As far as I remember, I heard from someone that the matrix [tex] \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3 [/tex] also known as the chirality operator in 3+1 dimension is not defined in odd dimensions. I do not understand why that should be the case. Suppose I am in the 4+1 dimension and I choose one more gamma matrix suitably to close the Clifford algebra in five dimensions and then define analogously to the 4 dimensional case, the operator [tex] \Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4 [/tex] as my chirality operator. Will that be a mistake? What about six dimensions? Can I define my chirality operator by multiplying the required no. of basic gamma matrices in that dimension?