View source for Wick product

''Wick monomial, Wick power''

The Wick products of random variables arise through an orthogonalization procedure.

Let $f_1,\ldots,f_n$ be (real-valued) random variables on some probability space $(\Omega,\mathcal{B},\mu)$. The Wick product
$$
:f_1^{k_1}\cdots f_n^{k_n}:
$$
is defined recursively as a polynomial in $f_1,\ldots,f_n$ of total degree $k_1+\cdots+k_n$ satisfying
$$
\left\langle { :f_1^{k_1}\cdots f_n^{k_n}: } \right\rangle = 0
$$
and for $k_i \ge 1$,
$$
\frac{\partial}{\partial f_i} \left( { :f_1^{k_1}\cdots f_n^{k_n}: } \right) = k_i  :f_1^{k_1}\cdots f_i^{k_i-1} \cdots f_n^{k_n}: 
$$
where $\langle {\cdot} \rangle$ denotes expectation. The $:\,:$ notation is traditional in physics.

For example,
$$
:f: = f - \langle f \rangle \ ,
$$
$$
:f^2: = f^2 - 2\langle f \rangle f  - \langle f^2 \rangle + 2\langle f \rangle^2 \ .
$$

There is a binomial theorem:
$$
:(af+bg)^n: = \sum_{m=0}^n \binom{n}{m} a^m b^{n-m} :f^m: :g^{n-m}:
$$
and a corresponding multinomial theorem. The Wick exponential is defined as
$$
:\exp(a f): = \sum_{m=0}^\infty \frac{a^m}{m!} :f^m:
$$
so that
$$
:\exp(af): = \langle \exp(af) \rangle^{-1} \exp(af) \ .
$$

The Wick products, powers and exponentials depend both on the variables involved and on the underlying measure.

Let $f,g$ be Gaussian random variables with mean zero. Then
$$
:\exp(af): = \exp\left({ af - \frac12 a^2 \langle f^2 \rangle }\right)
$$
$$
:f^n: = \sum_m (-1)^m \frac{ n! }{ m!(n-2m)! 2^m } f^{n-2m} ||f||^{2m} = ||f||^n h_n(||f||^{-1}f)
$$
where the
$$
h_n(x) = \sum_m (-1)^m \frac{ n! }{ m!(n-2m)! 2^m } x^{n-2m}
$$
are the [[Hermite polynomials]] with leading coefficient $1$ and $||f||^2 = \langle g \rangle^2$. Further,
$$
\langle :f:^m :g:^m \rangle = \delta_{mn} n! \langle fg \rangle^n \ .
$$

This follows from
$$
:\exp(af):\,:\exp(bg): = \exp(af+bg) \exp\left( { \frac{-1}{2} (a^2 \langle f^2 \rangle + b^2  \langle g^2 \rangle) } \right) \ ,
$$
a formula that contains a great deal of the combinatorics of Wick monomials.

If there are two measures $\mu$ and $\nu$ with respect to which $f$ is Gaussian of mean zero, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787032.png" /> be jointly Gaussian variabless with mean zero (not necessarily distinct). Then there is an explicit formula for the Wick monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787033.png" />, as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787035.png" /> runs over all pairings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787036.png" /> (sometimes called graphs), i.e. all sets of disjoint unordered pairs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787038.png" /> is the union of the unordered pairs making up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787039.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787040.png" /> is an unordered pair, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787041.png" /> is the set of vertices making up that pair.

For instance,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787043.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787044.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787046.png" />, be a collection of disjoint finite sets. A line on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787047.png" /> is by definition a pair of elements taken from different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787048.png" />. A graph on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787049.png" /> is a set of disjoint lines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787050.png" />. If each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787051.png" /> is seen as a vertex with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787052.png" />  "legs"  emanating from it, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787053.png" /> can be visualized as a set of lines joining legs from different vertices. A graph such that all legs are joined is a (certain special kind of) fully contracted graph, vacuum graph, Feynman graph, or Feynman diagram.

The case of  "pairings"  which occured above corresponds to a graph on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787054.png" /> where each vertex has precisely one leg. In terms of these Feynman diagrams a product of Wick monomials is expressed as a linear combination of Wick monomials as follows.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787056.png" />, be a collection of disjoint finite sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787057.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787058.png" /> a collection of jointly Gaussian random variables indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787059.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787061.png" /> runs over all graphs on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787063.png" /> is the union of all the disjoint unordered pairs making up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787064.png" />. More general Feynman graphs, such as graphs with also self-interaction lines, occur when several different covariances are involved, cf. [[#References|[a4]]].

