I am trying to write product of Pauli matrices in terms of its indicies. I am trying to find a proof of it.
$\sigma^{z}_{\mu \nu}\sigma^{z}_{\alpha \beta}=\delta_{\mu \beta}\delta_{\nu \alpha}$

$\sigma^{x}_{\mu \nu}\sigma^{x}_{\alpha \beta}=-\epsilon_{\mu \beta}\epsilon_{\nu \alpha}$

And the product of unlike Pauli matrices are

$\sigma^{z}_{\mu \nu}\sigma^{x}_{\alpha \beta}=i\delta_{\mu \beta}\epsilon_{\alpha \nu}$

Here $\sigma$'s are Pauli matrices and $\epsilon_{\alpha \beta}$ are Levi civita symbols.

Any help will be highly appreciated.

  • $\begingroup$ How about just verifying them using the explicit Pauli matrices? Aren’t these relations true only for the well-known matrices, not for other matrices equivalent through conjugation? $\endgroup$ – G. Smith Jun 26 at 2:07
  • $\begingroup$ Checked versus the completeness relations? $\endgroup$ – Cosmas Zachos Jun 26 at 2:55
  • $\begingroup$ $\sigma^{2}_{i} = I.$ You can test the others with $\sigma_{i}\sigma_{j}+\sigma_{i}\sigma_{j}=0.$ Note, $\sigma_{x}\sigma_{y} \neq \sigma_{y}\sigma_{x}$ so you're missing $3$ products. For instance, $\sigma_{x}\sigma_{y}=-\sigma_{y},\sigma_{x}\sigma_{z}=-i\sigma_{y},$ and $\sigma_{y}\sigma_{z}=i\sigma_{x}$ so it you know $3$ products where $i\neq j$ you know them all. $\endgroup$ – Cinaed Simson Jun 26 at 4:26
  • $\begingroup$ @ Cosmas Zachos Yes i have checked the completeness relation, but it speaks of sum on product of all $\sigma$s not individual products like $\sigma^{z}_{\mu \nu} \sigma^{z}_{\alpha \beta}$. I have $\endgroup$ – Hazoor Imran Jun 26 at 5:02

If you want to do it using only indices, maybe you can write the Pauli matrices in this way (making the indices vary in $\{0,1\}$ and $\epsilon_{01} = 1$) $$ \sigma^x_{\alpha\beta} = \epsilon_{\alpha\beta} (-1)^\alpha\,,\qquad \sigma^y_{\alpha\beta} = -i\epsilon_{\alpha\beta} \,,\qquad \sigma^z_{\alpha\beta} = \delta_{\alpha\beta} (-1)^\beta\,. $$ Then you'll need only the identities of the $\delta$ and $\epsilon$ tensors.

Anyway your formulas appear to be wrong. I checked two at random and I found $$ \begin{aligned} 1)&&\sigma^z_{01} \,\sigma^z_{10} &= 0 \neq \delta_{11}\delta_{00}\,. \\ \\ 4)&&\sigma^x_{01} \,\sigma^y_{01} &= -i \neq \delta_{10}\epsilon_{01}\,. \end{aligned} $$

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    $\begingroup$ Thanks MannyC. Yes there is mistake. $\endgroup$ – Hazoor Imran Jun 26 at 3:11
  • $\begingroup$ Did you figure out the correct relations you are supposed to prove now? $\endgroup$ – MannyC Jun 26 at 3:14
  • $\begingroup$ This is used in this article "journals.aps.org/prb/pdf/10.1103/PhysRevB.86.245309" Eq (5.18) and Eq. (5.24). It may seem some how obscure though. $\endgroup$ – Hazoor Imran Jun 26 at 3:24
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    $\begingroup$ You are trying verify the product $T_{\mu\nu} T_{\alpha\beta}$, right? I believe they are just using the completeness relations that @Cosmas Zachos linked in your post. $\endgroup$ – MannyC Jun 26 at 3:31

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