the Wick polynomials

Wick polynomials are an important concept in probability theory and quantum field theory, providing a systematic way to handle products of random variables or fields and their normal ordering. They play a crucial role in simplifying computations involving Gaussian processes and Feynman diagrams by eliminating redundancies and ensuring the correct handling of expectations in quantum systems.

1. Background and Definition

Origin: Wick polynomials were introduced by Gian-Carlo Wick in the context of quantum field theory for simplifying expressions involving products of operators.
Purpose: In essence, Wick polynomials reorganize products of random variables or fields to make them "centered" or "normalized" in terms of their probabilistic or quantum properties.

For a centered Gaussian random variable $X$ (i.e., $\mathbb{E}[X] = 0$ ), the Wick polynomial $:X^n:$ of degree $n$ is defined such that:

$:X^n: = H_n(X),$

where $H_n(X)$ is the Hermite polynomial of degree $n$ . The Hermite polynomial is specially tailored for Gaussian random variables and helps express products in a form that simplifies the computation of expectations.

2. Construction for Gaussian Processes

Consider a collection of random variables $X_1, X_2, \ldots, X_n$ from a Gaussian process. The Wick polynomial $:X_1 X_2 \ldots X_n:$ is the "normal-ordered" product defined such that:

$\mathbb{E}[:X_1 X_2 \ldots X_n:] = 0,$

when any of the $X_i$ are replaced with their respective expectations.

For example, for a second-order product:

$:X^2: = X^2 - \mathbb{E}[X^2],$

ensures that:

$\mathbb{E}[:X^2:] = 0.$

3. Properties of Wick Polynomials

Centering Property: By construction, $\mathbb{E}[:X^n:] = 0$ for any non-zero $n$ .
Orthogonality: Wick polynomials of different degrees are orthogonal with respect to the expectation:

$\mathbb{E}[:X^m: :X^n:] = 0, \quad \text{for } m \neq n.$

Additivity: For independent Gaussian random variables $X$ and $Y$ , the Wick product satisfies:

$:X Y: = X Y - \mathbb{E}[X] Y - X \mathbb{E}[Y] + \mathbb{E}[X] \mathbb{E}[Y].$

4. Applications in Quantum Field Theory

In quantum field theory, Wick polynomials are used to express operators in a form where normal ordering ensures that creation operators are placed to the left of annihilation operators, thus preventing vacuum expectation values from contributing non-zero terms:

$:\hat{\phi}^n(x): = \text{normal-ordered product of field operators}.$

This leads to simplifications in Feynman diagram calculations and allows for consistent handling of perturbation expansions.

5. Wick’s Theorem

Wick’s theorem states that any product of normally distributed random variables can be expressed in terms of sums of products of Wick polynomials and their expectations:

$X_1 X_2 \ldots X_n = :X_1 X_2 \ldots X_n: + \text{all possible pairings of } X_i.$

For instance, for a product of two variables:

$X_1 X_2 = :X_1 X_2: + \mathbb{E}[X_1 X_2].$

For a product of four variables:

$X_1 X_2 X_3 X_4 = :X_1 X_2 X_3 X_4: + \mathbb{E}[X_1 X_2] :X_3 X_4: + \mathbb{E}[X_1 X_3] :X_2 X_4: + \cdots,$

including all possible pairings.

6. Relation to Hermite Polynomials

The Wick polynomials for a standard Gaussian variable $X$ relate directly to Hermite polynomials $H_n(X)$ :

$:X^n: = H_n(X),$

where $H_n(X)$ satisfies the recurrence relation:

$H_{n+1}(X) = X H_n(X) - n H_{n-1}(X),$