Cadabra
a field theory motivated approach to computer algebra
{\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(_, $\theta, \epsilon, \psi_{\mu}, \psi_{\rho}$ );
\(\displaystyle{} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \psi_{\rho} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{m} \psi_{\rho} - \frac{1}{64}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{n m} \psi_{\rho} - \frac{1}{192}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{p n m} \psi_{\rho} - \frac{1}{768}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p q} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{q p n m} \psi_{\rho} - \frac{1}{3840}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \Gamma^{r s} \Gamma^{m n p q t} \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{t q p n m} \psi_{\rho}\)
join_gamma(_); distribute(_) eliminate_kronecker(_) join_gamma(_) distribute(_) eliminate_kronecker(_) canonicalise(_); rename_dummies(_);
\(\displaystyle{} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \left(\Gamma^{r s r}+\Gamma^{r} \delta^{r s}-\Gamma^{s} \delta^{r r}\right) \epsilon \bar{\psi_{\mu}} \psi_{\rho} - \frac{1}{32}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \left(\Gamma^{r s m}+\Gamma^{r} \delta^{m s}-\Gamma^{s} \delta^{m r}\right) \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{m} \psi_{\rho} - \frac{1}{64}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \left(\Gamma^{r s m n}+\Gamma^{r n} \delta^{m s}-\Gamma^{r m} \delta^{n s}-\Gamma^{s n} \delta^{m r}+\Gamma^{s m} \delta^{n r}+\delta^{m s} \delta^{n r}-\delta^{m r} \delta^{n s}\right) \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{n m} \psi_{\rho} - \frac{1}{192}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \left(\Gamma^{r s m n p}+\Gamma^{r n p} \delta^{m s}-\Gamma^{r n m} \delta^{p s}+\Gamma^{r p m} \delta^{n s}-\Gamma^{s n p} \delta^{m r}+\Gamma^{s n m} \delta^{p r}-\Gamma^{s p m} \delta^{n r}+\Gamma^{p} \delta^{m s} \delta^{n r}-\Gamma^{m} \delta^{p s} \delta^{n r}+\Gamma^{n} \delta^{p s} \delta^{m r}-\Gamma^{p} \delta^{m r} \delta^{n s}+\Gamma^{m} \delta^{p r} \delta^{n s}-\Gamma^{n} \delta^{p r} \delta^{m s}\right) \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{p n m} \psi_{\rho} - \frac{1}{768}T^{\mu \nu \rho} e_{\nu}\,^{s} \bar{\theta} \left(\Gamma^{r s m n p q}+\Gamma^{r n p q} \delta^{m s}-\Gamma^{r n p m} \delta^{q s}+\Gamma^{r n q m} \delta^{p s}-\Gamma^{r p q m} \delta^{n s}-\Gamma^{s n p q} \delta^{m r}+\Gamma^{s n p m} \delta^{q r}-\Gamma^{s n q m} \delta^{p r}+\Gamma^{s p q m} \delta^{n r}+\Gamma^{p q} \delta^{m s} \delta^{n r}-\Gamma^{p m} \delta^{q s} \delta^{n r}+\Gamma^{p n} \delta^{q s} \delta^{m r}+\Gamma^{q m} \delta^{p s} \delta^{n r}-\Gamma^{q n} \delta^{p s} \delta^{m r}+\Gamma^{m n} \delta^{p s} \delta^{q r}-\Gamma^{p q} \delta^{m r} \delta^{n s}+\Gamma^{p m} \delta^{q r} \delta^{n s}-\Gamma^{p n} \delta^{q r} \delta^{m s}-\Gamma^{q m} \delta^{p r} \delta^{n s}% +\Gamma^{q n} \delta^{p r} \delta^{m s}-\Gamma^{m n} \delta^{p r} \delta^{q s}\right) \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} \left(\Gamma^{r s m n p q t}+\Gamma^{r n p q t} \delta^{m s}-\Gamma^{r n p q m} \delta^{s t}+\Gamma^{r n p t m} \delta^{q s}-\Gamma^{r n q t m} \delta^{p s}+\Gamma^{r p q t m} \delta^{n s}-\Gamma^{s n p q t} \delta^{m r}+\Gamma^{s n p q m} \delta^{r t}-\Gamma^{s n p t m} \delta^{q