Graph Theory and Feynman Integrals (Mathematics and Its by Nakanish

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SalT)-1 d° (3-18) PT)I = - (s/ T)-'dT (3-19) Proof Any row of W can be expressed as a linear combination of rows of 9T because of Theorem 2-3 together with Definition 2-12 and Theorem 2-26. 'f = V'T. Since WT is nonsingular because of (3-7), (3-18) follows from (3-20). d. 3-2 Topological Formulas In this subsection, we calculate determinants of some matrices associated with a connected graph G. Since they are expressed as algebraic formulas on which some topological structures of G reflect directly, those formulas are called topological formulas.

4-3). We write one vertex of G inside each of those domains and link two vertices Fig. 4-3 An example of constructing a dual graph. Dotted lines indicate the lines of the dual graph. , circuits) of the corresponding two domains of G have r common lines of G. In particular, cut-lines in G correspond to loop lines in U. Since in general G may have another set of ,u(G) independent circuits satisfying the condition stated in Theorem 4-1, dual graphs of G will not necessarily be unique. 3 Nakanishi 34 GRAPH THEORY AND FEYNMAN INTEGRALS As seen from the above intuitive definition, a dual graph exists if and only if G is planar.

The axiomatic field theory [9] is a theory which starts from postulates such as the invariance under the Poincare group, the Hilbertspace properties of states, and microcausality and spectral conditions. In the axiomatic field theory, neither the existence of a Lagrangian nor detailed operator properties of fields are assumed. Therefore this theory cannot describe the system completely, but from it we can obtain certain amounts of information about the general framework of the quantum field theory.

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