How does one calculate the number of components a Dirac spinor in arbitrary dimensions? As far as I understand, the textbooks treat the four spacetime dimensions and here the spinor has four components because the gamma matrices must be 4x4 in nature to satisfy the required algebra. Now suppose I want to see what the case will be in five/six/arbitrarily higher dimensions. How should I proceed? Is the way to first identify dimensionality of gamma matrices which will satisfy the anticommutation algebra etc, and hence deduce the structure of the spinor? That will probably take a long time by trial and error. Is there any systematic approach? Second question: In four dimensions, the "minimum" dimensionality of gamma matrices required to satisfy the anticommutation relations, etc is four but that means, I can choose higher dimensional matrices also, if I can find suitably in four dimensions and then the spinor structure will be higher than four and the physics will be more complicated, I guess. But no textbook deals with such picture, and I think there must be something wrong with this argument. Can anybody help me with this issue?