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The formula that Ref. 1 uses is $$\tag{*} \exp\left(-\sum_j \eta_j a_j^{\dagger} \right) ~=~ \prod_j\exp\left( - \eta_j a_j^{\dagger} \right) ~=~\prod_j \left(1- \eta_j a_j^{\dagger}\right). $$ Ref. 1 correctly applies [the Hermitian conjugate of] eq. (*) to the bra in answer (a) on p. 181. There is no mistake on p. 181. Ref. 1 does not write a ...


For a univariable grassmann number $\eta_1$ it holds $\eta_1^2 = 0$ because of $ \{ \eta_1, \eta_1 \} = 0$. Hence all higher powers of this Grassmann number vanish. However, if there are multiple Grassmann numbers, linear combinations of these Grassmann numbers can be computed. It still holds $\eta_i^2 = 0$ for every $i$, but $\eta_i \eta_j \neq 0$ for $i ...

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