Integral Representation of Absorptive Part of Vertex Function

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Abstract
On the basis of the Lorentz invariance, local commutativity and mass spectral conditions, it is shown that the absorptive part of the vertex function A(z1, z2, σ2) has the integral representation in the form A(z1, z2, σ2) = \intdm1dm2dm3 φ(σ, m1, m2, m3) Ap (z1, z2, σ2; m1, m2, m3), provided that z1 and z2 are real negative, where Ap is that of the lowest order perturbation theory and mi is the mass of the virtual particle. The vanishing region of the weight function φ is determined by the mass spectral conditions. As an immediate consequence of this representation, the usual proof of the dispersion relation of the vertex function is given. If we add the information derivable from the perturbation theory to this representation, we can say that the dispersion relation always holds and the threshold is not lower than the lowest threshold of the vertex function in the lowest order perturbation theory which satisfies the mass spectral condition. It seems to us that Jost's example has not this integral representation. Finally it is conjectured that the non-vanishing region of the weight function is narrowed by introducing the conservation of the nucleon number.

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