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 $||f||^2 = \langle f^2 \rangle$. 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.

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