# The Polarization Identity and Wick's Formula

Posted on April 3, 2016
Tags: math

Let $$V$$ be a real vector space. Recall that a symmetric bilinear form on $$V^2$$ is a function $$T: V^2 \to \mathbf{R}$$ such that $$\label{eq:1} T(\alpha x + \beta y, z) = \alpha T(x,z) + \beta T(y,z) \quad \text{and} \quad T(x,y) = T(y,x)$$ for every $$\alpha, \beta \in \mathbf{R}$$ and $$x,y,z \in V$$. Define $$Q(x) = T(x,x)$$. Then, $$\label{eq:2} \frac{1}{8}[Q(x+y) - Q(x-y) - Q(-x+y) + Q(x+y)] = T(x,y)$$

for every $$x,y \in V$$. In particular, $$T$$ is determined completely by the values $$\{Q(x) = T(x,x) : x \in V\}$$. This is the general case of the following identity.

Theorem 1 (Polarization Identity). Let $$V$$ be a real vector space, and let $$T: V^n \to \mathbf{R}$$ be a symmetric $$n$$-linear form. Then, $$\label{eq:3} T(x_1, \dots, x_n) = \frac{1}{2^n n!} \sum_{(\epsilon_1, \dots, \epsilon_n) \in \{-1,1\}^n} \epsilon_1 \dots \epsilon_n Q(\epsilon_1 x_1 + \dots + \epsilon_n x_n)$$ for every $$x_1, \dots, x_n \in V$$, where $$\label{eq:4} Q(x) = T(x, \dots, x).$$

In particular, $$T$$ is completely determined by $$Q$$.

The Polarization Identity yields a quick proof of the following formula due to Wick.

Proposition (Wick’s Formula). Let $$(X_1, \dots, X_n)$$ be a centered Gaussian vector. Then $$\label{eq:5} \mathbf{E}[X_1 \dots X_n] = \sum \prod \mathbf{E}[X_i X_j]$$

where the RHS is taken over all partitions of $$\{1, \dots, n\}$$ into pairs $$\{(i_1, j_1), \dots, (i_{n/2}, j_{n/2})\}$$.

Proof. It is clear that both the LHS and RHS are symmetric and $$n$$-linear on the vector space $$V$$ of centered Gaussian variables. By Theorem 1, it suffices to establish (\ref{eq:5}) when $$X_1 = \ldots = X_n =: X \sim N(0,\sigma^2)$$. This follows from the familiar formula $$\label{eq:6} \mathbf{E}[X^n] = \begin{cases} 0 & \text{if n is odd}\\ (n-1)!! \sigma^2 & \text{if n is even} \end{cases}$$ where $$\label{eq:7} (n-1)!! = 1 \times 3 \times 5 \times \dots \times (n-1) = \frac{n!}{(n/2)!} \frac{1}{2^{n/2}}$$

is the number of partitions of $$\{1, \dots, n\}$$ into pairs (e.g. by induction on $$m = n/2$$).