Gell-Mann matrices

Murray Gell-Mann

The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the special unitary group called 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 linearly independent generators, which can be written as $g_{i}$ , with i taking values from 1 to 8.

Defining relations

These Lie Algebra elements obey the commutation relations

[g_{i},g_{j}]=if^{ijk}g_{k}\,

where a sum over the index k is implied.

The structure constants $f^{ijk}$ are completely antisymmetric in the three indices, and are analogous to the Levi-Civita symbol $\epsilon_{jkl}$ of $SU(2)$ . They have values

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}}\ .

In general, they vanish, unless they contain an odd number of indices from the set {2,5,7}, corresponding to the antisymmetric $λ$ s.

Any set of Hermitian matrices which obey these commutation relations qualifies. A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form $\mathrm {exp} (i\theta ^{j}g_{j})$ , where the eight $\theta ^{j}$ are real numbers and a sum over the index $j$ is implied. Given one representation, an equivalent one may be obtained by an arbitrary unitary similarity 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

$\lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}$	$\lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}$
$\lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}$	$\lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}$
$\lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}$	$\lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}$	$\lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}$

and $g_{i}=\lambda _{i}/2$ . These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation $\mathrm {tr} (\lambda _{i}\lambda _{j})=2\delta _{ij}$ , i.e., upon multiplying two of the above Gell-Mann matrices, the resultant matrix will have a trace equal to twice the Kronecker delta.

These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2). They further naturally extend to general SU(n), cf. Generalizations of Pauli matrices.

In this representation, it is evident that the Cartan subalgebra is the set of linear combinations (with real coefficients) of the two matrices $\lambda _{3}$ and $\lambda _{8}$ , which commute with each other. There are 3 independent SU(2) subalgebras: $\{\lambda _{1},\lambda _{2},\lambda _{3}\}$ , $\{\lambda _{4},\lambda _{5},x\}$ , and $\{\lambda _{6},\lambda _{7},y\}$ , where the x and y are linear combinations of $\lambda _{3}$ and $\lambda _{8}$ ; their SU(2) Casimirs commute. However, any SU(3) unitary similarity transformation of these SU(2)s will yield further SU(2) subalgebras—an infinity of them.

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

C=\sum _{i=1}^{8}\lambda _{i}\lambda _{i}=16/3

There is another, independent, cubic Casimir operator, as well.

These matrices serve to study the internal rotations among the different coloured quarks in quantum chromodynamics.

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 satisfied by the Pauli matrices. Namely, using the dot to sum over all adjoint indices $i$ and utilizing Greek indices for the matrices' row/column indices, the following identities hold,

\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

\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,

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

References

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.

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Gell-Mann matrices

Defining relations

Particular representations

See also

References