For the expection of a product of Wick monomials one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

and, in particular,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787067.png" /> runs over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787068.png" /> ways of splitting up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787069.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787070.png" /> unordered pairs. All of the formulas (a1)–(a4), (a7), (a8), especially (a8), generally go by the name Wick's formula or Wick's theorem.

In the setting of (Euclidean) quantum field theory, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787071.png" /> be the Schwartz space of rapidly-decreasing smooth functions and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787072.png" /> be the space of real-valued tempered distributions. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787073.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787074.png" /> be the linear function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787075.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787076.png" />. Then for any continuous positive scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787077.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787079.png" />, there is a unique countably-additive Gaussian measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787080.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787081.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787082.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787083.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787084.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787085.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787086.png" /></td> </tr></table>

So <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787087.png" />, and some of the formulas of Wick monomials, etc., now take the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787088.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3prm)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787089.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787090.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787091.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5prm)</td></tr></table>

Wick monomials have much to do with the [[Fock space|Fock space]] via the Itô–Wick–Segal isomorphism. This rest on either of two narrowly related uniqueness theorems: the uniqueness of standard Gaussian functions or the uniqueness of Fock representations.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787092.png" /> be a pre-Hilbert space. A representation of the canonical commutation relations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787093.png" /> is a pair of linear mappings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787094.png" /></td> </tr></table>

from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787095.png" /> to operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787097.png" /> defined on a dense domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787098.png" /> in a complex Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787099.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870100.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870101.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870102.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870103.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870105.png" />. The representation is called a Fock representation if there is moreover an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870106.png" />, called the vacuum vector, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870107.png" /></td> </tr></table>

and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870108.png" /> is the linear space span of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870111.png" />. There is an existence theorem (cf. [[Fock space|Fock space]] and [[Commutation and anti-commutation relationships, representation of|Commutation and anti-commutation relationships, representation of]]) and the uniqueness theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870112.png" /> are two Fock representations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870113.png" /> with vacuum vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870114.png" />, then they are unitarily equivalent and the unitary equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870115.png" /> is uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870116.png" />.

A standard Gaussian function on a real Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870117.png" /> (called a Gaussian random process indexed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870118.png" /> in [[#References|[a3]]]) is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870119.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870120.png" /> to the random variables on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870121.png" /> such that (almost everywhere)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870122.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870123.png" /></td> </tr></table>

such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870124.png" />-algebra generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870125.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870126.png" /> (up to the sets of measure zero) and such that

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870127.png" /> is a Gaussian random variable of mean zero, and

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870128.png" />.

For these objects there is an existence theorem, and also the uniqueness theorem that two standard Gaussian functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870130.png" /> on probability spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870132.png" /> are equivalent in the sense that there is an isomorphism of the two probability spaces under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870134.png" /> correspond for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870135.png" /> (cf. [[#References|[a1]]], §4, [[#References|[a3]]], Chap. 1). The uniqueness theorem is a special case of Kolmogorov's theorem that measure spaces are completely determined by consistent joint probability distributions.

Identifying the symmetric Fock space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870136.png" /> with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870137.png" /> realizing the standard Gaussian function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870138.png" />, the Wick products of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870139.png" /> are obtained by taking the usual products and then applying the orthogonal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870140.png" /> onto its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870141.png" />-particle subspace.

In the case of one Gaussian variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870142.png" /> with probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870143.png" />, the above works out as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870144.png" /></td> </tr></table>

A Fock representation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870145.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870146.png" /></td> </tr></table>

and, indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870147.png" />, which fits because the creation operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870148.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870149.png" />. In terms of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870150.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870151.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870152.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870153.png" /></td> </tr></table>

where in the  "binomial expansion of creation and annihilation operatorsbinomial expansion"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870154.png" /> on the right-hand side the annihilation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870155.png" /> all come before the creation operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870156.png" /> (Wick ordening). Suitably interpreted, the same formula holds in general, [[#References|[a3]]], p. 24.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R.L. Dobrushin,   R.A. Minlos,   "Polynomials in linear random functions"  ''Russian Math. Surveys'' , '''32'''  (1977)  pp. 71–127  ''Uspekhi Mat. Nauk'' , '''32'''  (1977)  pp. 67–122</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dimock,   J. Glimm,   "Measures on Schwartz distribution space and applications to $P(\phi)_2$ field theories"  ''Adv. in Math.'' , '''12'''  (1974)  pp. 58–83</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Simon,   "The $P(\phi)_2$ Euclidean (quantum) field theory" , Princeton Univ. Press  (1974)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Glimm,   A. Jaffe,   "Quantum physics, a functional integral point of view" , Springer  (1981)</TD></TR>
</table>

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