Cadabra manual: Expand

expand_delta

Expand generalised Kronecker delta symbols

In Cadabra the KroneckerDelta property indicates a generalised Kronecker delta symbol. In order to expand it into standard two-index Kronecker deltas, use expand_delta, as in the example below.

\delta{#}::KroneckerDelta;

\(\displaystyle{}\text{Attached property KroneckerDelta to }\delta\left(\#\right).\)

ex:=\delta^{a}_{b}^{c}_{d};

\(\displaystyle{}\delta^{a}\,_{b}\,^{c}\,_{d}\)

expand_delta(_);

\(\displaystyle{}\frac{1}{2}\delta^{a}\,_{b} \delta^{c}\,_{d} - \frac{1}{2}\delta^{c}\,_{b} \delta^{a}\,_{d}\)

ex:=\delta^{a}_{m}^{l}_{n} \delta_{a}^{c}_{b}^{d};

\(\displaystyle{}\delta^{a}\,_{m}\,^{l}\,_{n} \delta_{a}\,^{c}\,_{b}\,^{d}\)

expand_delta(_); distribute(_); eliminate_kronecker(_); canonicalise(_);

\(\displaystyle{}\left(\frac{1}{2}\delta^{a}\,_{m} \delta^{l}\,_{n} - \frac{1}{2}\delta^{l}\,_{m} \delta^{a}\,_{n}\right) \left(\frac{1}{2}\delta_{a}\,^{c} \delta_{b}\,^{d} - \frac{1}{2}\delta_{b}\,^{c} \delta_{a}\,^{d}\right)\)

\(\displaystyle{}\frac{1}{4}\delta^{a}\,_{m} \delta^{l}\,_{n} \delta_{a}\,^{c} \delta_{b}\,^{d} - \frac{1}{4}\delta^{a}\,_{m} \delta^{l}\,_{n} \delta_{b}\,^{c} \delta_{a}\,^{d} - \frac{1}{4}\delta^{l}\,_{m} \delta^{a}\,_{n} \delta_{a}\,^{c} \delta_{b}\,^{d}+\frac{1}{4}\delta^{l}\,_{m} \delta^{a}\,_{n} \delta_{b}\,^{c} \delta_{a}\,^{d}\)

\(\displaystyle{}\frac{1}{4}\delta^{l}\,_{n} \delta_{m}\,^{c} \delta_{b}\,^{d} - \frac{1}{4}\delta^{l}\,_{n} \delta_{b}\,^{c} \delta_{m}\,^{d} - \frac{1}{4}\delta^{l}\,_{m} \delta_{n}\,^{c} \delta_{b}\,^{d}+\frac{1}{4}\delta^{l}\,_{m} \delta_{b}\,^{c} \delta_{n}\,^{d}\)

\(\displaystyle{}\frac{1}{4}\delta_{b}\,^{d} \delta^{c}\,_{m} \delta^{l}\,_{n} - \frac{1}{4}\delta_{b}\,^{c} \delta^{d}\,_{m} \delta^{l}\,_{n} - \frac{1}{4}\delta_{b}\,^{d} \delta^{c}\,_{n} \delta^{l}\,_{m}+\frac{1}{4}\delta_{b}\,^{c} \delta^{d}\,_{n} \delta^{l}\,_{m}\)

Note that it is in principle possible to get a result similar to the expanded form by using the Young projector and then canonicalising, but this is more expensive:

ex:=\delta^{a}_{b}^{c}_{d};

\(\displaystyle{}\delta^{a}\,_{b}\,^{c}\,_{d}\)

young_project_tensor(_);

\(\displaystyle{}\delta^{a}\,_{b}\,^{c}\,_{d}\)