r}+\Gamma^{s n q t m} \delta^{p r}-\Gamma^{s p q t m} \delta^{n r}+\Gamma^{p q t} \delta^{m s} \delta^{n r}-\Gamma^{p q m} \delta^{s t} \delta^{n r}+\Gamma^{p q n} \delta^{s t} \delta^{m r}+\Gamma^{p t m} \delta^{q s} \delta^{n r}-\Gamma^{p t n} \delta^{q s} \delta^{m r}+\Gamma^{p m n} \delta^{q s} \delta^{r t}-\Gamma^{q t m} \delta^{p s} \delta^{n r}+\Gamma^{q t n} \delta^{p s} \delta^{m r}% -\Gamma^{q m n} \delta^{p s} \delta^{r t}+\Gamma^{t m n} \delta^{p s} \delta^{q r}-\Gamma^{p q t} \delta^{m r} \delta^{n s}+\Gamma^{p q m} \delta^{r t} \delta^{n s}-\Gamma^{p q n} \delta^{r t} \delta^{m s}-\Gamma^{p t m} \delta^{q r} \delta^{n s}+\Gamma^{p t n} \delta^{q r} \delta^{m s}-\Gamma^{p m n} \delta^{q r} \delta^{s t}+\Gamma^{q t m} \delta^{p r} \delta^{n s}-\Gamma^{q t n} \delta^{p r} \delta^{m s}+\Gamma^{q m n} \delta^{p r} \delta^{s t}-\Gamma^{t m n} \delta^{p r} \delta^{q s}\right) \Gamma^{r} \epsilon \bar{\psi_{\mu}} \Gamma_{t q p n m} \psi_{\rho}\)
\(\displaystyle{}\frac{9}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{s} \epsilon \bar{\psi_{\nu}} \Gamma^{s} \psi_{\rho} - \frac{1}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{r} \epsilon \bar{\psi_{\nu}} \Gamma^{r} \psi_{\rho}+\frac{5}{16}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho}+\frac{7}{64}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n s} \epsilon \bar{\psi_{\nu}} \Gamma_{n}\,^{s} \psi_{\rho} - \frac{1}{64}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n r} \epsilon \bar{\psi_{\nu}} \Gamma_{n}\,^{r} \psi_{\rho}+\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n} \epsilon \bar{\psi_{\nu}} \Gamma_{m n} \psi_{\rho}+\frac{1}{3840}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n p q s t} \epsilon \bar{\psi_{\nu}} \Gamma_{n p q}\,^{s}\,_{t} \psi_{\rho} - \frac{1}{3840}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n p q r t} \epsilon \bar{\psi_{\nu}} \Gamma_{n p q}\,^{r}\,_{t} \psi_{\rho}+\frac{1}{384}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n p q t} \epsilon \bar{\psi_{\nu}} \Gamma_{m n p q t} \psi_{\rho}\)
\(\displaystyle{}\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n} \epsilon \bar{\psi_{\nu}} \Gamma^{n} \psi_{\rho}+\frac{5}{16}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho}+\frac{3}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma_{m}\,^{n p} \epsilon \bar{\psi_{\nu}} \Gamma_{n}\,^{p} \psi_{\rho}+\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{n} \bar{\theta} \Gamma^{m} \epsilon \bar{\psi_{\nu}} \Gamma_{n m} \psi_{\rho}+\frac{1}{384}T^{\mu \nu \rho} e_{\mu}\,^{r} \bar{\theta} \Gamma^{m n p q} \epsilon \bar{\psi_{\nu}} \Gamma_{r m n p q} \psi_{\rho}\)
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};
\(\displaystyle{}\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{m n} \epsilon \bar{\psi_{\nu}} \Gamma_{n} \psi_{\rho}+\frac{5}{16}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \epsilon \bar{\psi_{\nu}} \Gamma_{m} \psi_{\rho}+\frac{3}{32}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{m n p} \epsilon \bar{\psi_{\nu}} \Gamma_{n p} \psi_{\rho}+\frac{1}{4}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n} \epsilon \bar{\psi_{\nu}} \Gamma_{m n} \psi_{\rho}+\frac{1}{384}T^{\mu \nu \rho} e_{\mu}\,^{m} \bar{\theta} \Gamma^{n p q r} \epsilon \bar{\psi_{\nu}} \Gamma_{m n p q r} \psi_{\rho}\)
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