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''Wick monomial, Wick power''
 
''Wick monomial, Wick power''
  
 
The Wick products of random variables arise through an orthogonalization procedure.
 
The Wick products of random variables arise through an orthogonalization procedure.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978701.png" /> be (real-valued) random variables on some probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978702.png" />. The Wick product
+
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.
  
<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/w0978703.png" /></td> </tr></table>
+
For example,
 +
$$
 +
:f: = f - \langle f \rangle \ ,
 +
$$
 +
$$
 +
:f^2: = f^2 - 2\langle f \rangle f  - \langle f^2 \rangle + 2\langle f \rangle^2 \ .
 +
$$
  
is defined recursively as a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978704.png" /> of total degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978705.png" /> satisfying
+
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
 +
$$
 +
:\exp(af): = \langle \exp(af) \rangle^{-1} \exp(af) \ .
 +
$$
  
<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/w0978706.png" /></td> </tr></table>
+
The Wick products, powers and exponentials depend both on the variables involved and on the underlying measure.
  
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978707.png" />,
+
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 $||f||^2 = \langle f^2 \rangle$. Further,
 +
$$
 +
\langle :f:^m :g:^m \rangle = \delta_{mn} n! \langle fg \rangle^n \ .
 +
$$
  
<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/w0978708.png" /></td> </tr></table>
+
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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w0978709.png" /> denotes expectation. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787010.png" /> notation is traditional in physics.
+
If there are two measures $\mu$ and $\nu$ with respect to which $f$ is Gaussian of mean zero, then
 +
$$
 +
:\exp(a f) :_\mu = :\exp af :_\nu \exp\left( { \frac{a^2}{2}\langle f^2 \rangle_\mu - \langle f^2 \rangle_\nu } \right) \ .
 +
$$
  
For example,
+
Let $f_1,\ldots,f_n$ be jointly Gaussian variables with mean zero (not necessarily distinct). Then there is an explicit formula for the Wick monomial $:f_1\cdots f_n:$, as follows:
 +
$$
 +
:f_1\cdots f_n: = \sum_G \prod_{e \in G} -\left\langle { f_{e_1} f_{e_2} } \right\rangle \prod_{i \not\in [G]} f_i \ .
 +
$$
  
<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/w09787011.png" /></td> </tr></table>
+
Here, $G$ runs over all pairings of $\{1,\ldots,n\}$ (sometimes called graphs), i.e. all sets of disjoint unordered pairs of $\{1,\ldots,n\}$, $[G]$ is the union of the unordered pairs making up $G$, and if $e$ is an unordered pair, then $\{e_1,e_2\}$ is the set of vertices making up that pair.
  
<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/w09787012.png" /></td> </tr></table>
+
For instance,
 
+
$$
There is a binomial theorem:
+
:fg^2: = fg^2 - 2\langle fg \rangle - \langle g^2 \rangle f \ ,
 +
$$
 +
$$
 +
:f^2g^2: = f^2g^2 - \langle f^2 \rangle g^2 - \langle g^2 \rangle f^2 - 4 \langle fg \rangle fg + 2\langle fg \rangle^2 + \langle f \rangle^2 \langle g \rangle^2 \ .
 +
$$
  
<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/w09787013.png" /></td> </tr></table>
+
Let $\{ I_\nu \}$, $\nu = 1,\ldots,n$, be a collection of disjoint finite sets. A line on $\{ I_\nu \}$ is by definition a pair of elements taken from different $I_\nu$. A graph on $\{ I_\nu \}$ is a set of disjoint lines on $\{ I_\nu \}$. If each $I_\nu$ is seen as a vertex with $|I_\nu|$  "legs" emanating from it, then $G$ 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.
  
and a corresponding multinomial theorem. The Wick exponential is defined as
+
The case of  "pairings"  which occurred above corresponds to a graph on $\{ I_\nu \}$ 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.
  
