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Appendix B
In calculations with the Levi-Civita symbol (tensor), it is often
convenient to express the result in terms of a so-called generalized
Kronecker delta defined by the determinant:
\[\delta
_{{b_1}{b_2}....{b_p}}^{{a_1}{a_2}...{a_p}}: = p!\delta
_{[{b_1}}^{{a_1}}...\delta _{{b_p}]}^{{a_p}} = \left|
{\begin{array}{*{20}{c}} {\delta
_{{b_1}}^{{a_1}}}&{..}&{\delta _{{b_p}}^{{a_1}}} \\
{..}&{}&{..} \\ {\delta
_{{b_1}}^{{a_p}}}&{..}&{\delta _{{b_p}}^{{a_p}}}
\end{array}} \right|\]
as a completely anti-symmetric unity tensor. [Wikipedia:
Kronecker delta]
For example, when $p = n$ (the dimension of the vector space), one may derive the compact
formula
\[{\varepsilon
^{{a_1}...{a_n}}}{\varepsilon _{{b_1}...{b_n}}} = (-1)^s \delta
_{{b_1}....{b_n}}^{{a_1}...{a_n}}\]
More generally
\[\frac{1}{{(n - p)!}}
{\varepsilon ^{{a_1}...{a_p}{c_{p + 1}}...{c_n}}} {\varepsilon
_{{b_1}...{b_p}{c_{p + 1}}...{c_n}}} = (-1)^s \delta
^{{a_1}...{a_p}}_{{b_1}...{b_p}}\]
The formulae for $p=2,3$, $n=4$
\[\frac{1}{{2!}}{\varepsilon
_{abcd}} {\varepsilon ^{abmn}} = (-1)^s \delta
_{cd}^{mn}{\qquad}\frac{1}{{1!}} {\varepsilon _{abcd}}{\varepsilon
^{akmn}} = (-1)^s \delta _{bcd}^{kmn}\]
prove to be particularly useful.