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I am reading a review about supersymmetry and in page 29 I have read that the mass dimension of the Grassmann anticommuting coordinates is $-1/2$. Why this? why don't they have the same mass dimensions as the bosonic coordinates?

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Suppose $\partial_\alpha$ is a derivative w.r.t. Grassmannian coordinate $\theta^\alpha$. We often define a super-derivative $$ {\cal D}_\alpha = \partial_\alpha + \cdots $$ where the $\cdots$ is chosen such that $$ \{ {\cal D}_\alpha , {\cal D}_\beta \} \sim \gamma^\mu_{\alpha\beta} \partial_\mu $$ Now, let's do some dimensional analysis, since $[x^\mu]=-1$, $[\partial_\mu]=+1$. Further, since the metric is dimensionless $[\gamma^\mu] = 0$ (from $\{ \gamma^\mu , \gamma^\nu \} = 2 g^{\mu\nu}$. Thus, $[{\cal D}_\alpha] = [\partial_\alpha] = +1/2$ and therefore $[\theta^\alpha]=-1/2$.

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