Table of Contents
Wick's theorem [Wic50] (aka Isserlis's theorem [Iss16; Iss18]) describes defining decomposition property of averages in Gaussian integrals.
Basic Gaussian integral (Time-independent 1-particle QM, QFT)
One-dimensional way of doing things
We are interested in finding all the moments
for the Gaussian distribution
The choice of the normalization coefficient is such that
First of all, we see that
since the integrand is odd.
To find even moments we need to introduce generating function
We can extract all the moments from it as follows
so the -th moment is the -th coefficient of Taylor expansion of multiplied by . In this sense it generate all the moments. On the other hand
We've described how to extract all the moments from this generating function, but haven't calculated what this is. Let's fill this gap. To achieve this, first thing to do is completing the square
Now, integral (5) can be explicitly evaluated
and Taylor expanded
Term-by-term comparison with (6) gives
As we expected all the odd moments are zero.
Any-dimensional way of doing things: sum over pairings
So far so good, but we haven't seen any Wick's theorem yet! For this we need to return to the expression
Let's look closer on it. What happens when we first differentiate exponent is that we just create factor in front of it
Next derivative has a choice: it can either annihilate just created factor ( remains), or create one more factor by acting on the exponent again
This or statement can be actually expressed as sign while acting on the proper vacuum. I mean that
What I've called vacuum is represented by here. The statement above is proved by induction by noting that pre-exponential factor is always polynomial in and this structure is preserved through differentiation steps. In sum of creation and annihilation operators one can recognize coordinate operator of QM harmonic oscillator. Our notation is taken from QFT and in zero dimensional case is exactly the field operator .
We haven't yet considered the last step of our procedure
It translates to the new language in the most straightforward way
This last step is the action on our conjugate vacuum which is annihilated by any state containing . We have the following commutation relations between our creation and annihilation operators
Using them we can drag all the to the right so they annihilate vacuum, and all the to the left, so they annihilate the conjugate one. Let's start
First term will vanish after action on conjugate vacuum, so
What we've actually proven is that
And the answer is already evident by recursion with known base
But we've seen more. We've seen that factor represents way to couple first with all except first producing multiplier as a result of this coupling. Next factor is due to pairing of one more with one of the rest and so on. multiplier can be represented through the propagator
so we can rewrite the answer in the form
where " " is the set of all ways to partition into unordered pairs. Here it is! Wick's theorem! It's the form of answer that can be generalized to more complex cases.
Pairings as fixed-point-free involutions of symmetric group
The nature of the set may be understood better in connection with the well-known set of all permutations . Conjugacy classes of are characterized by cycle types of these permutations, where is a Young diagram. In the conjugacy class we have only permutations that partition set of into pairs and swap entities in these pairs. Such permutaions themselves look like
where and all and are distinct and comprise the set (note that in permutations notation we can also write and so on). So we see that every pairing of elements has a permutation as a counterpart and vice versa. So we can rewrite our sum over pairings as a sum over symmetric group conjugacy class
One can refer to this sum as a sum over fixed-point-free involutions. The word involutions means
And fixed-point-free permutation is such that for any
These two characteristics unambiguously define our conjugacy class.
Pairings as canonical representatives in symmetric group
There is also one other way to write this sum. To pair elements one can start with the set and act with permutation on it obtaining
Next, the resulting set is split into pairs of neighbours
Nice! We've got some pairing. But if one wants to rewrite the sum above hove through sum over pairings obtained by this method, they first should deal with overcounting. We would get the same pairing if we switch ordering IN pairs, this amounts to factor. Moreover, the pairing would be the same if we switch ordering OF pairs, this gives us factor. As a result we have
The overcounting can be accounted by imposing a canonical ordering on obtained pairings. From all the permutations describing one pairing we can choose one in which elements in pairs are sorted in ascending order
and pairs are sorted by their first element
By summing only over such permutations we won't get any overcounting
Implicitly by that we've established a one-to-one correspondence between permutations , such that and for any , and pairings , which can also be thought as fixed-point-free involutions of permutation group .
Heisenberg algebraq of Gaussian integral
Now, when everything is rigorously enough splitted into pairs, we can rewrite the discussed quantities in maybe more familiar terms. Creation and annihilation operators are correspondingly
They have a commutation relations of Heisenberg algebra
Field (or coordinate) operator is
Vacuums are
And vacuum expectation values (VEVs) are then written as
Multivariate Gaussian integral (Time-independent -particle QM, QFT)
In a bit more sophisticated case of multivariate Gaussian distribution
generating function evaluates to
And moments are then splitted into pairs
with all odd moments vanishing. Comparing to the discussed univariate Gaussian integral case, pair indices as well as product limits have been specified. We need to take product over all -cycles in the permutation. One can name these cycles by their lowest element and take product over these names. This is exactly what we have done. Every individual transposition is of form , thus the indices.
Wick's theorem can also be written in a form
where, again, the new things are product limits and pair indices in accordance with the logic described in the previous section.
Infinite-dimensional Gaussian integral (QFT)
Here we have quadratic action
and source . The distribution is
and generating function reads as
2-point correlator:
Wick's theorem for -point correlator:
and again odd correlators vanish.
Gaussian Hermitian matrix model
In this case we have
It's now just the product of univariate Gaussian integrals. Generating function of matrix elements correlators is
Then the propagator is
and Wick's theorem writes as
Rectangular complex matrix model
The distribution for this model is
Let's rewrite it a bit more explicitly
so in essence this integral in nothing else than univariate Gaussian ones.
Generating function:
Propagators:
For the Wick's theorem combinatorics is a bit different. In the -point correlator
we have to pair all to all (all the other pairings vanish, see (58)). Each pairing is a bijection from onto itself, and we have to sum over all such bijections. Read this sentence again and recognize the sum over all permutations of elements. So we have