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By Alain Connes and Mathilde Marcolli

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We also write the term j∈X Mj in the form Mj = Mnι , The cardinality of X is n. / nι ! from the binomial expansion of ( I M)n . This gives an overall denominator δ = nι ! )nι , where dι is the order of the corresponding monomial M. 75) that respect the map X → I. In particular, this group ∆ acts on the pairings and the orbit ∆(π) of a given pairing gives all pairings of the same type. 81) σ(Γ) = #{g ∈ ∆ | g(π) = π}. 82) σ(Γ) = #Aut(Γ). 3. FEYNMAN DIAGRAMS 47 It accounts for repetitions, as usual in combinatorics.

Such a JE can be uniquely determined in perturbation theory. In the case of free fields, with S(φ) = S0 (φ), the effective action is the same as the original action, with the possible addition of an irrelevant constant. In the general case, where an interaction term Sint (φ) is present in the action, the effective action Seff (φ) is a non-linear functional of the classical fields, which constitutes the basic unknown in a given quantum field theory. The key result on the effective action is that it can be computed by dropping all graphs that can be disconnected by the removal of one edge, as in the case illustrated in Figure 12, where the shaded areas are a shorthand notation for an arbitrary graph with the specified external leg structure.

Even for the simplest case of the two-point Green’s function of the free field φ˜F some care is required in order to get the right answer. 32) GF2 (x, y) = N0 φ(x)φ(y) exp (i S0 (φ)) D[φ]. As a function of the classical field φ, the expression φ(x)φ(y) only depends upon the two-dimensional projection of φ but one cannot yet use the factorization property of the Gaussian integral since S0 (φ) involves the derivatives of φ. 34) S0 (φ) = (2π)−D 1 2 ˆ φ(−p) ˆ (p − m2 ) φ(p) dD p. 2 ˆ The reality condition for φ(x) shows that the φ(p) are not independent variables but fulfill the condition ˆ ˆ φ(−p) = φ(p).

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