# 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 $$\langle\,\Omega ,\big(c(f_1)+c(f_1)^*\big)...\big(c(f_{2k})+c(f_{2k})^*\big)\Omega\,\rangle_{\mathfrak{F}}$$ 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 $$f_i$$.

• 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... Aug 29, 2010 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? Aug 29, 2010 at 16:57
• The $f_i$ are any vectors in $H_\mathbb{C}$. I use the standard relations (q=-1): $$c(f)^{}c(e)+c(e)c(f)^{} = <f,e>Id$$ Aug 29, 2010 at 20:27

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).