Finite-dimensional Feynman Diagrams
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7. Correlation functions
The way path integrals are used in quantum field theory is, very roughlyspeaking, that the probability amplitude of a process going from point v1to point v2is an integral over all possible ways of getting fromv1 tov2. In our finite-dimensional model, each of these``ways'' is represented by a point v in Rn andthe probability measure assigned to that way is . The integral is what we called before a 2-point function
and what we will now call a correlation function.
We continue with the example of the cubic potential
By our previous calculations,
In terms of Wick's Theorem and our graph interpretation of pairings, thisbecomes:
where now the sum is over all graphs Gwith two single-valent vertices (the ends)labeled 1 and 2, and n 3-valent vertices.
This graph occurs in the calculation of the coefficientof in <v1,v2>.
The k-pointcorrelation functions are similarly defined and calculated. Hereis where we begin to see the usual ``Feynman diagrams.''
This graph occurs in the calculation of the coefficient of in<v1,v2,v3,v4>.