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Finite-dimensional Feynman Diagrams


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3. m-point functions

For any choice of m (not necessarily different) indices i1, ... , im between 1 and d, define the m-point function $ <v^{i_1}, \dots , v^{i_m}>$ as follows:

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

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 Zb

For example, note that

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

$\displaystyle \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}) = $

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

So the 1-point function <vi> is given by

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

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

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

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