Editing Special unitary group (section)

== SU(3) ==
{{see also|Clebsch–Gordan coefficients for SU(3)}}

The group {{math|SU(3)}} is an 8-dimensional [[simple Lie group]] consisting of all {{math|3&nbsp;×&nbsp;3}} [[Unitary matrix|unitary]] [[Matrix (mathematics)|matrices]] with [[determinant]] 1.

=== Topology ===
The group {{math|SU(3)}} is a simply-connected, compact Lie group.<ref>{{harvnb|Hall|2015}} Proposition 13.11</ref> Its topological structure can be understood by noting that {{math|SU(3)}} acts [[transitive action|transitively]] on the unit sphere <math>S^5</math> in <math>\mathbb{C}^3 \cong \mathbb{R}^6</math>. The [[Group action (mathematics)#Fixed points and stabilizer subgroups|stabilizer]] of an arbitrary point in the sphere is isomorphic to {{math|SU(2)}}, which topologically is a 3-sphere. It then follows that {{math|SU(3)}} is a [[fiber bundle]] over the base {{math|''S''<sup>5</sup>}} with fiber {{math|''S''<sup>3</sup>}}. Since the fibers and the base are simply connected, the simple connectedness of {{math|SU(3)}} then follows by means of a standard topological result (the [[Homotopy group#Long exact sequence of a fibration|long exact sequence of homotopy groups]] for fiber bundles).<ref>{{harvnb|Hall|2015}} Section 13.2</ref>

The {{math|SU(2)}}-bundles over {{math|''S''<sup>5</sup>}} are classified by <math>\pi_4\mathord\left(S^3\right) = \mathbb{Z}_2</math> since any such bundle can be constructed by looking at trivial bundles on the two hemispheres <math>S^5_\text{N}, S^5_\text{S}</math> and looking at the transition function on their intersection, which is a copy of {{math|''S''<sup>4</sup>}}, so

<math display="block">S^5_\text{N} \cap S^5_\text{S} \simeq S^4</math>

Then, all such transition functions are classified by homotopy classes of maps

<math display="block">\left[S^4, \mathrm{SU}(2)\right] \cong \left[S^4, S^3\right] = \pi_4\mathord\left(S^3\right) \cong \mathbb{Z}/2</math>

and as <math>\pi_4(\mathrm{SU}(3)) = \{0\}</math> rather than <math>\mathbb{Z}/2</math>, {{math|SU(3)}} cannot be the trivial bundle {{math|SU(2) × ''S''<sup>5</sup> ≅ ''S''<sup>3</sup> × ''S''<sup>5</sup>}}, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.

=== Representation theory ===
The representation theory of {{math|SU(3)}} is well-understood.<ref>{{harvnb|Hall|2015}} Chapter 6</ref> Descriptions of these representations, from the point of view of its complexified Lie algebra <math>\mathfrak{sl}(3; \mathbb{C})</math>, may be found in the articles on [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|Lie algebra representations]] or [[Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group|the Clebsch–Gordan coefficients for {{math|SU(3)}}]].

=== Lie algebra ===
The generators, {{mvar|T}}, of the Lie algebra <math>\mathfrak{su}(3)</math> of {{math|SU(3)}} in the defining (particle physics, Hermitian) representation, are

<math display="block">T_a = \frac{\lambda_a}{2}~,  </math>

where {{math|''λ''<sub>a</sub>}}, the [[Gell-Mann matrices]], are the {{math|SU(3)}} analog of the [[Pauli matrices]] for {{math|SU(2)}}:

<math display="block">\begin{align}
  \lambda_1 ={} &\begin{pmatrix} 0 &  1 &  0 \\ 1 &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & 
  \lambda_2 ={} &\begin{pmatrix} 0 & -i &  0 \\ i &  0 &  0 \\ 0 & 0 &  0 \end{pmatrix}, & 
  \lambda_3 ={} &\begin{pmatrix} 1 &  0 &  0 \\ 0 & -1 &  0 \\ 0 & 0 &  0 \end{pmatrix}, \\[6pt]
  \lambda_4 ={} &\begin{pmatrix} 0 &  0 &  1 \\ 0 &  0 &  0 \\ 1 & 0 &  0 \end{pmatrix}, & 
  \lambda_5 ={} &\begin{pmatrix} 0 &  0 & -i \\ 0 &  0 &  0 \\ i & 0 &  0 \end{pmatrix}, \\[6pt]
  \lambda_6 ={} &\begin{pmatrix} 0 &  0 &  0 \\ 0 &  0 &  1 \\ 0 & 1 &  0 \end{pmatrix}, & 
  \lambda_7 ={} &\begin{pmatrix} 0 &  0 &  0 \\ 0 &  0 & -i \\ 0 & i &  0 \end{pmatrix}, & 
  \lambda_8 = \frac{1}{\sqrt{3}}
                &\begin{pmatrix} 1 &  0 &  0 \\ 0 &  1 &  0 \\ 0 & 0 & -2 \end{pmatrix}.
\end{align}</math>

