Gell-Mann matrices

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3x3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in the defining representation.

Matrices

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

and ${\displaystyle g_{i}=\lambda _{i}/2}$.

Properties

These matrices are traceless, Hermitian (so they can generate unitary matrix group elements through exponentiation), and obey the extra trace orthonormality relation. 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. Gell-Mann's generalization further extends to general SU(n). For their connection to the standard basis of Lie algebras, see the 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 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 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 ${\displaystyle \lambda _{3}}$ and ${\displaystyle \lambda _{8}}$, which commute with each other.

There are 3 independent SU(2) subalgebras:

• ${\displaystyle \{\lambda _{1},\lambda _{2},\lambda _{3}\}}$
• ${\displaystyle \{\lambda _{4},\lambda _{5},x\}}$, and
• ${\displaystyle \{\lambda _{6},\lambda _{7},y\}}$,

where the x and y 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.

Commutation relations

The 8 infinitesimal generators of the Lie algebra are indexed by i satisfy the commutation relations

${\displaystyle [{\frac {\lambda _{i}}{2}},{\frac {\lambda _{j}}{2}}]=i{\sum \limits _{k}}f^{ijk}{\frac {\lambda _{k}}{2}}\,}$.

The structure constants ${\displaystyle f^{ijk}}$ are completely antisymmetric in the three indices, generalizing the antisymmetry of the Levi-Civita symbol ${\displaystyle \epsilon _{jkl}}$ of SU(2). They have values

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

In general, they evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric λs.

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 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,

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

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,

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

Representation theory

A particular choice of matrices is called a group representation, because any element of SU(3) can be written in the form ${\displaystyle \mathrm {exp} (i\theta ^{j}g_{j})}$, where the eight ${\displaystyle \theta ^{j}}$ 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.

The matrices can be realized as a 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 ${\displaystyle g_{i}}$, with i taking values from 1 to 8.

Casimir operators and invariants

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

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

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

Application to quantum chromodynamics

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