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The following article by E. R. Negrin provides the required formula for the antisymmetric Fock space in the corollary on page 3644.

I want to point out that the Wick products (for the antisymmetric Fock space) can be constructed from a Gaussian generating function which is Gaussian in (real) Grassmann variables, which is given for the case presented in the question by:

$G(\mathbf{\xi}) = exp((\Sigma_{i=0}^{2k} \xi_i f_i, \Sigma_{j=0}^{2k} \xi_j f_j))$

where $( , )$ denotes the Hilbert sapce $H_\mathbb{C}$ inner product.

The required Wick product is obtained as the coefficient of $\xi_1 \xi_2 . . .\xi_{2k}$.

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he following article by E. R. Negrin provides the required formula for the antisymmetric Fock space in the corollary on page 3644.

I want to point out that the Wick products (for the antisymmetric Fock space) can be constructed from a Gaussian generating function which is Gaussian in (real) Grassmann variables, which is given for the case presented in the question by:

$G(\mathbf{\xi}) = exp((\Sigma_{i=0}^{2k} \xi_i f_i, \Sigma_{j=0}^{2k} \xi_j f_j))$

where $( , )$ denotes the Hilbert sapce $H_\mathbb{C}$ inner product.

The required Wick product is obtained as the coefficient of $\xi_1 \xi_2 . . .\xi_{2k}$.