<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/w09787014.png" /></td> </tr></table>
+
Let  $  I _ \nu  $,
 +
$  \nu = 1 \dots n $,
 +
be a collection of disjoint finite sets,  $  I = \cup _ \nu  I _ \nu  $,
 +
and  $  f _{i} $
 +
a collection of jointly Gaussian random variables indexed by  $  I $.  
 +
Then
  
so that
+
$$ \tag{a6}
 +
\prod _ \nu  : \prod _ {i \in I _ \nu} f _{i} :\  = \
 +
\sum _{G} \prod _ {e \in G} < f _{ {e _ 1}} f _{ {e _ 2}} > : \prod _ {i \notin [G]} f _{i} : ,
 +
$$
  
<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/w09787015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
 
  
The Wick products, powers and exponentials depend both on the variables involved and on the underlying measure.
+
where  $  G $
 +
runs over all graphs on  $  \{ I _ \nu  \} $
 +
and $  [G] $
 +
is the union of all the disjoint unordered pairs making up  $  G $.
 +
More general Feynman graphs, such as graphs with also self-interaction lines, occur when several different covariances are involved, cf. [[#References|[a4]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787016.png" /> be Gaussian random variables with mean zero. Then
+
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/w09787017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a7}
 +
\left \langle  \prod _ \nu  : \prod _ {i \in I _ \nu} f _{i} : \right \rangle \  = \
 +
\sum _ {G \in \Gamma _{0} ( \{ I _ \nu  \} )} \
 +
\prod _ {e \in G} \langle  f _{ {e _ 1}} f _{ {e _ 2}} \rangle
 +
$$
  
<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/w09787018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</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/w09787019.png" /></td> </tr></table>
+
and, in particular,
  
where
+
$$ \tag{a8}
 +
\langle  f _{1} \dots f _{n} \rangle \  = \  \left \{
  
<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/w09787020.png" /></td> </tr></table>
+
\begin{array}{ll}
 +
0  &\textrm{ if } \  n \  \textrm{ is } \  \textrm{ odd } ,  \\
 +
\sum _ {G \in \Gamma _{0} (n)} \
 +
\prod _ {e \in G} \langle  f _{ {e _ 1}} f _{ {e _ 2}} \rangle  &\  \textrm{ if } \
 +
n=2k ,  \\
 +
\end{array}
  
is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787021.png" />-th Hermite polynomial with leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787023.png" /> (cf. [[Hermite polynomials|Hermite polynomials]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787024.png" />. Further,
+
\right .$$
  
<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/w09787025.png" /></td> </tr></table>
 
  
This follows from
+
where  $  \Gamma _{0} (2k) $
 +
runs over all  $  (2k)! 2 ^{-k} (k!) ^{-1} $
 +
ways of splitting up  $  \{ 1 \dots 2k \} $
 +
into  $  k $
 +
unordered pairs. All of the formulas (a1)–(a4), (a7), (a8), especially (a8), generally go by the name Wick's formula or Wick's theorem.
  
<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/w09787026.png" /></td> </tr></table>
+
In the setting of (Euclidean) quantum field theory, let  $  {\mathcal S} ( \mathbf R ^{n} ) $
 +
be the Schwartz space of rapidly-decreasing smooth functions and let  $  {\mathcal S} ^ \prime  ( \mathbf R ^{n} ) $
 +
be the space of real-valued tempered distributions. For  $  f \in {\mathcal S} ( \mathbf R ^{n} ) $,
 +
let  $  \phi (f \  ) $
 +
be the linear function on  $  {\mathcal S} ^ \prime  ( \mathbf R ^{n} ) $
 +
given by  $  \phi (f \  )(u) = u(f \  ) $.  
 +
Then for any continuous positive scalar product  $  C $
 +
on  $  {\mathcal S} ( \mathbf R ^{n} ) \times {\mathcal S} ( \mathbf R ^{n} ) $,
 +
$  (f,\  g) \mapsto \langle f,\  Cg\rangle $,
 +
there is a unique countably-additive Gaussian measure  $  d q _{C} $
 +
on  $  {\mathcal S} ^ \prime  ( \mathbf R ^{n} ) $
 +
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/w09787027.png" /></td> </tr></table>
+
$$
 +
\int\limits e ^ {\  i \phi (f \  )} \  dq _{C} \  = \
 +
\mathop{\rm exp}\nolimits \left ( -  
 +
\frac{1}{2}
 +
\langle  f ,\  C f \  \rangle \right ) ,\ \
 +
f \in {\mathcal S} ( \mathbf R ^{n} ) .
 +
$$
  
a formula that contains a great deal of the combinatorics of Wick monomials.
 
