a field-theory motivated approach to computer algebra

# Commutator algebra

An often asked question is how to handle commutator algebra with Cadabra. This requires a few steps which are perhaps not entirely transparent to a new user, hence the following simple example. In this notebook, we will verify the invariance of the two quadratic Casimirs of the PoincarĂ© algebra. That is, we will verify that $$[ J_{\mu\nu}, P^2 ] = 0\,,\quad{\rm and}\quad [ J_{\mu\nu}, W^2 ] = 0\,,$$ where $P^2 = P_\mu P_\mu$ is the momentum squared and $W^2 = W_\mu W_\mu$ is the square of $$W_\mu = \epsilon_{\mu\nu\rho\sigma} P_{\nu} J_{\rho\sigma}\,.$$ We first make some straightforward property assignments: declaration of indices, declaration of the operators $P_\mu$ and $J_{\mu\nu}$ and the fact that they do not commute, and so on.
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Indices. {\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Integer(0..3). \eta_{\mu\nu}::KroneckerDelta. \delta{#}::KroneckerDelta. e_{\mu\nu\lambda\rho}::EpsilonTensor(delta=\delta). J_{\mu\nu}::AntiSymmetric. J_{\mu\nu}::SelfNonCommuting. { J_{\mu\nu}, P_{\mu}, W_{\mu} }::NonCommuting.
{J_{\mu\nu}, P_{\mu}, W_{\mu} }::Depends(\commutator{#}).
For computations involving algebras like the one in this notebook, the following set of post processing rules is useful. They move all objects out of commutators and anti-commutators which commute with everything else, and then reorder and relabel indices to obtain a simple form.
def post_process(ex): unwrap(ex) eliminate_kronecker(ex) canonicalise(ex) rename_dummies(ex) collect_terms(ex)

We now input the rules which define the PoincarĂ© algebra. These are simply substitution rules, to be used later in explicit substition commands.
poincare:= { \commutator{J_{\mu\nu}}{P_{\rho}} -> \eta_{\mu\rho} P_{\nu} - \eta_{\nu\rho} P_{\mu}, \commutator{J_{\mu\nu}}{J_{\rho\sigma}} -> \eta_{\mu\rho} J_{\nu\sigma} - \eta_{\mu\sigma} J_{\nu\rho} - \eta_{\nu\rho} J_{\mu\sigma} + \eta_{\nu\sigma} J_{\mu\rho} };
$$\displaystyle{}\left\{{}\left[J_{\mu \nu}, P_{\rho}\right]{} \rightarrow \eta_{\mu \rho} P_{\nu}-\eta_{\nu \rho} P_{\mu}, \mmlToken{mo}[linebreak="goodbreak"]{} {}\left[J_{\mu \nu}, J_{\rho \sigma}\right]{} \rightarrow \eta_{\mu \rho} J_{\nu \sigma}-\eta_{\mu \sigma} J_{\nu \rho}-\eta_{\nu \rho} J_{\mu \sigma}+\eta_{\nu \sigma} J_{\mu \rho}\right\}$$

## The $P^2$ Casimir

We know that $P^2$ is a Casimir, so the following should vanish:
Psq:= \commutator{J_{\mu\nu}}{ P_{\rho}P_{\rho} };
$$\displaystyle{}{}\left[J_{\mu \nu}, P_{\rho} P_{\rho}\right]{}$$
product_rule(_);
$$\displaystyle{}{}\left[J_{\mu \nu}, P_{\rho}\right]{} P_{\rho}+P_{\rho} {}\left[J_{\mu \nu}, P_{\rho}\right]{}$$
substitute(_, poincare);
$$\displaystyle{}\left(\eta_{\mu \rho} P_{\nu}-\eta_{\nu \rho} P_{\mu}\right) P_{\rho}+P_{\rho} \left(\eta_{\mu \rho} P_{\nu}-\eta_{\nu \rho} P_{\mu}\right)$$
distribute(_);
$$\displaystyle{}0$$

