Talk:Wick product

The Definition

I'm sorry to be a bother, but I'm finding the definition of this product difficult to follow (I'm just an undergraduate.. hehe). If we have $\langle \rangle =1$ and ${\partial \langle X_{1}\rangle \over \partial X_{1}}=\langle \rangle ,$ and the constraint that $\operatorname {E} \langle X_{1}\rangle =\int _{\Omega (X_{1})}\langle X_{1}\rangle d?=0$, and if it follows (as in the example) that $\int _{?someSet}1d{X_{1}}=X_{1}-\operatorname {E} X_{1}$, what is the set we are integrating over, and also, how is the expectation of $\operatorname {E} \langle X_{1}\rangle$ defined? Sorry again, this is probably a really silly question ^_^ Yesitsapril 04:24, 25 September 2006 (UTC)

The Wick product is a polynomial function of the random variables and their expected values and expected values of their products. The partial derivative notation just means the derivative of that polynomial with respect to one of the variables, treating not only the others as constant, but also all expected values as constant. Thus

${\partial \langle X_{1}\rangle \over \partial X_{1}}=\langle \rangle =1$

entails that $\langle X_{1}\rangle =X_{1}+\mathrm {constant} ,\,$ and then the constraint $E\langle X_{1}\rangle =0\,$ entails that the "constant" must be $-E(X_{1}).\,$ When we get to the next step—finding the Wick product of a pair of random variables, and we differenatiate it with respect to X₁, the "constant" may depend on X₂ and also on the expected values of X₁, X₂, and X₁X₂. And so on.

It may not be the most felicitous notation. I followed fairly closely the notation of the paper I cited. Michael Hardy 20:33, 25 September 2006 (UTC)

Aah, I see now! Oh dear, my question was a bit silly, thank you very much for the clarification Michael :) Yesitsapril 04:11, 26 September 2006 (UTC)

In the third order Product, subtraction of the expectation value seems to have been forgotten. —Preceding unsigned comment added by 80.201.76.195 (talk • contribs)

Also noncommutative

More generally, Wick product is treated in the noncommutative probability theory; see for instance: John C. Baez, "Wick Products of the Free Bose Field", JOURNAL OF FUNCTIONAL ANALYSIS 86, 21 l-225 (1989). Boris Tsirelson (talk) 18:03, 22 November 2016 (UTC)