8.11: Appendix II- Distributions and Functionals

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\( \newcommand\VTheta{\vec\Theta}\)
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\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
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\( \newcommand\BA{\mib A}\)
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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.

[Fdiscretize] Discretization of a continuous function \(\eta(t)\). Upon discretization, a functional \(\Phi\big[\eta(t)\big]\) becomes an ordinary multivariable function \(\Phi(\{\eta_j\})\).

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\,$\cdots$$2n$}} = (2n-1)(2n-3)\cdots 3\cdot 1 = {(2n)!\over 2^n\,n!}\ .\]