Cadabra manual: Fierz

{\mu,\nu,\rho}::Indices(curved, position=fixed). {m,n,p,q,r,s,t,u,v}::Indices(flat, position=independent). {m,n,p,q,r,s,t,u,v}::Integer(0..10). T^{#{\mu}}::AntiSymmetric. \psi_{\mu}::SelfAntiCommuting. \psi_{\mu}::Spinor(dimension=11, type=Majorana). \theta::Spinor(dimension=11, type=Majorana). \epsilon::Spinor(dimension=11, type=Majorana). {\theta,\epsilon,\psi_{\mu}}::AntiCommuting. \bar{#}::DiracBar. \delta^{m n}::KroneckerDelta. \Gamma^{#{m}}::GammaMatrix(metric=\delta).

obj:= T^{\mu\nu\rho} e_{\nu}^{s} \bar{\theta} \Gamma^{r s} \psi_{\rho} \bar{\psi_{\mu}} \Gamma^{r} \epsilon;

$\displaystyle{}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \psi_{\rho} \bar{\psi_{\mu}} \Gamma^{r} \epsilon$

fierz(_, $\theta, \epsilon, \psi_{\mu}, \psi_{\rho}$ );

$\displaystyle{} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \psi_{\rho} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{m} \psi_{\rho} - \frac{1}{64}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{n m} \psi_{\rho} - \frac{1}{192}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{p n m} \psi_{\rho} - \frac{1}{768}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p q} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{q p n m} \psi_{\rho} - \frac{1}{3840}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p q t} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{t q p n m} \psi_{\rho}$

converge(obj): join_gamma(_) distribute(_) eliminate_kronecker(_) canonicalise(_) rename_dummies(_);

$\displaystyle{}\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n} \epsilon \bar{\psi_{\nu}} \Gamma^{n} \psi_{\rho}+\frac{5}{16}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho}+\frac{3}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n p} \epsilon \bar{\psi_{\nu}} \Gamma_{n}\,^{p} \psi_{\rho}+\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{n} \bar{\theta} \Gamma^{m} \epsilon \bar{\psi_{\nu}} \Gamma_{n m} \psi_{\rho}+\frac{1}{384}T^{\mu \nu \rho} e_{\mu}\,^{r} \bar{\theta} \Gamma^{m n p q} \epsilon \bar{\psi_{\nu}} \Gamma_{r m n p q} \psi_{\rho}$

tst:=1/4 T^{\mu \nu \rho} e_{\mu}^{m} \bar{\theta} \Gamma^{m n} \epsilon \bar{\psi_{\nu}} \Gamma_{n} \psi_{\rho} + 5/16 T^{\mu \nu \rho} e_{\mu}^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho} + 3/32 T^{\mu \nu \rho} e_{\mu}^{m} \bar{\theta} \Gamma^{m n p} \epsilon \bar{\psi_{\nu}} \Gamma_{n p} \psi_{\rho} + 1/4 T^{\mu \nu \rho} e_{\mu}^{m} \bar{\theta} \Gamma^{n} \epsilon \bar{\psi_{\nu}} \Gamma_{m n} \psi_{\rho} + 1/384 T^{\mu \nu \rho} e_{\mu}^{m} \bar{\theta} \Gamma^{n p q r} \epsilon \bar{\psi_{\nu}} \Gamma_{m n p q r} \psi_{\rho};

$\displaystyle{}\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{m n} \epsilon \bar{\psi_{\nu}} \Gamma_{n} \psi_{\rho}+\frac{5}{16}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho}+\frac{3}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{m n p} \epsilon \bar{\psi_{\nu}} \Gamma_{n p} \psi_{\rho}+\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n} \epsilon \bar{\psi_{\nu}} \Gamma_{m n} \psi_{\rho}+\frac{1}{384}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n p q r} \epsilon \bar{\psi_{\nu}} \Gamma_{m n p q r} \psi_{\rho}$