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\cdbalgorithm{dualise\_tensor}{}
Dualises tensors which have been declared \subsprop{SelfDual}
according to the formula
\begin{equation}
F_{\mu_{n+1}\cdots \mu_{d}}\;\; \rightarrow\;\; *F_{\mu_{n+1}\cdots \mu_{d}} = \frac{1}{n!}
\epsilon_{\mu_{n+1}\cdots\mu_d}{}^{\mu_1\cdots\mu_n} F_{\mu_1\cdots
\mu_n}\, .
\end{equation}
In order for this to work the indices on the tensor have to be
declared with \subsprop{Indices} and their range should have been
specified with \subsprop{Integer}.
\begin{screen}{1,2,3,4,5}
{m,n,p,q,r#}::Indices.
{m,n,p,q,r#}::Integer(0..5).
F_{m n p}::SelfDual.
A_{q} F_{m n p} F_{m n p};
@dualise_tensor!(%);
1/36 A_{q} \epsilon_{m n p r1 r2 r3} F_{r1 r2 r3}
\epsilon_{m n p r4 r5 r6} F_{r4 r5 r6};
\end{screen}
As indices are by default position-free, all indices on the epsilon
tensors are generated as lower indices in the example above. Here is a
modification which takes index positions into account:
\begin{screen}{1,2,3,4,5}
{m,n,p,q,r#}::Indices(position=fixed).
{m,n,p,q,r#}::Integer(0..5).
F_{m n p}::SelfDual.
F^{m n p}::SelfDual.
A_{q} F_{m n p} F^{m n p};
@dualise_tensor!(%);
1/36 A_{q} \epsilon_{m n p}^{r1 r2 r3} F_{r1 r2 r3}
\epsilon^{m n p r4 r5 r6} F_{r4 r5 r6};
\end{screen}
Products of epsilon tensors can be expanded in terms of Kronecker
delta symbols with \subscommand{epsprod2gendelta}.
\cdbseeprop{SelfDual}
\cdbseeprop{Indices}
\cdbseeprop{Integer}
\cdbseealgo{epsprod2gendelta}
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