Study on Fermions in d Dimensions

Introducing d dimensional space, with NxN Lorentz invariant gamma matrices, the connection between N and d is N=2d/2 and since N is an integer, d must be an even number. Thus, the only candidates for a higher dimension universe for fermions may be SO(8) , SO(16) etc. SO(7), SO(11) etc. are ruled out in case of fermions. Moreover, it is shown that d > 4 is impossible since already for d = 6 , the representation is reducible. Thus, our universe, at least as far as Fermions goes, must be a (3+1) 4-dimensional.