This corresponds to an 8-dimensional ("octet") representation of the group SU(3). Since ''A'' is an approximate symmetry, all the particles in this octet have similar mass.<ref name=Griffiths-2008/>

Every [[Lie group]] has a corresponding [[Lie algebra]], and each [[group representation]] of the Lie group can be mapped to a corresponding [[Lie algebra representation]] on the same vector space. The Lie algebra <math>\mathfrak{su}</math>(3) can be written as the set of 3×3 traceless [[Hermitian matrices]]. Physicists generally discuss the representation theory of the Lie algebra <math>\mathfrak{su}</math>(3) instead of the Lie group SU(3), since the former is simpler and the two are ultimately equivalent.