# Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$(\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger$$ and why not $\phi^{\dagger} \psi^\dagger$. I want to know the motivation behind this. In the case of multidimensional vectors, it makes sense, but why in the case of a Grassmann Variable ?

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Maybe this is relevant:en.wikipedia.org/wiki/Grassmann_number#Matrix_representations –  Andrew Sep 29 '12 at 18:29

If $\eta$ is a complex Grassman variable then we require $\eta^*\eta=x$ to be a real (non-Grassmanian) variable. In particular it means that $$(\eta^*\eta)^*=x^*\overset{!}{=}x=\eta^*\eta$$ Write $\eta$ in terms of two real Grassman variables $\eta=a+ib$, then $$\eta^*\eta=(a-ib)(a+ib)=iab-iba$$ and $$(\eta^*\eta)^*=(iab-iba)^*=-i(ab)^*+i(ba)^*\overset{!}{=}iab-iba$$ It follows immedeately that for two real Grassman variables $a$ and $b$ we must have $$(ab)^* = ba$$ It also follows that for two complex Grassman variables $\eta=a+ib$ and $\xi=c+id$ we must have $$(\eta\xi)^*=[(a+ib)(c+id)]^*=(ac-bd+ibc+iad)^*=ca-db-icb-ida=(c-id)(a-ib)=\xi^*\eta^*$$