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Finite-dimensional Feynman Diagrams

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5. The first appearance of graphs

In the last section we calculated some 2 and 4-point functions:

<v¹,v²> = A^-1_1,2

<v¹,v¹> = A^-1_1,1

<v¹,v²,v³,v⁴> = A^-1_2,3A^-1_1,4 + A^-1_2,4A^-1_1,3 + A^-1₃₄A^-1_1,2

<v¹,v¹,v³,v⁴> = 2 A^-1_1,4A^-1_1,3 + A^-1_3,4A^-1_1,1

<v¹,v¹,v¹,v⁴> = 3 A^-1_1,4A^-1_1,1

<v¹,v¹,v⁴,v⁴> = 2A^-1_1,4A^-1_1,4 + A^-1_4,4A^-1_1,1

<v¹,v¹,v¹,v¹> = 3 A^-1_1,1A^-1_1,1

It is convenient to represent each of products appearing on the right as a graph,where the vertices represent the indices of the coordinatesvⁱ appearing in the m-pointfunction, and eachA^-1_i,jbecomes an edge from vertex ito vertex j. Here are the graphs correspondingto the terms in the 4-point functions above.

1. What every Freshman should know
2. Facts from calculus and their d-dimensional analogues
3. m-point functions
4. Wick's Theorem
5. The first appearance of graphs
6. Calculations with a potential function, ``Feynman Rules''
7. Correlation functions