  
If there are two measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787029.png" /> with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w09787030.png" /> is Gaussian of mean zero, then
+
Then  $  \phi (f \  ) \in L _{p} ( {\mathcal S} ^ \prime  ( \mathbf R ^{n} ) ,\  d q _{C} ) $
 +
for all  $  p \in [1,\  \infty ) $
 +
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/w09787031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$
 +
\int\limits \phi (f \  ) \  dq _{C} \  = 0 ,
 +
$$
  
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>
+
$$
 +
\int\limits \phi (f _{1} ) \phi (f _{2} ) \  d q _{C} \  = \  \langle  f _{1} ,\  Cf _{2} \rangle .
 +
$$
  
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,
+
So  $  \langle  \phi (f _{1} ) \phi (f _{2} ) \rangle = \langle  f _{1} ,\  Cf _{2} \rangle $,
 +
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/w09787042.png" /></td> </tr></table>
+
$$ \tag{a3\prime}
 +
: \phi (f \  ) ^{n} :=
 +
$$
  
<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>
+
$$
 +
= \
 +
\sum _{j}
 +
\frac{n!}{(n-2j)! j! 2 ^ j}
 +
(-1)
 +
^{j} \langle f,\  Cf \  \rangle ^{j} \phi (f \  ) ^{n-2j\ } =
 +
$$
  
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.
+
$$
 +
= \
 +
\langle f,\ Cf \ \rangle ^{n/2} h _{n} \left (
 +
\frac{\phi (f \  )}{\langle  f,\  Cf \  \rangle ^ 1/2}
 +
\right ) ,
 +
$$
  
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>
+
$$ \tag{a5\prime}
 +
: \prod _ {\nu =1} ^ n \phi (f _ \nu  ) : = \  \sum
 +
_{G} \prod _ {e \in G} < f _{ {e _ 1}} ,\  - Cf
 +
_{ {e _ 2}} > \prod _ {i \notin [G]} \phi (f _{i} ) .
 +
$$
  
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
+
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.
  
<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>
+
Let  $  {\mathcal S} $
 +
be a pre-Hilbert space. A representation of the canonical commutation relations over  $  {\mathcal S} $
 +
is a pair of linear mappings
  
and, in particular,
+
$$
 +
f \  \mapsto \  a(f \  ) ,\ \  g \  \mapsto \  a ^{*} (g)
 +
$$
  
<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.
+
from  $  {\mathcal S} $
 +
to operators  $  a(f \  ) $,  
 +
$  a ^{*} (g) $
 +
defined on a dense domain  $  D $
 +
in a complex Hilbert space  $  H $
 +
such that
  
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
+
$$
 +
a(f \  ) D \  \subset \  D ,\ \  a ^{*} (g) D \  \subset \  D ,
 +
$$
  
<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
+
$$
 +
\langle  x _{1} ,\  a (f \  )x _{2} \rangle \  = \  \langle  a ^{*} (f \  )x _{1} ,\  x _{2} \rangle ,
 +
$$
  
<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>
+
$$
 +
[a(f \  ),\  a(g)] \  = \  [a ^{*} (f \  ),\  a ^{*} (g)] \  = 0,
 +
$$
  
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>
+
$$
 +
[a(f \  ),\  a ^{*} (g)] x \  = \  \langle  f,\  g\rangle x ,
 +
$$
  
<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>
+
for all  $  x,\  x _{1} ,\  x _{2} \in D $,
 +
$  f ,\  g \in {\mathcal S} $.  
 +
The representation is called a Fock representation if there is moreover an  $  \Omega \in D $,
 +
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/w09787091.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5prm)</td></tr></table>
+
$$
 +
a(f \  ) \Omega \  = 0 ,\ \  f \in {\mathcal S} ,
 +
$$
  
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
+
and such that  $  D $
 +
is the linear space span of the vectors  $  a ^{*} (g _{1} ) \dots a ^{*} (g _{k} ) \Omega $,
 +
$  g _{i} \in {\mathcal S} $,
 +
$  k = 0,\  1,\dots $.  
 +
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  $  (a _{i} ,\  a _{i} ^{*} ) $
 +
are two Fock representations over $  {\mathcal S} $
 +
with vacuum vectors  $  \Omega _{i} $,
 +
then they are unitarily equivalent and the unitary equivalence  $  U $
 +
is uniquely determined by  $  U \Omega _{1} = \Omega _{2} $.
  
