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I am importing this question from https://www.physicsoverflow.org/39342/equivalence-between-grassmann-fermion-mathbb-fermion-coherent

Cahill and Glauber in the paper 'Density operators for Fermions' construct Fermionic coherent state as the displaced vacuum state

$\vert\boldsymbol{\gamma}\rangle=D\left(\boldsymbol{\gamma}\right)\vert 0\rangle=\exp\left(\sum_{i}c_{i}^{\dagger}\gamma_{i}-\gamma_{i}^{\star}c_{i}\right)\vert 0\rangle$

where $\boldsymbol{\gamma}$ is the set of the Grassmann variables $\gamma_i$, one for each fermionic mode $c_i$.

Perelomov in the book 'Generalized Coherent States and Their Applications' constructs the Fermionic coherent state in a different manner. He first shows that it is the group $SO(2N,\mathbb{R})$ that preserves the canonical anti-commutation relations obeyed by the fermionic modes $c_i$ and that the representation $T(g)$ of an arbitrary element of this group $g\in SO(2N,\mathbb{R})$ is

$T\left(g\right)={\cal N}\exp\left(-\frac{1}{2}\xi_{ij}X^{ij}\right)\exp\left(\alpha_{k}^{l}X_{l}^{k}\right)\exp\left(-\frac{1}{2}\eta^{ij}X_{ij}\right)$

where $X_{ij}=a_{i}a_{j}$, $X^{ij}=a_{j}^{\dagger}a_{i}^{\dagger}$ and $X_{l}^{k}=\frac{1}{2}\left(a_{k}^{\dagger}a_{l}+a_{l}^{\dagger}a_{k}\right)$. Coherent states are constructed by acting an element $g$ of this group on a vector $\vert \phi_0 \rangle$ that is annihilated by $X_{ij}$. For the Fock subspace of states with even number of fermionic modes occupied, this vector is just $\vert 0 \rangle = \vert 0,0,....0 \rangle$. And we get the coherent state to be

$\vert\xi\rangle={\cal N}\exp\left(-\frac{1}{2}\xi_{ij}X^{ij}\right)\vert 0 \rangle$.

In the book 'Coherent States and Applications in Mathematical Physics' by Monique Combescure and Didier Robert, it is claimed that the two points of view are equivalent but I am not sure where exactly they prove the equivalence. They also claim, perhaps in support of the same point, that the model of spin states (which, I believe, is used by Perelomov to construct the coherent states because it is analogous to the construction of spin coherent states) is unitarily equivalent to the fermionic Fock model using which they construct the Grassmann coherent states. Does this equivalence mean that there is a mapping between the Grassmann coherent states and the $SO(2N,\mathbb{R})$ coherent states? I can see that in the former, the displacement operation $\exp\left(\sum_{i}c_{i}^{\dagger}\gamma_{i}-\gamma_{i}^{\star}c_{i}\right)\vert 0\rangle$ has an exponent which is linear in the fermionic modes while in the latter, the operator ${\cal N}\exp\left(-\frac{1}{2}\xi_{ij}X^{ij}\right)$ is quadratic in the fermionic modes. Mathematically, Grassmann variables are 'fermions', so even the former is quadratic in 'fermionic modes'. Is that the mapping between the two sets of coherent states? Or are the two sets of coherent states unitarily related? How would that be possible- since the former is a displaced vacuum state while the latter is constructed by the analog of squeezing and beam splitters in fermionic phase space?

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