MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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 $f_i$.

share|cite|improve this answer
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 $\xi_1 \xi_2 . . .\xi_{2k}$.

share|cite|improve this answer

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

share|cite|improve this answer
up vote 1 down vote accepted

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.