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The '''Gell-Mann matrices''', named for [[Murray Gell-Mann]], are one possible 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#n_.3D_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.  
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==Defining relations==
 
These Lie Algebra elements  obey the [[Commutator|commutation]] relations
:<math>[g_i, g_j] = if^{ijk} g_k \,</math>
where a sum over the index ''k'' [[Einstein notation|is implied]]. The structure constants <math>f^{ijk}</math> are completely antisymmetric in the three indices and have 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>
Any set of [[Hermitian matrices]] which obey these relations qualifies. 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>, where <math>\theta_j</math> are real numbers and a sum over the index ''j'' is implied. Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.
 
==Particular representations==
 
An important representation involves 3×3 matrices, because the group elements then act on complex vectors with 3 entries, i.e., on the [[fundamental representation]] of the group. A particular choice of this representation is
 
:{| 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>
|}
and <math>g_i = \lambda_i/2</math>.
 
These matrices are [[traceless]], Hermitian, and obey the extra relation <math>\mathrm{tr}(\lambda_i \lambda_j) = 2\delta_{ij}</math>. These properties were chosen by Gell-Mann because they then generalize the [[Pauli matrices]] for [[SU(2)]]. They also naturally extend to general [[SU(n)]], cf. [[Generalizations of Pauli matrices]].
 
In this representation, it is clear that 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 3 independent [[SU(2)]] subgroups: <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 ''x'' and ''y'' are linear combinations of <math>\lambda_3</math> and <math>\lambda_8</math>.
 
The squared sum of the Gell-Mann matrices gives the [[Casimir operator]]:
 
:<math> C = \sum_{i=1}^8 \lambda_i \lambda_i = 16/3 </math>.
 
These matrices form a useful model to study the internal rotations in the SU(3) space in the [[quark model]] (mixing up of quark colors), and, to a lesser extent, in [[quantum chromodynamics]].
 
==See also==
*[[Generalizations of Pauli matrices]]
*[[Unitary group]]s and [[group representation]]s
*[[Quark model]], [[color charge]] and [[quantum chromodynamics]]
*[[Gluon#Eight_gluon_colors|colors of the Gluon]]
 
==References==
* {{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
|publisher=[[W. A. Benjamin]]
|lccn=69014391
}}
 
[[Category:Lie groups]]
[[Category:Matrices]]
[[Category:Quantum chromodynamics]]

Latest revision as of 02:04, 10 December 2014

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