Emil
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I'm trying to explicitly show that
[tex]\varepsilon^{0 i j k} \varepsilon_{0 i j l} = - 2 \delta^k_l[/tex]
I sort of went off the deep end and tried to express everything instead of using snazzy tricks and ended up with
[tex] \begin{eqnarray*}<br /> \delta^{\mu \rho}_{\nu \sigma} & = & \delta^{\mu}_{\nu}<br /> \delta^{\rho}_{\sigma} - \delta^{\mu}_{\sigma} \delta^{\rho}_{\nu}\\<br /> & & \\<br /> \delta^{\mu \rho_1 \rho_2}_{\nu \sigma_1 \sigma_2} & = & \delta^{\mu}_{\nu}<br /> \delta^{\rho_1 \rho_2}_{\sigma_1 \sigma_2} - \delta^{\mu}_{\sigma_1}<br /> \delta^{\rho_1 \rho_2}_{\nu \sigma_2} + \delta^{\mu}_{\sigma_1}<br /> \delta^{\rho_1 \rho_2}_{\sigma_2 \nu}\\<br /> & & \\<br /> \delta^{\mu \rho_1 \rho_2 \rho_3}_{\nu \sigma_1 \sigma_2 \sigma_3} & = &<br /> \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_1 \sigma_2<br /> \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\nu<br /> \sigma_2 \sigma_3} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2<br /> \rho_3}_{\sigma_2 \nu \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1<br /> \rho_2 \rho_3}_{\sigma_2 \sigma_3 \nu}\\<br /> & & \\<br /> \varepsilon^{0 i j k} \varepsilon_{0 i j l} = \delta^{0 i j k}_{0 i j l} & =<br /> & \delta^0_0 \delta^{i j k}_{i j l} - \delta^0_i \delta^{i j k}_{0 j l} +<br /> \delta^0_i \delta^{i j k}_{j 0 l} - \delta^0_i \delta^{i j k}_{j l 0}\\<br /> & & \\<br /> & = & \delta^0_0 \left( \delta^i_i \delta^{j k}_{j l} - \delta^i_j<br /> \delta^{j k}_{i l} + \delta^i_i \delta^{j k}_{l j} \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_0 \delta^{j k}_{j l} - \delta^i_j<br /> \delta^{j k}_{0 l} + \delta^0_j \delta^{j k}_{l 0} \right) \ldots\\<br /> & & + \delta^0_i \left( \delta^i_j \delta^{j k}_{0 l} - \delta^i_0<br /> \delta^{j k}_{j l} + \delta^0_0 \delta^{j k}_{l j} \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_j \delta^{j k}_{l 0} - \delta^i_l<br /> \delta^{j k}_{j 0} + \delta^0_l \delta^{j k}_{0 j} \right)\\<br /> & & \\<br /> & = & \delta^0_0 \left( \delta^i_i \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) - \delta^i_j \left( \delta^j_i \delta^k_l -<br /> \delta^j_l \delta^k_i \right) + \delta^i_i \left( \delta^j_l \delta^k_j -<br /> \delta^j_j \delta^k_l \right) \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_0 \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) - \delta^i_j \left( \delta^j_0 \delta^k_l -<br /> \delta^j_l \delta^k_0 \right) + \delta^0_j \left( \delta^j_l \delta^k_0 -<br /> \delta^j_0 \delta^k_l \right) \right) \ldots\\<br /> & & + \delta^0_i \left( \delta^i_j \left( \delta^j_0 \delta^k_l -<br /> \delta^j_l \delta^k_0 \right) - \delta^i_0 \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) + \delta^0_0 \left( \delta^j_l \delta^k_j -<br /> \delta^j_j \delta^k_l \right) \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_j \left( \delta^j_l \delta^k_0 -<br /> \delta^j_0 \delta^k_l \right) - \delta^i_l \left( \delta^j_j \delta^k_0 -<br /> \delta^j_0 \delta^k_j \right) + \delta^0_l \left( \delta^j_0 \delta^k_j -<br /> \delta^j_j \delta^k_0 \right) \right)\\<br /> & & \\<br /> & & 0 = i = j\\<br /> & & \\<br /> & = & \delta^0_0 \delta^i_i \delta^j_j \delta^k_l - \delta^0_0 \delta^i_j<br /> \delta^j_i \delta^k_l - \delta^0_0 \delta^i_i \delta^j_j \delta^k_l \ldots\\<br /> & & - \delta^0_i \delta^i_0 \delta^j_j \delta^k_l + \delta^0_i \delta^i_j<br /> \delta^j_0 \delta^k_l + \delta^0_i \delta^0_j \delta^j_0 \delta^k_l \ldots\\<br /> & & + \delta^0_i \delta^i_j \delta^j_0 \delta^k_l - \delta^0_i \delta^i_0<br /> \delta^j_j \delta^k_l - \delta^0_0 \delta^j_j \delta^k_l \ldots\\<br /> & & + \delta^0_i \delta^i_j \delta^j_0 \delta^k_l\\<br /> & & \\<br /> & = & \delta^k_l - \delta^k_l\\<br /> \end{eqnarray*} [/tex]
The bottom line is that all I want for christmas is to get [tex]- 2 \delta^k_l[/tex] from
[tex]\varepsilon^{0 i j k} \varepsilon_{0 i j l} = \delta^{0 i j k}_{0 i j l} =<br /> \left|\begin{array}{cccc}<br /> \delta^0_0 & \delta^0_i & \delta^0_j & \delta^0_l\\<br /> \delta^i_0 & \delta^i_i & \delta^i_j & \delta^i_l\\<br /> \delta^j_0 & \delta^j_i & \delta^j_j & \delta^j_l\\<br /> \delta^k_0 & \delta^k_i & \delta^k_j & \delta^k_l<br /> \end{array}\right| =[/tex]
in a way that doesn't involve 100000 kronecker deltas. THAAAAANKS
[tex]\varepsilon^{0 i j k} \varepsilon_{0 i j l} = - 2 \delta^k_l[/tex]
I sort of went off the deep end and tried to express everything instead of using snazzy tricks and ended up with
[tex] \begin{eqnarray*}<br /> \delta^{\mu \rho}_{\nu \sigma} & = & \delta^{\mu}_{\nu}<br /> \delta^{\rho}_{\sigma} - \delta^{\mu}_{\sigma} \delta^{\rho}_{\nu}\\<br /> & & \\<br /> \delta^{\mu \rho_1 \rho_2}_{\nu \sigma_1 \sigma_2} & = & \delta^{\mu}_{\nu}<br /> \delta^{\rho_1 \rho_2}_{\sigma_1 \sigma_2} - \delta^{\mu}_{\sigma_1}<br /> \delta^{\rho_1 \rho_2}_{\nu \sigma_2} + \delta^{\mu}_{\sigma_1}<br /> \delta^{\rho_1 \rho_2}_{\sigma_2 \nu}\\<br /> & & \\<br /> \delta^{\mu \rho_1 \rho_2 \rho_3}_{\nu \sigma_1 \sigma_2 \sigma_3} & = &<br /> \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_1 \sigma_2<br /> \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\nu<br /> \sigma_2 \sigma_3} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2<br /> \rho_3}_{\sigma_2 \nu \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1<br /> \rho_2 \rho_3}_{\sigma_2 \sigma_3 \nu}\\<br /> & & \\<br /> \varepsilon^{0 i j k} \varepsilon_{0 i j l} = \delta^{0 i j k}_{0 i j l} & =<br /> & \delta^0_0 \delta^{i j k}_{i j l} - \delta^0_i \delta^{i j k}_{0 j l} +<br /> \delta^0_i \delta^{i j k}_{j 0 l} - \delta^0_i \delta^{i j k}_{j l 0}\\<br /> & & \\<br /> & = & \delta^0_0 \left( \delta^i_i \delta^{j