Finite-dimensional Feynman Diagrams

7. Correlation functions

The way path integrals are used in quantum field theory is, very roughly speaking, that the probability amplitude of a process going from point v₁ to point v₂ is an integral over all possible ways of getting from v₁ to v₂. In our finite-dimensional model, each of these ``ways'' is represented by a point v in Rⁿ and the probability measure assigned to that way is $\frac{1}{Z_U}\exp(-{\scriptstyle\frac{1}{2}}{\bf v}^tA~{\bf v} +\hbar U({\bf v}))v^1v^2~d{\bf v}$ . The integral is what we called before a 2-point function

$\displaystyle <v^1,v^2> = \frac{1}{Z_U}\int_{{\bf R}^d}d{\bf v}~ \exp(-{\scriptstyle\frac{1}{2}}{\bf v}^tA~{\bf v} +\hbar U({\bf v}))v^1v^2,$

and what we will now call a correlation function.

We continue with the example of the cubic potential

$\displaystyle U({\bf v}) = \sum_{i,j,k} u_{ijk}v^iv^jv^k$ .

By our previous calculations,

$\displaystyle <v^1,v^2> = \frac{1}{Z_U}\partial_1\partial_2 \exp(\hbar \sum_{i,... ...iptstyle\frac{1}{2}}{\bf b}^tA^{-1}{\bf b})) _{\textstyle \vert _{{\bf b} =0}}.$

In terms of Wick's Theorem and our graph interpretation of pairings, this becomes:

$\displaystyle \sum_G \frac{\textstyle \hbar^n}{\textstyle \vert{\rm Aut~}G\vert... ... edge~labellings} \prod_v u_{\rm vertex~label} \prod_e A^{-1}_{\rm edge~label},$

where now the sum is over all graphs G with two single-valent vertices (the ends) labeled 1 and 2, and n 3-valent vertices.

This graph occurs in the calculation of the coefficient of $\hbar^6$ in <v¹,v²>.

The k-point correlation functions are similarly defined and calculated. Here is where we begin to see the usual ``Feynman diagrams.''

This graph occurs in the calculation of the coefficient of $\hbar^2$ in <v¹,v²,v³,v⁴>.