<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
+
A standard Gaussian function on a real Hilbert space  $  V $(
 +
called a Gaussian random process indexed by  $  V $
 +
in [[#References|[a3]]]) is a mapping  $  \phi $
 +
from $  V $
 +
to the random variables on a probability space $  (X ,\  {\mathcal B} ,\  \mu ) $
 +
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/w097870100.png" /></td> </tr></table>
+
$$
 +
\phi (v+w) \  = \  \phi (v)+ \phi (w) ,\ \  v,\  w \in V ,
 +
$$
  
<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>
+
$$
 +
\phi ( \alpha v ) \  = \  \alpha \phi ( v) ,\ \  \alpha \in \mathbf R ,\ \  v \in V ,
 +
$$
  
<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
+
such that the  $  \sigma $-
 +
algebra generated by the  $  \phi (f \  ) $
 +
is $  {\mathcal B} $(
 +
up to the sets of measure zero) and 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>
+
$  \phi (v) $
 +
is a Gaussian random variable of mean zero, and
  
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" />.
+
$  \langle  \phi (v) \phi (w)\rangle = \langle  v,w\rangle $.
  
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>
+
For these objects there is an existence theorem, and also the uniqueness theorem that two standard Gaussian functions  $  \phi $
 +
and  $  \phi ^ \prime  $
 +
on probability spaces  $  (X,\  {\mathcal B} , \mu ) $,
 +
$  (X ^ \prime  ,\  {\mathcal B} ^ \prime  , \mu ^ \prime  ) $
 +
are equivalent in the sense that there is an isomorphism of the two probability spaces under which  $  \phi (v) $
 +
and  $  \phi ^ \prime  (v) $
 +
correspond for all  $  v \in V $(
 +
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.
  
<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>
+
Identifying the symmetric Fock space  $  F(V) $
 +
with the space  $  L _{2} (X,\  {\mathcal B} ,\  \mu ) $
 +
realizing the standard Gaussian function on  $  H $,
 +
the Wick products of the  $  \phi (v) $
 +
are obtained by taking the usual products and then applying the orthogonal projection of  $  F(V) $
 +
onto its  $  n $-
 +
particle subspace.
  
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
+
In the case of one Gaussian variable  $  x $
 +
with probability measure  $  \pi ^ {- 1/2} e ^ {- x ^{2} /2} \  dx $,
 +
the above works out as follows:
  
<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
+
$$
 +
: x ^{n} :\  = \  h _{n} (x).
 +
$$
  
<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.
+
A Fock representation in $  L _{2} ( \mathbf R ,\  (2 \pi ) ^ {- 1/2} e ^ {- x ^{2} /2} \  dx ) $
 +
is
  
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.
+
$$
 +
\Omega \  = \  1 ,\ \
 +
a \  =
 +
\frac{d}{dx}
 +
,\ \
 +
a ^{*} \  = \  x -  
 +
\frac{d}{dx}
 +
,
 +
$$
  
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>
+
and, indeed,  $  h _{n} (x) = (x- d / dx ) ^{n} (1) $,
 +
which fits because the creation operator on  $  F ( \mathbf R ) $
 +
is  $  a ^{*} (e ^ {\otimes n} ) = e ^ {\otimes (n+1)} $.
 +
In terms of the variable  $  y = x / \sqrt 2 $,
  
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>
+
$$
 +
\Omega \  = \  1,\ \
 +
a \  =
 +
\frac{1}{\sqrt 2}
 +
 +
\frac{d}{dy}
 +
,\ \
 +
a ^{*} \  = \  \sqrt 2 y -  
 +
\frac{1}{\sqrt 2}
 +
 +
\frac{d}{dy}
 +
,
 +
$$
  
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>
+
$$
 +
y \  =
 +
\frac{1}{\sqrt 2}
 +
(a + a ^{*} ),
 +
$$
  
<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
 
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>
+
$$
 +
: y ^{n} := \  ( \sqrt 2 ) ^{-n} h _{n} ( \sqrt 2 y ) \  = \  ( \sqrt 2 ) ^{-n} \sum _{k=0} ^ n
 +
\binom{n}{k} a ^{*k} a ^{n-k} ,
 +
$$
 +
 
