# Number of Grassmann generators for Dirac field?

How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? i.e. The Dirac field is a map to $\Lambda_N$, the space of supernumbers with $N$ real Grassmann generators. What is $N$?

This is a follow up question to my previous question Grassmann Paradox Weirdness. I am following Prakash's book Mathemaical Perspectives on Theoretical Physics, where they say a supernumber $z\in\Lambda_N$ can be thought of the extension of complex numbers by the addition of $N$ Grassmann generators $\zeta^1,\, \zeta^2,\,\ldots \zeta^N$. The most general supernumber is written $$z = z_0+z_i\zeta^i+\textstyle\frac{1}{2!}z_{ij}\zeta^i\zeta^j+\ldots,$$ where $z_i$, $z_{ij}$, $\ldots$, are complex-valued and antisymmetric. The odd part of this is anticommuting and is used to describe Fermion fields. The book says that for finite $N$, it takes $2^{N-1}$ complex numbers to specify an anticommuting number.

How do I figure out how many Grassmann generators $\zeta^i$ I need to specify a Dirac spinor in 4 dimensions.

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I would say that you need uncountable infinite number of Grassmann generators. One for each spacetime point and spinor component, then double everything up for the conjugate field. Then the Dirac field is an element of the subspace of the Grassmann algebra with only one generator. So the only product of Dirac fields which is zero is $\psi_\alpha(x)\psi_\alpha(x)=0$, where both are the same spinor component and at the same spacetime point. –  Heidar Oct 22 '12 at 18:03
I don't agree with the answer. It's very likely because I don't fully understand your logic. Would you flesh it out in detail as an answer? –  QuantumDot Oct 25 '12 at 1:31