These {{math|''λ''<sub>a</sub>}} span all [[trace (linear algebra)|traceless]] [[Hermitian matrix|Hermitian matrices]] {{mvar|H}} of the [[Lie algebra]], as required. Note that {{math|''λ''<sub>2</sub>, ''λ''<sub>5</sub>, ''λ''<sub>7</sub>}} are antisymmetric.

They obey the relations

<math display="block">\begin{align}
    \left[T_a, T_b\right] &= i \sum_{c=1}^8 f_{abc} T_c, \\
  \left\{T_a, T_b\right\} &= \frac{1}{3} \delta_{ab} I_3 + \sum_{c=1}^8 d_{abc} T_c,
\end{align}</math>

or, equivalently,

<math display="block">\begin{align}
            \left[\lambda_a, \lambda_b\right] &= 2i \sum_{c=1}^8 f_{abc} \lambda_c, \\
             \{\lambda_a, \lambda_b\} &= \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c}.
\end{align}</math>

The {{mvar|f}} are the [[structure constants]] of the Lie algebra, given by

<math display="block">\begin{align}
                                                      f_{123} &= 1, \\
  f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} &= \frac{1}{2}, \\
                                           f_{458} =  f_{678} &= \frac{\sqrt{3}}{2},
\end{align}</math>

while all other {{math|''f<sub>abc</sub>''}} not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set {{math|{{mset|2, 5, 7}}}}.{{efn|So fewer than {{frac|1|6}} of all {{math|''f<sub>abc</sub>''}}s are non-vanishing.}}

The symmetric coefficients {{math|''d''}} take the values

<math display="block">\begin{align}
                                             d_{118} = d_{228} = d_{338} = -d_{888} &=  \frac{1}{\sqrt{3}} \\
                                             d_{448} = d_{558} = d_{668} =  d_{778} &= -\frac{1}{2\sqrt{3}} \\
  d_{344} = d_{355} = -d_{366} = -d_{377} = -d_{247} = d_{146} = d_{157} =  d_{256} &=  \frac{1}{2} ~.
\end{align}</math>

They vanish if the number of indices from the set {{math|{{mset|2, 5, 7}}}} is odd.

A generic {{math|SU(3)}} group element generated by a traceless 3×3 Hermitian matrix {{mvar|H}}, normalized as {{math|tr(''H''<sup>2</sup>) {{=}} 2}}, can be expressed as a ''second order'' matrix polynomial in {{mvar|H}}:<ref>{{cite journal|last1=Rosen|first1=S P|title=Finite Transformations in Various Representations of SU(3)|journal=Journal of Mathematical Physics|volume=12|issue=4|year=1971|pages=673–681 |doi=10.1063/1.1665634|bibcode=1971JMP....12..673R}};  {{cite journal|doi=10.1016/S0034-4877(15)30040-9|title= Elementary results for the fundamental representation of SU(3)|author1= Curtright, T L|author2=Zachos, C K|year=2015|journal=Reports on Mathematical Physics|volume=76|issue=3|pages=401–404|bibcode=2015RpMP...76..401C|arxiv=1508.00868|s2cid= 119679825}}</ref>

<math display="block">\begin{align}
  \exp(i\theta H) ={}
          &\left[-\frac{1}{3} I\sin\left(\varphi + \frac{2\pi}{3}\right) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin(\varphi) - \frac{1}{4}~H^2\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta\sin(\varphi)\right)}
                {\cos\left(\varphi + \frac{2\pi}{3}\right) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt]
    & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{4}~H^{2}\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi + \frac{2\pi}{3}\right)\right)}
                {\cos(\varphi) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\[6pt]
    & {} + \left[-\frac{1}{3}~I\sin(\varphi) \sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{4}~H^2\right]
           \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi - \frac{2\pi}{3}\right)\right)}
                {\cos(\varphi)\cos\left(\varphi + \frac{2\pi}{3}\right)}
\end{align}</math>
LP
where

<math display="block">\varphi \equiv \frac{1}{3}\left[\arccos\left(\frac{3\sqrt{3}}{2}\det H\right) - \frac{\pi}{2}\right].</math>