TheGell-Mann matrices, developed byMurray Gell-Mann, are a set of eightlinearly independent 3×3tracelessHermitian matrices used in the study of thestrong interaction inparticle physics.They span theLie algebra of theSU(3) group in the defining representation.

These matrices aretraceless,Hermitian, and obey the extra trace orthonormality relation, so they can generateunitary matrix group elements ofSU(3) throughexponentiation.^[1] These properties were chosen by Gell-Mann because they then naturally generalize thePauli matrices forSU(2) to SU(3), which formed the basis for Gell-Mann'squark model.^[2] Gell-Mann's generalization furtherextends to general SU(n). For their connection to thestandard basis of Lie algebras, see theWeyl–Cartan basis.

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, thetrace of the pairwise product results in the ortho-normalization condition

${\displaystyle \mathrm{tr}({\lambda }_i{\lambda }_j)=2{\delta }_{ij},}$

where${\displaystyle {\delta }_{ij}}$ is theKronecker delta.

This is so the embedded Pauli matrices corresponding to the three embedded subalgebras ofSU(2) are conventionally normalized. In this three-dimensional matrix representation, theCartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices${\displaystyle {\lambda }_3}$ and${\displaystyle {\lambda }_8}$, which commute with each other.

There are threesignificantSU(2) subalgebras:

• ${\displaystyle \{{\lambda }_1,{\lambda }_2,{\lambda }_3\}}$
• ${\displaystyle \{{\lambda }_4,{\lambda }_5,x\},}$ and
• ${\displaystyle \{{\lambda }_6,{\lambda }_7,y\},}$

where thex andy are linear combinations of${\displaystyle {\lambda }_3}$ and${\displaystyle {\lambda }_8}$. 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.

The 8 generators of SU(3) satisfy thecommutation and anti-commutation relations^[3]

with thestructure constants

Thestructure constants${\displaystyle d_{abc}}$ are completely symmetric in the three indices. Thestructure constants${\displaystyle f_{abc}}$ are completely antisymmetric in the three indices, generalizing the antisymmetry of theLevi-Civita symbol${\displaystyle {ϵ}_{jkl}}$ ofSU(2). For the present order of Gell-Mann matrices they take the values

${\displaystyle f_{123}=1,f_{147}=f_{165}=f_{246}=f_{257}=f_{345}=f_{376}=\frac{1}{2},f_{458}=f_{678}=\frac{\sqrt{3}}{2}.}$

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)λs.

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

${\displaystyle {\lambda }_a{\lambda }_b=\frac{1}{2}([{\lambda }_a,{\lambda }_b]+\{{\lambda }_a,{\lambda }_b\})=\frac{2}{3}{\delta }_{ab}I+\sum _c\left(d_{abc}+i{f}_{abc}\right){\lambda }_c,}$

whereI is the 3×3 identity matrix.

Since the eight matrices and the identity are a complete trace-orthogonal set spanning all 3×3 matrices, it is straightforward to find two Fierzcompleteness relations, (Li & Cheng, 4.134), analogous to thatsatisfied 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,

${\displaystyle {\delta }_{\beta }^{\alpha }{\delta }_{\delta }^{\gamma }=\frac{1}{3}{\delta }_{\delta }^{\alpha }{\delta }_{\beta }^{\gamma }+\frac{1}{2}{\lambda }_{\delta }^{\alpha }\cdot {\lambda }_{\beta }^{\gamma }}$

and

${\displaystyle {\lambda }_{\beta }^{\alpha }\cdot {\lambda }_{\delta }^{\gamma }=\frac{16}{9}{\delta }_{\delta }^{\alpha }{\delta }_{\beta }^{\gamma }-\frac{1}{3}{\lambda }_{\delta }^{\alpha }\cdot {\lambda }_{\beta }^{\gamma }.}$

One may prefer the recast version, resulting from a linear combination of the above,

${\displaystyle {\lambda }_{\beta }^{\alpha }\cdot {\lambda }_{\delta }^{\gamma }=2{\delta }_{\delta }^{\alpha }{\delta }_{\beta }^{\gamma }-\frac{2}{3}{\delta }_{\beta }^{\alpha }{\delta }_{\delta }^{\gamma }.}$

A particular choice of matrices is called agroup representation, because any element of SU(3) can be written in the form${\displaystyle \mathrm{e}\mathrm{x}\mathrm{p}(i{\theta }^j{g}_j)}$ using theEinstein notation, where the eight${\displaystyle {\theta }^j}$ are real numbers and a sum over the indexj 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 theinfinitesimal generators of thespecial unitary group calledSU(3). TheLie algebra of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eightlinearly independent generators, which can be written as${\displaystyle g_i}$, withi taking values from 1 to 8.^[1]

The squared sum of the Gell-Mann matrices gives the quadraticCasimir operator, a group invariant,

${\displaystyle C=\sum _{i=1}^8{\lambda }_i{\lambda }_i=\frac{16}{3}I}$

where${\displaystyle I}$is 3×3 identity matrix. There is another, independent,cubic Casimir operator, as well.

These matrices serve to study the internal (color) rotations of thegluon fields associated with the coloured quarks ofquantum chromodynamics (cf.colours of the gluon). A gauge colour rotation is a spacetime-dependent SU(3) group element${\displaystyle U=\exp \left(\frac{i}{2}{\theta }^k(\mathbf{r},t){\lambda }_k\right),}$ where summation over the eight indicesk is implied.

• Casimir element
• Clebsch–Gordan coefficients for SU(3)
• Generalizations of Pauli matrices
• Group representations
• Killing form
• Pauli matrices
• Qutrit
• SU(3)

• Gell-Mann, Murray (1962-02-01)."Symmetries of Baryons and Mesons".Physical Review.125 (3). American Physical Society (APS):1067–1084.Bibcode:1962PhRv..125.1067G.doi:10.1103/physrev.125.1067.ISSN 0031-899X.
• Cheng, T.-P.; Li, L.-F. (1983).Gauge Theory of Elementary Particle Physics.Oxford University Press.ISBN 0-19-851961-3.
• Georgi, H. (1999).Lie Algebras in Particle Physics (2nd ed.).Westview Press.ISBN 978-0-7382-0233-4.
• Arfken, G. B.; Weber, H. J.; Harris, F. E. (2000).Mathematical Methods for Physicists (7th ed.).Academic Press.ISBN 978-0-12-384654-9.
• Kokkedee, J. J. J. (1969).The Quark Model.W. A. Benjamin.LCCN 69014391.

• Matrices (mathematics)
• Quantum chromodynamics
• Mathematical physics
• Lie algebras
• Representation theory of Lie algebras
• Murray Gell-Mann

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