Some useful identities

In this Section we collect some identities that we shall often use in the calculations. They concern mainly the antisymmetric tensor $\epsilon^{ijkl}$, the differential $1$-forms $\omega^i$ and the Dirac matrices $\gamma^i$.

A first set of identities is

$$\epsilon^{ijkl} \epsilon_{ijkl} = -24, \qquad \epsilon^{mijk} \epsil... ...n^{ijmn} \epsilon_{klmn} = - 2 (\delta^i_k \delta^j_l - \delta^j_k \delta^i_l),$$

$$\epsilon^{ijkl} \epsilon_{imnp} = - \delta^j_m \delta^k_n \delta^l_p... ...elta^l_p + \delta^l_m \delta^k_n \delta^j_p + \delta^j_m \delta^l_n \delta^k_p.$$ (1)

Since an expression completely antisymmetric with respect to 5 indices that can take only 4 values must vanish, we have the useful identity

$$\epsilon_{ijkl} x_m - \epsilon_{mjkl} x_i - \epsilon_{imkl} x_j - \epsilon_{ijml} x_k - \epsilon_{ijkm} x_l = 0.$$ (2)

The differential forms

$$\eta = \omega^0 \wedge \omega^1 \wedge \omega^2 \wedge \omeg... ...jkl} \omega^i \wedge \omega^j \wedge \omega^k \wedge \omega^l,$$ (3)

$$\eta_i = 6^{-1} \epsilon_{ijkl} \omega^j \wedge \omega^k \wedge \omega^l = i(A_i) \eta$$ (4)

appear in many formulas. They have the properties

$$\omega^i \wedge \omega^j \wedge \omega^k \wedge \omega^l = -... ...ijkl} \eta_i, \qquad \omega^k \wedge \eta_i = \delta^k_i \eta.$$ (5)

The Dirac matrices, characterized by the equation

$$\gamma^i \gamma^k + \gamma^k\gamma^i = 2 g^{ik},$$ (6)

have the properties

$$\mathrm{Tr}\,(\gamma^i \gamma^k) = 4 g^{ik}, \qquad \mathrm{... ...gamma^l) = 4 (g^{ij} g^{kl} - g^{ik} g^{jl} + g^{il} g^{jk}),$$ (7)

$$\gamma_5 = - \gamma^5 = \gamma_0 \gamma_1 \gamma_2 \gamma_3 ... ...\gamma^i \gamma^j \gamma^k \gamma^l, \qquad (\gamma_5)^2 = -1,$$ (8)

$$\mathrm{Tr}\,(\gamma_5 \gamma^i \gamma^j \gamma^k \gamma^l) = -4 \epsilon^{ijkl},$$ (9)

$$(\gamma^i \gamma^k - \gamma^k \gamma^i) \gamma_5 = \epsilon^{ik}{}_{jl} \gamma^j \gamma^l.$$ (10)

$$\gamma_i \gamma_j \gamma_k = \epsilon_{lijk} \gamma^l \gamma_5.$$ (11)