  
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.
+
where in the  "binomial expansion of creation and annihilation operatorsbinomial expansion"  of $  ( (a+a ^{*} ) / \sqrt 2 ) ^{n} $
 +
on the right-hand side the annihilation operators $  a $
 +
all come before the creation operators $  a ^{*} $(
 +
Wick ordening). Suitably interpreted, the same formula holds in general, [[#References|[a3]]], p. 24.
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870157.png" /> field theories"  ''Adv. in Math.'' , '''12'''  (1974)  pp. 58–83</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Simon,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097870/w097870158.png" /> 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>
+
<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>

Latest revision as of 18:50, 11 December 2020


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 $||f||^2 = \langle f^2 \rangle$. 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 $$ :\exp(a f) :_\mu = :\exp af :_\nu \exp\left( { \frac{a^2}{2}\langle f^2 \rangle_\mu - \langle f^2 \rangle_\nu } \right) \ . $$

Let $f_1,\ldots,f_n$ be jointly Gaussian variables with mean zero (not necessarily distinct). Then there is an explicit formula for the Wick monomial $:f_1\cdots f_n:$, as follows: $$ :f_1\cdots f_n: = \sum_G \prod_{e \in G} -\left\langle { f_{e_1} f_{e_2} } \right\rangle \prod_{i \not\in [G]} f_i \ . $$

Here, $G$ runs over all pairings of $\{1,\ldots,n\}$ (sometimes called graphs), i.e. all sets of disjoint unordered pairs of $\{1,\ldots,n\}$, $[G]$ is the union of the unordered pairs making up $G$, and if $e$ is an unordered pair, then $\{e_1,e_2\}$ is the set of vertices making up that pair.

For instance, $$ :fg^2: = fg^2 - 2\langle fg \rangle - \langle g^2 \rangle f \ , $$ $$ :f^2g^2: = f^2g^2 - \langle f^2 \rangle g^2 - \langle g^2 \rangle f^2 - 4 \langle fg \rangle fg + 2\langle fg \rangle^2 + \langle f \rangle^2 \langle g \rangle^2 \ . $$

Let $\{ I_\nu \}$, $\nu = 1,\ldots,n$, be a collection of disjoint finite sets. A line on $\{ I_\nu \}$ is by definition a pair of elements taken from different $I_\nu$. A graph on $\{ I_\nu \}$ is a set of disjoint lines on $\{ I_\nu \}$. If each $I_\nu$ is seen as a vertex with $|I_\nu|$ "legs" emanating from it, then $G$ 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 occurred above corresponds to a graph on $\{ I_\nu \}$ 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 $ I _ \nu $, $ \nu = 1 \dots n $, be a collection of disjoint finite sets, $ I = \cup _ \nu I _ \nu $, and $ f _{i} $ a collection of jointly Gaussian random variables indexed by $ I $. Then

$$ \tag{a6} \prod _ \nu : \prod _ {i \in I _ \nu} f _{i} :\ = \ \sum _{G} \prod _ {e \in G} < f _{ {e _ 1}} f _{ {e _ 2}} > : \prod _ {i \notin [G]} f _{i} : , $$


where $ G $ runs over all graphs on $ \{ I _ \nu \} $ and $ [G] $ is the union of all the disjoint unordered pairs making up $ G $. More general Feynman graphs, such as graphs with also self-interaction lines, occur when several different covariances are involved, cf. [a4].

For the expection of a product of Wick monomials one has

$$ \tag{a7} \left \langle \prod _ \nu : \prod _ {i \in I _ \nu} f _{i} : \right \rangle \ = \ \sum _ {G \in \Gamma _{0} ( \{ I _ \nu \} )} \ \prod _ {e \in G} \langle f _{ {e _ 1}} f _{ {e _ 2}} \rangle $$


and, in particular,

$$ \tag{a8} \langle f _{1} \dots f _{n} \rangle \ = \ \left \{ \begin{array}{ll} 0 &\textrm{ if } \ n \ \textrm{ is } \ \textrm{ odd } , \\ \sum _ {G \in \Gamma _{0} (n)} \ \prod _ {e \in G} \langle f _{ {e _ 1}} f _{ {e _ 2}} \rangle &\ \textrm{ if } \ n=2k , \\ \end{array} \right .$$


