Let $H$ be a real Hilbert space with complexification $H_{\mathbb{C}}$. We denote by $\mathfrak{F}$ the antisymmetric Fock space over $H_\mathbb{C}$ ("fermions"). A creation operator is denoted by $c(f)$. I need a reference for the calculus of $$ \langle\,\Omega ,\big(c(f_1)+c(f_1)^*\big)...\big(c(f_{2k})+c(f_{2k})^*\big)\Omega\,\rangle_{\mathfrak{F}} $$ where $\Omega$ the unit vector, called vacuum.
Thank you.
The original reference for Wick's theorem is, not surprisingly, Wick's original 1950 paper: The Evaluation of the Collision Matrix published in the Physical Review 80 (2) pp. 268-272. He also shows how to compute it and it is surprisingly readable 60 years on.
Of course, depending on your background, this may be too physical. A more mathematical reference are the Bombay Lectures by Kac and Raina Highest-weight representations of infinite-dimensional Lie algebras, particularly the 5th lecture on the Bose-Fermi correspondence.
The basic idea is to think of $\mathfrak{F}$ as the space of semi-infinite forms. The vacuum vector would be given by $$\Omega = f_1^* \wedge f_2^* \wedge \cdots$$ and $c(f_i)^*$ acts by wedging with $f_i^*$ whereas $c(f_i)$ acts by contracting with $f_i$.
The article Wick products of the CAR algebra 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 $\xi_1 \xi_2 . . .\xi_{2k}$.
One should be able to obtain the formula from the appendix of:
They have a formula for all vectors, to the vacuum expectation just the summand with $2p=n$ contributes. They using Arakis self dual CAR algebra, and if you consider $a(f)$ for $f=\Gamma f$ it should equal your $c(f)+c(f)^\ast$.
The answer is provided by the article
However, the authors work in the context of $q$-Fock space. I does not know if there exists an older paper which provides the answer in the less general context of antisymmetric Fock space (i.e. q=1).
