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Is there any way to write the Pauli matrices: $$\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_2=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ (more precisely the component $(\sigma_k)_{ij}$) concisely in summation notation?

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  • $\begingroup$ More on Pauli matrices: physics.stackexchange.com/q/21595/2451 $\endgroup$ – Qmechanic Jan 27 '16 at 17:09
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    $\begingroup$ What do you mean by "summation notation"? $\endgroup$ – DanielSank Jan 27 '16 at 18:41
  • $\begingroup$ @DanielSank Einstein summation notation e.g. $(\vec a\times \vec b)_i=\varepsilon_{ijk} a_jb_k$ $\endgroup$ – Quantum spaghettification Jan 27 '16 at 19:04
  • $\begingroup$ Well, it's true that $[\sigma_a, \sigma_b] = 2 i \varepsilon_{abc} \sigma_c$ and that $\{\sigma_a, \sigma_b \} = 2 \delta_{ab} I$. For all I know the Pauli matrices are uniquely determined by those properties and the fact that they're Hermitian, or something. $\endgroup$ – DanielSank Jan 27 '16 at 19:08
  • $\begingroup$ Those conditions are not enough to uniquely determine the Pauli matrices, because they are all left invariant by a unitary basis change $\sigma^a \mapsto P\sigma^a P^†$. To determine them uniquely you must additionally demand that $\sigma^3$ is diagonal. $\endgroup$ – Luke Pritchett Jan 27 '16 at 20:05

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