Editing Gell-Mann matrices

{{short description|A basis for the SU(3) Lie algebra}}
The '''Gell-Mann matrices''', developed by [[Murray Gell-Mann]], are a set of eight [[Linear independence|linearly independent]] 3×3 [[matrix trace|traceless]] [[Hermitian matrices]]  used in the study of the [[strong interaction]] in [[particle physics]].
They span the  [[Lie group#The Lie algebra associated with a Lie group|Lie algebra]] of the [[Special_unitary_group#SU(3)|SU(3)]] group in the defining representation.

==Matrices==

:{| border="0" cellpadding="8" cellspacing="0"
|<math>\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math> 
|<math>\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math> 
|<math>\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}</math> 
|-
|<math>\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}</math> 
|<math>\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}</math> 
|
|-
|<math>\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}</math> 
|<math>\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}</math> 
|<math>\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}  .</math>
|}

==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>

==Representation theory==
{{main|Clebsch–Gordan coefficients for SU(3)}}
A particular choice of matrices is called a [[group representation]], because any element of SU(3) can be written in the form <math>\mathrm{exp}(i \theta^j  g_j)</math> using the [[Einstein notation]], where the eight <math>\theta^j</math> are real numbers and a sum over the index {{mvar|j}} is implied. Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity transformation, since that leaves the commutator unchanged.

The matrices can be realized as a representation of the  [[Lie group#The Lie algebra associated with a Lie group|infinitesimal generator]]s of the [[special unitary group]] called [[Special_unitary_group#The_group_SU(3)|SU(3)]]. The [[Lie algebra]] of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight [[Linear independence|linearly independent]] generators, which can be written as <math>g_i</math>, with ''i'' taking values from 1 to 8.<ref name="Scherer-Schindler"/>

===Casimir operators and invariants===

The squared sum of the Gell-Mann matrices gives the quadratic [[Casimir operator]], a group invariant, 
:<math> C = \sum_{i=1}^8 \lambda_i \lambda_i = \frac{16} 3 I </math>

where <math> I\, </math>is 3×3 identity matrix. There is another, independent, [[Clebsch–Gordan coefficients for SU(3)#Casimir operators|cubic Casimir operator]], as well.

==Application to quantum chromodynamics==
{{main|Color charge|Quantum chromodynamics}}
 
These matrices serve to study the internal (color) rotations of the [[gluon field]]s associated with the coloured quarks of [[quantum chromodynamics]] (cf. [[Gluon#Eight gluon colours|colours of the gluon]]). A gauge colour rotation is a spacetime-dependent SU(3) group element 
<math>\; U = \exp \left(\frac{\ i\ }{2}\ \theta^k({\mathbf r},t)\ \lambda_k\right) \;,</math> where summation over the eight indices {{mvar|k}} is implied.

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

==See also==
* [[Casimir element]]
* [[Clebsch–Gordan coefficients for SU(3)]]
* [[Generalizations of Pauli matrices]]
* [[Group representation]]s
* [[Killing form]]
* [[Pauli matrices]]
* [[Qutrit]]
* [[Special unitary group#The group SU(3)|SU(3)]]

==References==
{{reflist}}
* {{cite journal | last=Gell-Mann | first=Murray | title=Symmetries of Baryons and Mesons | journal=Physical Review | publisher=American Physical Society (APS) | volume=125 | issue=3 | date=1962-02-01 | issn=0031-899X | doi=10.1103/physrev.125.1067 | pages=1067–1084| bibcode=1962PhRv..125.1067G |doi-access=free}}
*{{cite book |first1=T.-P. |last1=Cheng |first2=L.-F. |last2=Li |title=Gauge Theory of Elementary Particle Physics |publisher=[[Oxford University Press]] |year=1983 |isbn=0-19-851961-3 }}
* {{cite book
 |last=Georgi |first=H.
 |year=1999
 |title=Lie Algebras in Particle Physics
 |edition=2nd
 |publisher=[[Westview Press]]
 |isbn=978-0-7382-0233-4
}}
* {{cite book
 |last1=Arfken |first1=G. B.
 |last2=Weber |first2=H. J.
 |last3=Harris |first3=F. E.
 |year=2000
 |title=Mathematical Methods for Physicists
 |edition=7th
 |publisher=[[Academic Press]]
 |isbn=978-0-12-384654-9
}}
* {{cite book
 |last=Kokkedee |first=J. J. J.
 |year=1969
 |title=The Quark Model
 |url=https://archive.org/details/quarkmodel0000kokk |url-access=registration |publisher=[[W. A. Benjamin]]
 |lccn=69014391
}}

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:Quantum chromodynamics]]
[[Category:Mathematical physics]]
[[Category:Lie algebras]]
[[Category:Representation theory of Lie algebras]]
[[Category:Murray Gell-Mann]]