Editing Gell-Mann matrices (section)

==Properties==
{{main|Generalizations of Pauli matrices}}
These matrices are [[traceless]], [[Hermitian matrix|Hermitian]], and obey the extra trace orthonormality relation, so they can generate [[unitary matrix]] group elements of [[SU(3)]] through [[Matrix exponential|exponentiation]].<ref name="Scherer-Schindler">{{cite arXiv |author=Stefan Scherer |author2=Matthias R. Schindler |title=A Chiral Perturbation Theory Primer|eprint=hep-ph/0505265|date=31 May 2005|page=1–2}}</ref> These properties were chosen by Gell-Mann because they then naturally generalize the [[Pauli matrices]] for [[SU(2)]] to SU(3), which formed the basis for Gell-Mann's [[quark model]].<ref>{{cite book|author=David Griffiths|title=Introduction to Elementary Particles (2nd ed.)|publisher=[[John Wiley & Sons]]|isbn=978-3-527-40601-2|date=2008|pages=283–288,366–369}}</ref> Gell-Mann's generalization further [[Generalizations of Pauli matrices#Construction|extends to general SU(''n'')]]. For their connection to the [[Root system|standard basis]] of Lie algebras, see  the [[Clebsch–Gordan coefficients for SU(3)#Standard basis|Weyl–Cartan basis]].

===Trace orthonormality===

In mathematics, orthonormality typically implies a norm which has a value of unity (1).  Gell-Mann matrices, however, are normalized to a value of 2.  Thus, the [[trace (linear algebra)|trace]] of the pairwise product results in the ortho-normalization condition

:<math>\operatorname{tr}(\lambda_i \lambda_j) = 2\delta_{ij},</math>

where <math>\delta_{ij}</math> is the [[Kronecker delta]].

This is so the embedded Pauli matrices corresponding to the  three embedded subalgebras of ''SU''(2) are conventionally normalized. In this three-dimensional matrix representation,  the [[Cartan subalgebra]] is the set of linear combinations (with real coefficients) of the two matrices <math>\lambda_3</math> and <math>\lambda_8</math>, which commute with each other.

There are three [[Clebsch–Gordan_coefficients_for_SU(3)#Standard_basis|significant]] [[SU(2)]] subalgebras:

*<math>\{\lambda_1, \lambda_2, \lambda_3\}</math>
*<math>\{\lambda_4, \lambda_5, x\},</math> and
*<math>\{\lambda_6, \lambda_7, y\},</math>
where the {{mvar|x}} and {{mvar|y}} are linear combinations of <math>\lambda_3</math> and <math>\lambda_8</math>. The  SU(2) Casimirs of these subalgebras mutually commute.

However, any unitary similarity transformation of these subalgebras will yield SU(2) subalgebras. There is an uncountable number of such transformations.

===Commutation relations===

The 8 generators of SU(3) satisfy the [[commutator|commutation and anti-commutation relations]]<ref name="gellmann17">{{cite web |last1=Haber |first1=Howard |title=Properties of the Gell-Mann matrices |url=http://scipp.ucsc.edu/~haber/ph251/gellmann17.pdf |website=Physics 251 Group Theory and Modern Physics |publisher=U.C. Santa Cruz |access-date=1 April 2019}}</ref>

: <math> \begin{align}
    \left[ \lambda_a, \lambda_b \right] &= 2 i \sum_c f^{abc} \lambda_c, \\
    \{ \lambda_a, \lambda_b \} &= \frac{4}{3} \delta_{ab} I + 2 \sum_c d^{abc} \lambda_c,
\end{align} </math>

with the [[structure constant]]s

: <math> \begin{align}
    f^{abc} &= -\frac{1}{4} i \operatorname{tr}(\lambda_a [ \lambda_b, \lambda_c ]), \\
    d^{abc} &= \frac{1}{4} \operatorname{tr}(\lambda_a \{ \lambda_b, \lambda_c \}).
\end{align} </math>

The [[structure constant]]s <math>d^{abc}</math> are completely symmetric in the three indices. The [[structure constant]]s <math>f^{abc}</math> are completely antisymmetric in the three indices, generalizing the antisymmetry of the [[Levi-Civita symbol]] <math>\epsilon_{jkl}</math> of {{math|''SU''(2)}}.  For the present order of Gell-Mann matrices they take the values

:<math>f^{123} = 1 \ , \quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \frac{1}{2} \ , \quad f^{458} = f^{678} = \frac{\sqrt{3}}{2} \ . </math>

In general, they evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric (imaginary) {{mvar|λ}}s.

Using these commutation relations, the product of Gell-Mann matrices can be written as

: <math> \lambda_a \lambda_b
  = \frac{1}{2} ([\lambda_a,\lambda_b] + \{\lambda_a,\lambda_b\})
  = \frac{2}{3} \delta_{ab} I + \sum_c \left(d^{abc} + i f^{abc}\right) \lambda_c , </math>

where {{mvar|I}} is the identity matrix.

===Fierz completeness relations===

Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierz '''''completeness relations''''', (Li & Cheng, 4.134), analogous to that [[Pauli matrices#Completeness relation 2|satisfied by the Pauli matrices]]. Namely, using the dot to sum over the eight matrices and using Greek indices for their row/column indices, the following identities hold,
:<math>\delta^\alpha _\beta  \delta^\gamma  _\delta   = \frac{1}{3} \delta^\alpha_\delta  \delta^\gamma _\beta  +\frac{1}{2} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta </math>
and 
:<math>\lambda^\alpha _\beta \cdot  \lambda^\gamma  _\delta   = \frac{16}{9} \delta^\alpha_\delta  \delta^\gamma _\beta  -\frac{1}{3} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta ~.</math>
One may prefer the recast version, resulting from a linear combination of the above,
:<math>\lambda^\alpha _\beta \cdot  \lambda^\gamma  _\delta   = 2 \delta^\alpha_\delta  \delta^\gamma _\beta  -\frac{2}{3}  \delta^\alpha_\beta  \delta^\gamma _\delta    ~.</math>