Gell-Mann matrices

The Gell-Mann matrices, developed by, are a set of eight 3×3    used in the study of the  in. They span the  of the  group in the defining representation.

Matrices

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and $$g_i = \lambda_i/2$$.
 * $$\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$
 * $$\lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$
 * $$\lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$
 * $$\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$
 * $$\lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}$$
 * $$\lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$$
 * $$\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}$$
 * $$\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}$$
 * }
 * $$\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}$$
 * $$\lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}$$
 * }

Properties
These matrices are, Hermitian (so they can generate group elements through exponentiation), and obey the extra trace orthonormality relation. These properties were chosen by Gell-Mann because they then naturally generalize the for  to, which formed the basis for Gell-Mann's. Gell-Mann's generalization further. For their connection to the of Lie algebras, see  the.

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 of the pairwise product results in the ortho-normalization condition


 * $$\operatorname{tr}(\lambda_i \lambda_j) = 2\delta_{ij},$$

where $$\delta_{ij}$$ is the.

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  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 three independent subalgebras:

where the $x$ and $y$ are linear combinations of $$\lambda_3$$ and $$\lambda_8$$. The SU(2) Casimirs of these subalgebras mutually commute.
 * $$\{\lambda_1, \lambda_2, \lambda_3\}$$
 * $$\{\lambda_4, \lambda_5, x\},$$ and
 * $$\{\lambda_6, \lambda_7, y\},$$

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


 * $$ \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} $$

with the s


 * $$ \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} $$

The s $$f^{abc}$$ are completely antisymmetric in the three indices, generalizing the antisymmetry of the $$\epsilon_{jkl}$$ of $SU(2)$. For the present order of Gell-Mann matrices they take the 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 evaluate to zero, unless they contain an odd count of indices from the set {2,5,7}, corresponding to the antisymmetric (imaginary) $λ$s.

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


 * $$ \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, $$

where 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. Namely, using the dot to sum over the eight matrices and using Greek indices for the their row/column indices, the following identities hold,
 * $$\delta^\alpha _\beta \delta^\gamma  _\delta   = \frac{1}{3} \delta^\alpha_\delta  \delta^\gamma _\beta  +\frac{1}{2} \lambda^\alpha _\delta \cdot \lambda^\gamma _\beta $$

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

One may prefer the recast version, resulting from a linear combination of the above,
 * $$\lambda^\alpha _\beta \cdot \lambda^\gamma  _\delta   = 2 \delta^\alpha_\delta  \delta^\gamma _\beta  -\frac{2}{3}  \delta^\alpha_\beta  \delta^\gamma _\delta    ~.$$

Representation theory
A particular choice of matrices is called a, 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.

The matrices can be realized as a representation of the s of the  called. The of this group (a real Lie algebra in fact) has dimension eight and therefore it has some set with eight  generators, which can be written as $$g_i$$, with i taking values from 1 to 8.

Casimir operators and invariants
The squared sum of the Gell-Mann matrices gives the quadratic, a group invariant,
 * $$ C = \sum_{i=1}^8 \lambda_i \lambda_i = \frac{16} 3 I $$

where $$ I\, $$is 3×3 identity matrix. There is another, independent,, as well.

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