# 8.11: Appendix II- Distributions and Functionals


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Let $$x\in{\mathbb R}$$ be a random variable, and $$P(x)$$ a probability distribution for $$x$$. The average of any function $$\phi(x)$$ is then $\blangle\phi(x)\brangle = \int\limits_{-\infty}^\infty\!\!dx\>P(x)\,\phi(x)\Bigg/\!\!\int\limits_{-\infty}^\infty\!\!dx\>P(x)\ .$

Let $$\eta(t)$$ be a random function of $$t$$, with $$\eta(t)\in{\mathbb R}$$, and let $$P\big[\eta(t)\big]$$ be the probability distribution functional for $$\eta(t)$$. Then if $$\Phi\big[\eta(t)\big]$$ is a functional of $$\eta(t)$$, the average of $$\Phi$$ is given by $\int\!\!D\eta\>P\big[\eta(t)\big]\>\Phi\big[\eta(t)\big]\Bigg/\!\!\int\!\!D\eta\,P\big[\eta(t)\big]$ The expression $$\int\!D\eta\>P[\eta]\,\Phi[\eta]$$ is a functional integral. A functional integral is a continuum limit of a multivariable integral. Suppose $$\eta(t)$$ were defined on a set of $$t$$ values $$t\ns_n=n\tau$$. A functional of $$\eta(t)$$ becomes a multivariable function of the values $$\eta\ns_n\equiv \eta(t\ns_n)$$. The metric then becomes $D\eta\longrightarrow \prod_n d\eta\ns_n\ .$

In fact, for our purposes we will not need to know any details about the functional measure $$D\eta$$; we will finesse this delicate issue18. Consider the generating functional, $Z\big[J(t)\big]=\int\!\!D\eta\>P[\eta]\,\exp\Bigg(\int\limits_{-\infty}^\infty\!\!\!dt\>J(t)\,\eta(t)\Bigg)\ .$ It is clear that ${1\over Z[J]}\,{\delta^n\!Z[J]\over\delta J(t\ns_1)\cdots\delta J(t\ns_n)}\Bigg|\nd_{J(t)=0}= \blangle \eta(t\ns_1)\cdots\eta(t\ns_n)\brangle\ .$ The function $$J(t)$$ is an arbitrary source function. We differentiate with respect to it in order to find the $$\eta$$-field correlators.

Let’s compute the generating function for a class of distributions of the Gaussian form, \begin{aligned} P[\eta]&=\exp\!\Bigg(\!-{1\over 2\Gamma}\!\!\int\limits_{-\infty}^\infty\!\!\!dt\> \big(\tau^2\,{\dot\eta}^2 + \eta^2\big)\Bigg)\\ &=\exp\!\Bigg(\!-{1\over 2\Gamma}\!\!\int\limits_{-\infty}^\infty\!\!{d\omega\over 2\pi}\> \big( 1 + \omega^2\tau^2\big)\, \big|{\hat\eta}(\omega)\big|^2\Bigg)\ .\end{aligned} Then Fourier transforming the source function $$J(t)$$, it is easy to see that $Z[J]=Z[0]\cdot\exp\!\Bigg({\Gamma\over 2}\!\!\int\limits_{-\infty}^\infty\!\!{d\omega\over 2\pi}\> { \big|{\hat J}(\omega)\big|^2\over 1+\omega^2\tau^2}\Bigg)\ .$ Note that with $$\eta(t)\in {\mathbb R}$$ and $$J(t)\in {\mathbb R}$$ we have $$\eta^*(\omega)=\eta(-\omega)$$ and $$J^*(\omega)=J(-\omega)$$. Transforming back to real time, we have $Z[J]=Z[0]\cdot\exp\Bigg({1\over 2}\!\int\limits_{-\infty}^\infty\!\!\!dt\!\!\int\limits_{-\infty}^\infty\!\!\!dt'\> J(t)\,G(t-t')\,J(t')\Bigg)\ ,$ where $G(s)={\Gamma\over 2\tau}\,e^{-|s|/\tau} \qquad,\qquad {\widehat G}(\omega)={\Gamma\over 1+\omega^2\tau^2}$ is the Green’s function, in real and Fourier space. Note that $\int\limits_{-\infty}^\infty\!\!\!ds\>G(s)={\widehat G}(0)=\Gamma\ .$ We can now compute \begin{aligned} \blangle \eta(t\ns_1)\,\eta(t\ns_2)\brangle&=G(t\ns_1-t\ns_2)\bvph\\ \blangle \eta(t\ns_1)\,\eta(t\ns_2)\,\eta(t\ns_3)\,\eta(t\ns_4)\brangle&=G(t\ns_1-t\ns_2)\,G(t\ns_3-t\ns_4)+ G(t\ns_1-t\ns_3)\,G(t\ns_2-t\ns_4)\\ &\qquad\qquad + G(t\ns_1-t\ns_4)\,G(t\ns_2-t\ns_3)\ .\nonumber\end{aligned} The generalization is now easy to prove, and is known as Wick’s theorem: $\blangle \eta(t\ns_1)\cdots\eta(t\ns_{2n})\brangle=\sum_{contractions}\!\!\! G(t\ns_{i\ns_1}-t\ns_{i\ns_2})\cdots G(t\ns_{i\ns_{2n-1}}-t\ns_{i\ns_{2n}})\ ,$ where the sum is over all distinct contractions of the sequence $$1$$-$$2\cdots 2n$$ into products of pairs. How many terms are there? Some simple combinatorics answers this question. Choose the index $$1$$. There are $$(2n-1)$$ other time indices with which it can be contracted. Now choose another index. There are $$(2n-3)$$ indices with which that index can be contracted. And so on. We thus obtain $C(n)\equiv{\hbox{\# of contractions}\atop \hbox{of 1-2-3\,\cdots2n}} = (2n-1)(2n-3)\cdots 3\cdot 1 = {(2n)!\over 2^n\,n!}\ .$

This page titled 8.11: Appendix II- Distributions and Functionals is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.