where $ \Gamma _{0} (2k) $ runs over all $ (2k)! 2 ^{-k} (k!) ^{-1} $ ways of splitting up $ \{ 1 \dots 2k \} $ into $ k $ 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 $ {\mathcal S} ( \mathbf R ^{n} ) $ be the Schwartz space of rapidly-decreasing smooth functions and let $ {\mathcal S} ^ \prime ( \mathbf R ^{n} ) $ be the space of real-valued tempered distributions. For $ f \in {\mathcal S} ( \mathbf R ^{n} ) $, let $ \phi (f \ ) $ be the linear function on $ {\mathcal S} ^ \prime ( \mathbf R ^{n} ) $ given by $ \phi (f \ )(u) = u(f \ ) $. Then for any continuous positive scalar product $ C $ on $ {\mathcal S} ( \mathbf R ^{n} ) \times {\mathcal S} ( \mathbf R ^{n} ) $, $ (f,\ g) \mapsto \langle f,\ Cg\rangle $, there is a unique countably-additive Gaussian measure $ d q _{C} $ on $ {\mathcal S} ^ \prime ( \mathbf R ^{n} ) $ such that

$$ \int\limits e ^ {\ i \phi (f \ )} \ dq _{C} \ = \ \mathop{\rm exp}\nolimits \left ( - \frac{1}{2} \langle f ,\ C f \ \rangle \right ) ,\ \ f \in {\mathcal S} ( \mathbf R ^{n} ) . $$


Then $ \phi (f \ ) \in L _{p} ( {\mathcal S} ^ \prime ( \mathbf R ^{n} ) ,\ d q _{C} ) $ for all $ p \in [1,\ \infty ) $ and

$$ \int\limits \phi (f \ ) \ dq _{C} \ = \ 0 , $$


$$ \int\limits \phi (f _{1} ) \phi (f _{2} ) \ d q _{C} \ = \ \langle f _{1} ,\ Cf _{2} \rangle . $$


So $ \langle \phi (f _{1} ) \phi (f _{2} ) \rangle = \langle f _{1} ,\ Cf _{2} \rangle $, and some of the formulas of Wick monomials, etc., now take the form

$$ \tag{a3\prime} : \phi (f \ ) ^{n} :\ = $$


$$ = \ \sum _{j} \frac{n!}{(n-2j)! j! 2 ^ j} (-1) ^{j} \langle f,\ Cf \ \rangle ^{j} \phi (f \ ) ^{n-2j\ } = $$


$$ = \ \langle f,\ Cf \ \rangle ^{n/2} h _{n} \left ( \frac{\phi (f \ )}{\langle f,\ Cf \ \rangle ^ 1/2} \right ) , $$


$$ \tag{a5\prime} : \prod _ {\nu =1} ^ n \phi (f _ \nu ) : \ = \ \sum _{G} \prod _ {e \in G} < f _{ {e _ 1}} ,\ - Cf _{ {e _ 2}} > \prod _ {i \notin [G]} \phi (f _{i} ) . $$


Wick monomials have much to do with the 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 $ {\mathcal S} $ be a pre-Hilbert space. A representation of the canonical commutation relations over $ {\mathcal S} $ is a pair of linear mappings

$$ f \ \mapsto \ a(f \ ) ,\ \ g \ \mapsto \ a ^{*} (g) $$


from $ {\mathcal S} $ to operators $ a(f \ ) $, $ a ^{*} (g) $ defined on a dense domain $ D $ in a complex Hilbert space $ H $ such that

$$ a(f \ ) D \ \subset \ D ,\ \ a ^{*} (g) D \ \subset \ D , $$


$$ \langle x _{1} ,\ a (f \ )x _{2} \rangle \ = \ \langle a ^{*} (f \ )x _{1} ,\ x _{2} \rangle , $$


$$ [a(f \ ),\ a(g)] \ = \ [a ^{*} (f \ ),\ a ^{*} (g)] \ = \ 0, $$


$$ [a(f \ ),\ a ^{*} (g)] x \ = \ \langle f,\ g\rangle x , $$


for all $ x,\ x _{1} ,\ x _{2} \in D $, $ f ,\ g \in {\mathcal S} $. The representation is called a Fock representation if there is moreover an $ \Omega \in D $, called the vacuum vector, such that

$$ a(f \ ) \Omega \ = \ 0 ,\ \ f \in {\mathcal S} , $$


and such that $ D $ is the linear space span of the vectors $ a ^{*} (g _{1} ) \dots a ^{*} (g _{k} ) \Omega $, $ g _{i} \in {\mathcal S} $, $ k = 0,\ 1,\dots $. There is an existence theorem (cf. Fock space and Commutation and anti-commutation relationships, representation of) and the uniqueness theorem: If $ (a _{i} ,\ a _{i} ^{*} ) $ are two Fock representations over $ {\mathcal S} $ with vacuum vectors $ \Omega _{i} $, then they are unitarily equivalent and the unitary equivalence $ U $ is uniquely determined by $ U \Omega _{1} = \Omega _{2} $.


