I'm trying to show:

$\overline{\sigma}^\mu\sigma^\nu\overline{\sigma}^\rho = \eta^{\mu\nu}\overline{\sigma}^\rho+\eta^{\nu\rho}\overline{\sigma}^\mu-\eta^{\mu\rho}\overline{\sigma}^\nu+i\epsilon^{\mu\nu\rho\lambda}\overline{\sigma}_\lambda$

Where $\sigma^\mu = (1,-\sigma_i)$ and $\overline{\sigma}^\mu=(1,\sigma_i)$ with $1$ the 2x2 Identity matrix and $\sigma_i$ the Pauli matrices and $\eta^{\mu\nu} = diag(1,-1,-1,-1)$.

The trouble that I'm having is that when checking one of the possibilities, for example $\mu=0, \nu=1, \rho=2$, the relation doesn't seem to work:

The left hand side would be:

$\overline{\sigma}^0\sigma^1\overline{\sigma}^2 = -\sigma^1\sigma^2 = -i\sigma^3$ (where the minus sign comes from the definition of $\sigma^\mu$)·

While the right hand side I think is:

$i \sigma^3$

Therefore I have a sign somewhere missing.

Could someone tell me where am I being mistaken?

  • 2
    $\begingroup$ $\bar\sigma_3=-\sigma^3$ $\endgroup$ – AccidentalFourierTransform Jun 7 '18 at 20:54
  • $\begingroup$ Everthing is fine, apply the relation proposed by AccidentalFourierTransform and you're done. $\endgroup$ – Frederic Thomas Jun 8 '18 at 12:36

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