Fierz identity

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Majorana Fermions in 2 Dimensions

  • \chi_\alpha \bar \chi_\beta= -\frac{1}{2}\delta_{\alpha\beta} \bar\chi_\gamma \chi_\gamma\,
  • \chi_\alpha (\bar\xi \eta) = -\xi_\alpha (\bar\eta\chi) - \eta_\alpha (\bar\chi\xi)\,

Proof:

Using \bar\chi\xi = \bar\xi\chi\,,

\xi_\alpha (\bar\eta\chi) + \eta_\alpha (\bar\chi\xi)\, = \left( \xi\bar\eta + \eta\bar\xi\right)_{\alpha\gamma}\chi_\gamma\,

Then, e.g., if \chi_\gamma = \chi_\lambda \delta_{\lambda\gamma}\, for some \lambda\, and denoting \bar\lambda\, by its complement,

\xi_\alpha (\bar\eta\chi) + \eta_\alpha (\bar\chi\xi)\, = -i\left( \xi_\alpha \varepsilon^{\gamma\beta}\eta_\beta + \eta_\alpha \varepsilon^{\gamma\beta} \xi_\beta\right)\chi_\gamma\,
= -i\left( \xi_\alpha \varepsilon^{\lambda \bar\lambda}\eta_{\bar\lambda} + \eta_\alpha \varepsilon^{\lambda \bar\lambda} \xi_{\bar\lambda}\right)\chi_{\lambda} \, (no implicit summation)

Then, e.g., if \bar\lambda = \alpha\,, this expression vanishes. If \bar\lambda = \bar\alpha\,, i.e., \lambda = \alpha\,, however,

\xi_\alpha (\bar\eta\chi) + \eta_\alpha (\bar\chi\xi)\, = -i\left( \xi_\lambda\varepsilon^{\lambda \bar\lambda}\eta_{\bar\lambda} + \eta_\lambda\varepsilon^{\lambda \bar\lambda} \xi_{\bar\lambda}\right)\chi_{\lambda} \,
= -i \chi_{\lambda} \left( \xi_\lambda\varepsilon^{\lambda \bar\lambda}\eta_{\bar\lambda} + \xi_{\bar\lambda}\varepsilon^{\bar\lambda\lambda }\eta_\lambda \right) \,
= -\chi_{\lambda} (\bar\xi \eta)\,

Hence the RHS is only non-zero when \lambda= \alpha\,. Summing the original expression over \gamma\,, we get a single contribution from \gamma = \alpha\,, so that

\chi_\alpha (\bar\xi \eta) = -\xi_\alpha (\bar\eta\chi)-\eta_\alpha (\bar\chi\xi)\,

References

[1] [2]

  1. ↑ M. Fierz (1937). "Zur Fermischen Theorie des β-Zerfalls (The Fermi theory of β-decay)". Z. Physik 104: 553. DOI:10.1007/BF01330070. 
  2. ↑ W. Pauli (1936). "Contributions mathématiques à la théorie des matrices de Dirac". Ann. Inst. Henri Poincaré 6: 109. 
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