A standard Gaussian function on a real Hilbert space $ V $( called a Gaussian random process indexed by $ V $ in [a3]) is a mapping $ \phi $ from $ V $ to the random variables on a probability space $ (X ,\ {\mathcal B} ,\ \mu ) $ such that (almost everywhere)

$$ \phi (v+w) \ = \ \phi (v)+ \phi (w) ,\ \ v,\ w \in V , $$


$$ \phi ( \alpha v ) \ = \ \alpha \phi ( v) ,\ \ \alpha \in \mathbf R ,\ \ v \in V , $$


such that the $ \sigma $- algebra generated by the $ \phi (f \ ) $ is $ {\mathcal B} $( up to the sets of measure zero) and such that

$ \phi (v) $ is a Gaussian random variable of mean zero, and

$ \langle \phi (v) \phi (w)\rangle = \langle v,\ w\rangle $.


For these objects there is an existence theorem, and also the uniqueness theorem that two standard Gaussian functions $ \phi $ and $ \phi ^ \prime $ on probability spaces $ (X,\ {\mathcal B} , \mu ) $, $ (X ^ \prime ,\ {\mathcal B} ^ \prime , \mu ^ \prime ) $ are equivalent in the sense that there is an isomorphism of the two probability spaces under which $ \phi (v) $ and $ \phi ^ \prime (v) $ correspond for all $ v \in V $( cf. [a1], §4, [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 $ F(V) $ with the space $ L _{2} (X,\ {\mathcal B} ,\ \mu ) $ realizing the standard Gaussian function on $ H $, the Wick products of the $ \phi (v) $ are obtained by taking the usual products and then applying the orthogonal projection of $ F(V) $ onto its $ n $- particle subspace.

In the case of one Gaussian variable $ x $ with probability measure $ \pi ^ {- 1/2} e ^ {- x ^{2} /2} \ dx $, the above works out as follows:

$$ : x ^{n} :\ = \ h _{n} (x). $$


A Fock representation in $ L _{2} ( \mathbf R ,\ (2 \pi ) ^ {- 1/2} e ^ {- x ^{2} /2} \ dx ) $ is

$$ \Omega \ = \ 1 ,\ \ a \ = \ \frac{d}{dx} ,\ \ a ^{*} \ = \ x - \frac{d}{dx} , $$


and, indeed, $ h _{n} (x) = (x- d / dx ) ^{n} (1) $, which fits because the creation operator on $ F ( \mathbf R ) $ is $ a ^{*} (e ^ {\otimes n} ) = e ^ {\otimes (n+1)} $. In terms of the variable $ y = x / \sqrt 2 $,


$$ \Omega \ = \ 1,\ \ a \ = \ \frac{1}{\sqrt 2} \frac{d}{dy} ,\ \ a ^{*} \ = \ \sqrt 2 y - \frac{1}{\sqrt 2} \frac{d}{dy} , $$


$$ y \ = \ \frac{1}{\sqrt 2} (a + a ^{*} ), $$


and

$$ : y ^{n} :\ = \ ( \sqrt 2 ) ^{-n} h _{n} ( \sqrt 2 y ) \ = \ ( \sqrt 2 ) ^{-n} \sum _{k=0} ^ n \binom{n}{k} a ^{*k} a ^{n-k} , $$


where in the "binomial expansion of creation and annihilation operatorsbinomial expansion" of $ ( (a+a ^{*} ) / \sqrt 2 ) ^{n} $ on the right-hand side the annihilation operators $ a $ all come before the creation operators $ a ^{*} $( Wick ordening). Suitably interpreted, the same formula holds in general, [a3], p. 24.

References

[a1] 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
[a2] 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
[a3] B. Simon, "The $P(\phi)_2$ Euclidean (quantum) field theory" , Princeton Univ. Press (1974)
[a4] J. Glimm, A. Jaffe, "Quantum physics, a functional integral point of view" , Springer (1981)
How to Cite This Entry:
Wick product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wick_product&oldid=12724