Products of gamma matrices in n dimensions

[Michi123]

Hello,

i have here some identities for gamma matrices in n dimensions to prove and don't know how to do this. My problem is that I am not very familiar with the ⊗ in the equations. I think it should be the Kronecker-product. If someone could give me a explanation of how to work with this stuff it would be great.
here the exercise:
Let γ⁽ⁿ⁾_{μ1... μn} be the totally antisymmetric products of n γ-matrices and γ⁽ⁿ⁾⊗γ⁽ⁿ⁾ = γ⁽ⁿ⁾_{μ1... μn} ⊗ γ⁽ⁿ⁾_{μ1... μn}.

it should hold that:
1.)γ^μ γ^ν γ⁽²⁾⊗γ^μ γ^ν γ⁽²⁾ = γ⁽⁴⁾⊗γ⁽⁴⁾ +2(5μ -4) γ⁽²⁾⊗γ⁽²⁾ +4μ(2μ-1) id⊗id
2.)γ^ργ^μγ^σγ^ν⊗γ^ργ^νγ^σγ^μ= -γ⁽⁴⁾⊗γ⁽⁴⁾ +4γ⁽²⁾⊗γ⁽⁴⁾ + 4μ(3μ-1) id⊗id

id is the n dimensional identity matrix and μ =d/2 where d is the dimension
for the gamma matrices in n dimensions also holds the basic anticommutation relation and η_μνη^μν = d

greetz mk

 

[Michi123]

Ok, I think the antisymmetric product of the gamma-matrices is defined by:
$\gamma^{(n)} =\gamma^{[\mu_1\mu_2\ldots\mu_n]}= \frac{1}{n!}\epsilon_{\mu_1 \mu_2 \ldots \mu_n}\gamma^{\mu_1}\gamma^{\mu_2}\ldots\gamma^{\mu_n}$
It would be good to show that $\gamma^\nu\cdot\gamma^{[\mu_1\mu_2\ldots\mu_n]} = \gamma^{[\nu\mu_1\mu_2\ldots\mu_n]} + \sum\limits_{i=1} ^n (-1)^{i+1} g^{\nu\mu_i}\gamma^{[\mu_1\mu_2\ldots\hat\mu_i\ldots\mu_n]}.$The $\hat\mu_i$ means that this indice is deleted from the product because $\nu$ and $\mu_i$ were equal. So if $\nu$ is different than all other indices, I'm left with a n+1 matrices product. If $\nu$ matches with one indice, I'm left with a n-1 matrices product. I am not pretty sure how to do this. For the cases n=2 or n=3 one can do this simply and just form an antisymmetric product to see how this works. But I can't do this for the general case, i.e. for arbitrary n. Perhaps one can do this by induction or just by using some combinatorial stuff?! If one has this identity, the tensor-product identities in the first post should follow by using this relation.