# Representation of spin-$1/2$ operators in terms of Majorana fermions

I am reading Quantum Field Theory in Condensed Matter Physics by A.M. Tsvelik. In Chapter 20, it is claimed that introducing three Majorana fermions $$\gamma^\mu_i$$ on each site $$i$$ of the lattice (such that $$\{\gamma^\mu_i,\gamma^\nu_j\}=\delta^{\mu,\nu}\delta_{i,j}$$), spin-$$1/2$$ operators can be expressed as $$S_i^\lambda=-{i\over 2}\epsilon_{\lambda\mu\nu} \gamma_i^\mu\gamma_i^\nu$$ I am trying to show the commutations relations $$[S_i^\lambda,S_j^\rho]=i\epsilon_{\lambda\rho\sigma}S^\sigma_i \delta_{i,j}$$ Unfortunately, my calculation is wrong (by a factor 2). I would be very grateful if somebody could find the mistake.

I start with $$[S_i^\lambda,S_j^\rho] =-{1\over 4}\epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \big(\gamma_i^\mu\gamma_i^\nu\gamma_j^\zeta\gamma_j^\xi -\gamma_j^\zeta\gamma_j^\xi\gamma_i^\mu\gamma_i^\nu\big)$$ Using the anti-commutation relations between Majorana, I get \eqalign{ \gamma_j^\zeta\gamma_j^\xi\gamma_i^\mu\gamma_i^\nu &=\gamma_j^\zeta\big(\delta^{\xi,\mu}\delta_{i,j} -\gamma_i^\mu\gamma_j^\xi\big)\gamma_i^\nu \cr &=\delta^{\xi,\mu}\delta_{i,j}\gamma_j^\zeta\gamma_i^\nu -\big(\delta^{\mu,\zeta}\delta_{i,j}-\gamma_i^\mu\gamma_j^\zeta\big) \big(\delta^{\nu,\xi}\delta_{i,j}-\gamma_i^\nu\gamma_j^\xi\big)\cr &=\delta^{\xi,\mu}\delta_{i,j}\gamma_j^\zeta\gamma_i^\nu +\delta^{\mu,\zeta}\delta_{i,j}\gamma_i^\nu\gamma_j^\xi +\delta^{\nu,\xi}\delta_{i,j}\gamma_i^\mu\gamma_j^\zeta -\gamma_i^\mu\gamma_j^\zeta\gamma_i^\nu\gamma_j^\xi -\delta^{\mu,\zeta}\delta^{\nu,\xi}\delta_{i,j} \cr &=\big(\delta^{\xi,\mu}\gamma_j^\zeta\gamma_i^\nu \!+\!\delta^{\mu,\zeta}\gamma_i^\nu\gamma_j^\xi \!+\!\delta^{\nu,\xi}\gamma_i^\mu\gamma_j^\zeta\big)\delta_{i,j} -\gamma_i^\mu\big(\delta^{\nu,\zeta}\delta_{i,j} -\gamma_i^\nu\gamma_j^\zeta\big)\gamma_j^\xi -\delta^{\mu,\zeta}\delta^{\nu,\xi}\delta_{i,j} \cr &=\big(\delta^{\xi,\mu}\gamma_j^\zeta\gamma_i^\nu +\delta^{\mu,\zeta}\gamma_i^\nu\gamma_j^\xi +\delta^{\nu,\xi}\gamma_i^\mu\gamma_j^\zeta -\delta^{\nu,\zeta}\gamma_i^\mu\gamma_j^\xi\big) +\gamma_i^\mu\gamma_i^\nu\gamma_j^\zeta\gamma_j^\xi -\delta^{\mu,\zeta}\delta^{\nu,\xi}\delta_{i,j} \cr } so that the commutator reads $$[\gamma_i^\mu\gamma_i^\nu,\gamma_j^\zeta\gamma_j^\xi] =-\big(\delta^{\xi,\mu}\gamma_j^\zeta\gamma_i^\nu +\delta^{\mu,\zeta}\gamma_i^\nu\gamma_j^\xi +\delta^{\nu,\xi}\gamma_i^\mu\gamma_j^\zeta -\delta^{\nu,\zeta}\gamma_i^\mu\gamma_j^\xi\big) +\delta^{\mu,\zeta}\delta^{\nu,\xi}\delta_{i,j}$$ Since $$(\gamma_i^\mu)^2=1/2$$ for Majorana fermions, \eqalign{ \epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \delta^{\xi,\mu}\gamma_i^\zeta\gamma_i^\nu &=-\epsilon_{\lambda\nu\mu}\epsilon_{\rho\zeta\mu}\gamma_i^\zeta\gamma_i^\nu\cr &=-\big(\delta^{\lambda,\rho}\delta^{\nu,\zeta} -\delta^{\lambda,\zeta}\delta^{\nu,\rho}\big)\gamma_i^\zeta\gamma_i^\nu\cr &=-\delta^{\lambda,\rho}\big(\gamma_i^\zeta\big)^2 +\gamma_i^\lambda\gamma_i^\rho \cr &=-{3\over 2}\delta^{\lambda,\rho}+\gamma_i^\lambda\gamma_i^\rho \cr } because of the implicit sum over $$\zeta$$ in the first term. Similarly, \eqalign{ &\epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \delta^{\mu,\zeta}\gamma_i^\nu\gamma_j^\xi ={3\over 2}\delta^{\lambda,\rho}-\gamma_i^\rho\gamma_i^\lambda\cr &\epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \delta^{\nu,\xi}\gamma_i^\mu\gamma_j^\zeta ={3\over 2}\delta^{\lambda,\rho}-\gamma_i^\rho\gamma_i^\lambda\cr &\epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \delta^{\nu,\zeta}\gamma_i^\mu\gamma_j^\xi =-{3\over 2}\delta^{\lambda,\rho}+\gamma_i^\rho\gamma_i^\lambda\cr } so that \eqalign{ \delta^{\xi,\mu}\gamma_j^\zeta\gamma_i^\nu +\delta^{\mu,\zeta}\gamma_i^\nu\gamma_j^\xi +\delta^{\nu,\xi}\gamma_i^\mu\gamma_j^\zeta -\delta^{\nu,\zeta}\gamma_i^\mu\gamma_j^\xi &=3\delta^{\lambda,\rho}+\gamma_i^\lambda\gamma_i^\rho -3\gamma_i^\rho\gamma_i^\lambda\cr &=3\delta^{\lambda,\rho}+\gamma_i^\lambda\gamma_i^\rho -3\big(\delta^{\lambda,\rho}-\gamma_i^\lambda\gamma_i^\rho\big)\cr &=4\gamma_i^\lambda\gamma_i^\rho\cr } The last term of the commutator is $$\epsilon_{\lambda\mu\nu}\epsilon_{\rho\zeta\xi} \delta^{\mu,\zeta}\delta^{\nu,\xi} =\epsilon_{\lambda\mu\nu}\epsilon_{\rho\mu\nu} =\delta^{\lambda,\rho}$$ so \eqalign{ [S_i^\lambda,S_j^\rho] &={1\over 4}\big(4\gamma_i^\lambda\gamma_i^\rho -\delta^{\lambda,\rho}\big)\delta_{i,j} \cr } This result is unfortunately wrong! I expect instead \eqalign{ i\epsilon_{\lambda\rho\sigma}S_i^\sigma &={1\over 2}\epsilon_{\lambda\rho\sigma} \epsilon_{\sigma\zeta\xi} \gamma_i^\zeta\gamma_i^\xi\cr &={1\over 2}\epsilon_{\lambda\rho\sigma} \epsilon_{\zeta\xi\sigma} \gamma_i^\zeta\gamma_i^\xi\cr &={1\over 2}\big(\gamma_i^\lambda\gamma_i^\rho -\gamma_i^\rho\gamma_i^\lambda\big) \cr &={1\over 2}\big(2\gamma_i^\lambda\gamma_i^\rho-\delta^{\lambda,\rho}\big) }

When you dealt with the last term, product of deltas, you missed a factor of two: $$$$\epsilon_{\lambda\mu\nu} \epsilon_{\rho\mu\nu} = {\color{red}2}\delta^{\lambda,\rho}$$$$