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Is there any special reason why Pauli matrices are:
$\sigma _1=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$, $\sigma _2=\left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right)$, $\sigma _3=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)$
and not for example
$\sigma _1=\left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right)$, $\sigma _2=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$, $\sigma _3=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)$ ?
Will the reason be applicable to Gell-Mann matrices ?

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    $\begingroup$ The two sets do not share the same structure constants. Can you flip signs so they would? $\endgroup$ May 23 '20 at 2:47
  • $\begingroup$ Which one is done first, fixing the structure constant, or defining the order/index of the matrices ? $\endgroup$
    – Hantarto
    May 23 '20 at 2:49
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Under the convention that $1 \leftrightarrow x, 2 \leftrightarrow y, 3 \leftrightarrow z$, the ordering we have ties to the actions performed about $x, y, z$ axes.

  • In the context of Pauli's work, $σ_k$ represents the observable corresponding to spin along the $k^{\text{th}}$ coordinate axis in three-dimensional Euclidean space $\mathbb{R}^3$.
  • Also, the Bloch sphere rotations follow that $\exp(i\sigma_{k}\theta)$ rotates the sphere by $\theta/2$ about $k^{\text{th}}$ axis.

Again it is purely conventional, we might have reordered the axes themselves. What you should be wary of though, are the signs as you flip the ordering. Many algebraic properties including commutation relationships, relations to dot and cross products, structure constants in $SU(2)$ and so on may no longer hold and it is better not to change that since many QM and QFT identities look nicer under the normal convention.

Will the reason be applicable to Gell-Mann matrices ?

Same as above. There is no new physics with reordering, it is a convenient choice.

Though it may seem arbitrary to name the only two diagonal matrices as $\lambda_3$ and $\lambda_8$, notice some nice properties. For example,

  • $\lambda_1, \lambda_2, \lambda_3$ are just $2 \times 2$ Pauli matrices embedded in $3$ dimensions.
  • There are three independent $SU(2)$ subalgebras whose Casimirs mutually commute:

    1. $\{\lambda_1, \lambda_2, \lambda_3\}$
    2. $\{\lambda_4, \lambda_5, x\}$
    3. $\{\lambda_6, \lambda_7, y\}$ where the $x$ and $y$ are linear combinations of $\lambda_3$ and $\lambda_8$.

This is not to say there are no other orderings yielding similar easy-to-remember niceties, but we happen to use this one more.

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Your first set of $\sigma_a$ satisfies the standard angular momentum algebra, $$[\sigma_a, \sigma_b] = 2 i \varepsilon_{a b c}\,\sigma_c .$$

Your second set (name it differently!!), $\Sigma_a: ~~ \{\Sigma_1=\sigma_2, \Sigma_2=\sigma_1, \Sigma_3=\sigma_3\}$, satisfies an algebra with the opposite structure constants, $$[\Sigma_a, \Sigma_b] = -2 i \varepsilon_{a b c}\,\Sigma_c .$$

To get an alternate representation of your exact first algebra, you need $s_a\equiv -\Sigma_a$, $$[s_a, s_b] = 2 i \varepsilon_{a b c}\,s_c .$$

This amounts to a succession of two rotations (around z by π/2 and around x by π) of your axes from a right-handed coordinate system to a likewise right-handed one; hence a similarity transformation, equivalent to a basis change, preserving the original Lie algebra, $$s_a= (-i\sigma_3) \sigma_a (i\sigma_3) .$$

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