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## Homework Statement

Using

[tex] (\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \epsilon_{\alpha \gamma} \epsilon^{\beta \delta} + \delta^{\delta}_{\alpha} \delta^{\beta}_{\gamma}[/tex]

show that

[tex] \Psi_{\alpha} X_{\beta} = \frac{1}{2} \epsilon_{\alpha \beta} (\Psi X) + \frac{1}{2} (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) [/tex]

## Homework Equations

## The Attempt at a Solution

if I do [tex] (\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} \Psi _{\beta} X_{\delta} [/tex]

I can get

[tex]\Psi_{\beta} X_{\alpha} = \epsilon_{\alpha \beta} (\Psi X) + (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) [/tex]

so i don't know where the factors of a half come from and how to get the right index order

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