## The $W^2$ Casimir

Next we do the same thing with $W^2$, the other PoincarĂ© Casimir...
Wsq:=\commutator{J_{\mu\nu}}{W_\mu W_\mu};
$$\displaystyle{}{}\left[J_{\mu \nu}, W_{\rho} W_{\rho}\right]{}$$
substitute(_, $W_\mu -> e_{\mu\nu\lambda\rho} P_\nu J_{\lambda\rho}$);
$$\displaystyle{}e_{\gamma \alpha \lambda \kappa} e_{\gamma \rho \sigma \beta} {}\left[J_{\mu \nu}, P_{\alpha} J_{\rho \sigma} P_{\beta} J_{\lambda \kappa}\right]{}$$
epsilon_to_delta(_);
$$\displaystyle{}6\delta_{\alpha \rho \lambda \sigma \kappa \beta} {}\left[J_{\mu \nu}, P_{\alpha} J_{\rho \sigma} P_{\beta} J_{\lambda \kappa}\right]{}$$
expand_delta(_);
$$\displaystyle{}-6\left(\frac{1}{6}\delta_{\alpha \rho} \delta_{\beta \lambda} \delta_{\kappa \sigma} - \frac{1}{6}\delta_{\alpha \rho} \delta_{\beta \kappa} \delta_{\lambda \sigma} - \frac{1}{6}\delta_{\alpha \sigma} \delta_{\beta \lambda} \delta_{\kappa \rho}+\frac{1}{6}\delta_{\alpha \beta} \delta_{\lambda \sigma} \delta_{\kappa \rho}+\frac{1}{6}\delta_{\alpha \sigma} \delta_{\beta \kappa} \delta_{\lambda \rho} - \frac{1}{6}\delta_{\alpha \beta} \delta_{\lambda \rho} \delta_{\kappa \sigma}\right) {}\left[J_{\mu \nu}, P_{\alpha} J_{\rho \sigma} P_{\beta} J_{\lambda \kappa}\right]{}$$
product_rule(_);
$$\displaystyle{}-6\left(\frac{1}{6}\delta_{\alpha \rho} \delta_{\beta \kappa} \delta_{\lambda \sigma} - \frac{1}{6}\delta_{\alpha \rho} \delta_{\beta \lambda} \delta_{\kappa \sigma} - \frac{1}{6}\delta_{\alpha \sigma} \delta_{\beta \kappa} \delta_{\lambda \rho}+\frac{1}{6}\delta_{\alpha \beta} \delta_{\lambda \rho} \delta_{\kappa \sigma}+\frac{1}{6}\delta_{\alpha \sigma} \delta_{\beta \lambda} \delta_{\kappa \rho} - \frac{1}{6}\delta_{\alpha \beta} \delta_{\lambda \sigma} \delta_{\kappa \rho}\right) \left(-{}\left[J_{\mu \nu}, P_{\alpha}\right]{} J_{\rho \sigma} P_{\beta} J_{\lambda \kappa}-P_{\alpha} {}\left[J_{\mu \nu}, J_{\rho \sigma}\right]{} P_{\beta} J_{\lambda \kappa}-P_{\alpha} J_{\rho \sigma} {}\left[J_{\mu \nu}, P_{\beta}\right]{} J_{\lambda \kappa}+P_{\alpha} J_{\rho \sigma} P_{\beta} {}\left[J_{\mu \nu}, J_{\kappa \lambda}\right]{}\right)$$
substitute(_, poincare);
$$\displaystyle{}-6\left(\frac{1}{6}\delta_{\beta \rho} \delta_{\alpha \kappa} \delta_{\lambda \sigma} - \frac{1}{6}\delta_{\beta \rho} \delta_{\alpha \lambda} \delta_{\kappa \sigma} - \frac{1}{6}\delta_{\beta \sigma} \delta_{\alpha \kappa} \delta_{\lambda \rho}+\frac{1}{6}\delta_{\beta \alpha} \delta_{\kappa \sigma} \delta_{\lambda \rho}+\frac{1}{6}\delta_{\beta \sigma} \delta_{\alpha \lambda} \delta_{\kappa \rho} - \frac{1}{6}\delta_{\beta \alpha} \delta_{\kappa \rho} \delta_{\lambda \sigma}\right) \left(-\left(\eta_{\beta \mu} P_{\nu}-\eta_{\beta \nu} P_{\mu}\right) J_{\rho \sigma} P_{\alpha} J_{\lambda \kappa}-P_{\beta} \left(\eta_{\mu \rho} J_{\nu \sigma}-\eta_{\mu \sigma} J_{\nu \rho}-\eta_{\nu \rho} J_{\mu \sigma}+\eta_{\nu \sigma} J_{\mu \rho}\right) P_{\alpha} J_{\lambda \kappa}-P_{\beta} J_{\rho \sigma} \left(\eta_{\alpha \mu} P_{\nu}-\eta_{\alpha \nu} P_{\mu}\right) J_{\lambda \kappa}+P_{\beta} J_{\rho \sigma} P_{\alpha} \left(-\eta_{\kappa \mu} J_{\lambda \nu}+\eta_{\lambda \mu} J_{\kappa \nu}+\eta_{\kappa \nu} J_{\lambda \mu}-\eta_{\lambda \nu} J_{\kappa \mu}\right)\right)$$
distribute(_);
$$\displaystyle{}0$$