k}_{j l} - \delta^i_j<br /> \delta^{j k}_{i l} + \delta^i_i \delta^{j k}_{l j} \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_0 \delta^{j k}_{j l} - \delta^i_j<br /> \delta^{j k}_{0 l} + \delta^0_j \delta^{j k}_{l 0} \right) \ldots\\<br /> & & + \delta^0_i \left( \delta^i_j \delta^{j k}_{0 l} - \delta^i_0<br /> \delta^{j k}_{j l} + \delta^0_0 \delta^{j k}_{l j} \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_j \delta^{j k}_{l 0} - \delta^i_l<br /> \delta^{j k}_{j 0} + \delta^0_l \delta^{j k}_{0 j} \right)\\<br /> & & \\<br /> & = & \delta^0_0 \left( \delta^i_i \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) - \delta^i_j \left( \delta^j_i \delta^k_l -<br /> \delta^j_l \delta^k_i \right) + \delta^i_i \left( \delta^j_l \delta^k_j -<br /> \delta^j_j \delta^k_l \right) \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_0 \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) - \delta^i_j \left( \delta^j_0 \delta^k_l -<br /> \delta^j_l \delta^k_0 \right) + \delta^0_j \left( \delta^j_l \delta^k_0 -<br /> \delta^j_0 \delta^k_l \right) \right) \ldots\\<br /> & & + \delta^0_i \left( \delta^i_j \left( \delta^j_0 \delta^k_l -<br /> \delta^j_l \delta^k_0 \right) - \delta^i_0 \left( \delta^j_j \delta^k_l -<br /> \delta^j_l \delta^k_j \right) + \delta^0_0 \left( \delta^j_l \delta^k_j -<br /> \delta^j_j \delta^k_l \right) \right) \ldots\\<br /> & & - \delta^0_i \left( \delta^i_j \left( \delta^j_l \delta^k_0 -<br /> \delta^j_0 \delta^k_l \right) - \delta^i_l \left( \delta^j_j \delta^k_0 -<br /> \delta^j_0 \delta^k_j \right) + \delta^0_l \left( \delta^j_0 \delta^k_j -<br /> \delta^j_j \delta^k_0 \right) \right)\\<br /> & & \\<br /> & & 0 = i = j\\<br /> & & \\<br /> & = & \delta^0_0 \delta^i_i \delta^j_j \delta^k_l - \delta^0_0 \delta^i_j<br /> \delta^j_i \delta^k_l - \delta^0_0 \delta^i_i \delta^j_j \delta^k_l \ldots\\<br /> & & - \delta^0_i \delta^i_0 \delta^j_j \delta^k_l + \delta^0_i \delta^i_j<br /> \delta^j_0 \delta^k_l + \delta^0_i \delta^0_j \delta^j_0 \delta^k_l \ldots\\<br /> & & + \delta^0_i \delta^i_j \delta^j_0 \delta^k_l - \delta^0_i \delta^i_0<br /> \delta^j_j \delta^k_l - \delta^0_0 \delta^j_j \delta^k_l \ldots\\<br /> & & + \delta^0_i \delta^i_j \delta^j_0 \delta^k_l\\<br /> & & \\<br /> & = & \delta^k_l - \delta^k_l\\<br /> \end{eqnarray*} [/tex]
The bottom line is that all I want for christmas is to get [tex]- 2 \delta^k_l[/tex] from
[tex]\varepsilon^{0 i j k} \varepsilon_{0 i j l} = \delta^{0 i j k}_{0 i j l} =<br /> \left|\begin{array}{cccc}<br /> \delta^0_0 & \delta^0_i & \delta^0_j & \delta^0_l\\<br /> \delta^i_0 & \delta^i_i & \delta^i_j & \delta^i_l\\<br /> \delta^j_0 & \delta^j_i & \delta^j_j & \delta^j_l\\<br /> \delta^k_0 & \delta^k_i & \delta^k_j & \delta^k_l<br /> \end{array}\right| =[/tex]
in a way that doesn't involve 100000 kronecker deltas. THAAAAANKS