$m$-point Functions

For any choice of $m$ (not necessarily different) indices $i_1, \dots, i_m$ between 1 and $d$, define the $m$-point function $\< v^{i_1}, \dots , v^{i_m}\>$ as follows:

$$\< v^{i_1} ,\dots , v^{i_m}\> = \frac{1}{Z_0}\int_{{\bf R}^d} d{\... ...} ~~\exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v})v^{i_1},\dots , v^{i_m}.$$

The $m$-point functions are a step towards the ultimate aim of our calculation. They enter at this moment because they can be calculated by repeated differentiation of $Z_{\bf b}$.

For example, note that

$$\frac{\partial Z_{\bf b}}{\partial b^i} = \frac{\partial}{\partia... ...f v} ~~\exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) =$$

$$\int_{{\bf R}^d} d{\bf v} ~~ \frac{\partial}{\partial b^i}\exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) =$$

$$\int_{{\bf R}^d} d{\bf v} ~~ \exp({\scriptstyle\frac{ 1}{ 2}}{\bf v}^tA~{\bf v} + {\bf b}^t{\bf v}) v^i.$$

So the 1-point function $\< v^i\>$ is given by

$$\< v^i\> = \frac{1}{Z_0} \frac{\partial Z_{\bf b}}{\partial b^i}\vert _{{\bf b} =0}.$$

Similarly the $m$-point function $\< v^{i_1}, \dots , v^{i_m}\>$ is given by

$$\< v^{i_1} ,\dots , v^{i_m}\> =\frac{1}{Z_0} (\frac{\partial}{\pa... ...\frac{\partial}{\partial b^{i_m}}Z_{\bf b})_{\textstyle \vert _{{\bf b} =0}} =$$

$$\frac{\partial}{\partial b^{i_1}}\cdots \frac{\partial}{\partial ... ...criptstyle\frac{1}{2}}{\bf b}^tA^{-1}{\bf b})_{\textstyle \vert _{{\bf b} =0}}.$$