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The definition (pay attention to not confuse generic tensor with tensor components) of the antisymmetric gamma tensor is: $$\gamma^{\mu_1 \mu_2 \dots \mu_r}=\gamma^{[\mu_1 \mu_2 \dots \mu_r]}$$ For the highest rank you have $r=D$, so you have to use all the possible indices. For example, in components, you have the identity: $$\gamma^{1 2 3}=\frac{1}{3!} ...


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Maybe I am missing something, but in your eq. (2) there are two equal terms on the right-hand side, and the factor is 1/2 to take that into account. In your eq.(1) there are also two equal terms on the right-hand side, and the factor is 1/2 to take that into account. There is no summation over all permutations in your formulas.



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