# reference for wick product

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 $$<\Omega ,\big(c(f_1)+c(f_1)^*\big)...\big(c(f_{2k})+c(f_{2k})^*\big)\Omega>_{\mathfrak{F}}$$ where $\Omega$ the unit vector, called vacuum.

Thank you.

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## 4 Answers

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 $f_i$.

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Thank you very much for your answer. In fact, I only need the result of the calculation (and locate the reference). Indeed, I believe that the calculus of this quantity is classical. My background is a very basic knowledge of $q$-fock space ($-1\leq q \leq 1$). The first reference is too physical. The context of the second is unfortunately far from my knowledge... –  BigBill Aug 29 '10 at 16:36
It would perhaps help to know precisely what you are asking. What are the $f_i$? Are they linearly independent, or simply any vectors in $H_{\mathbb{C}}$? And what canonical anticommutation relations are you using? The $q$-deformed ones or the standard ones? –  José Figueroa-O'Farrill Aug 29 '10 at 16:57
The $f_i$ are any vectors in $H_\mathbb{C}$. I use the standard relations (q=-1): $$c(f)^{}c(e)+c(e)c(f)^{} = <f,e>Id$$ –  BigBill Aug 29 '10 at 20:27

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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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 $c(f)+c(f)^\ast$.

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The answer is provided by the article

http://www.pnas.org/content/100/15/8629.full

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).

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