Fierz (Request)

Question

Hi everyone,

Is there a way to perform a Fierz transformation on a product of two spinors?

Thanks.

Edit:

For example, if we have the majonara spinors $\chi^{\sigma} = \bar{\chi} = \bar{\lambda} P_{L}$ and $\varphi_{\alpha} = \varphi = P_{L} \chi$, where $P_{L} = \frac{1}{2} (1 + \gamma_{*}) $, the fierz identity for $\varphi \bar{\chi}$ is $ \varphi \bar{\chi} = -\frac{1}{2} P{L} (\bar{\lambda} P_{L} \chi) + \frac{1}{8} P_{L} (\gamma_{\mu \nu} \bar{\lambda} \gamma^{\mu \nu} P\{L} \chi)$.

Comments

Do you mean that the two other spinor indices are 'free', not implicitly contracted with spinor objects?

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For example, if we have the majonara spinors $\chi^{\sigma} = \bar{\chi} = \bar{\lambda} P{L}$ and $\varphi{\alpha} = \varphi = P{L} \chi$, where $P{L} = \frac{1}{2} (1 + \gamma{*}) $, the fierz identity for $\varphi \bar{\chi}$ is $ \varphi \bar{\chi} = -\frac{1}{2} P{L} (\bar{\lambda} P{L} \chi) + \frac{1}{8} P{L} (\gamma{\mu \nu} \bar{\lambda} \gamma^{\mu \nu} P{L} \chi)$.