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Grassmann-odd variables provide a classical description of Grassmann-odd quantum operators in the same way that Grassmann-even variables provide a classical description of Grassmann-even quantum operators. The classical super-Poisson bracket $$\{\psi^i,\psi^j\}_{PB} ~=~ -i (T^{-1})^{ij} \tag{A}$$ is related to the super-commutator$^1$ $$\hat{\psi}^i\hat{\... 3 Yes. OP is right. There is a minus. Since by convention the complex conjugation obeys$$ (z w)^{\ast} ~=~ w^{\ast}z^{\ast}~=~(-1)^{|z|~|w|} z^{\ast}w^{\ast} \tag{1}$$for any two supernumbers z, w (of definite Grassmann parities |z|,|w|), we should also have$$ (A f)^{\ast} ~=~(-1)^{|A| ~|f|} A^{\ast}f^{\ast} \tag{2} for an operator $A$ and ...
Normally the notation $(n_b|n_f)$ denotes the dimension of a super vector space of Grassmann-even dimension $n_b$ and Grassmann-odd dimension $n_f$. When writing a super vector as a column vector, it is standard to order the Grassmann-even sector before the Grassmann-odd sector. However, the authors introduce a non-standard ordering $(n_{b_1}|n_f|n_{b_2})$ ...