Some Converging Examples of the Perturbation Series in the Quantum Field Theory

Abstract

Properties of the Hamiltonian operator of the quantized field is studied in the framework of the theory of Hilbert space. The fixed source theory and the boson-fermion interaction are mainly investigated in the cases of both discrete and continuous spectra. The total Hamiltonian operator is defined first in a domain dense in the Hilbert space, under the con· clition that the interaction form factor in the momentum space is square integrable. Then it is shown that the total Hamiltonian as a self-adjoint operator can be determined in terms of the perturbation series for every finite value of the coupling constant if the boson mass is not zero and that it is the unique self-adjoint extension of the symmetric operator defined initially. For the boson-fermion interaction with continuous spectrum, some additional con· clition on the interaction form factor is needed. Further the perturbation series of the S-matrix element for scattering is discussed in the framework of the wave packet for· mulation. In the quantum :field theory the problem of the convergence of the pertur­ bation series expressing the solution of the fundamental equation is of both practical and theoretical importance and has been studied by many authors from various points of view. The difficulty of the diverging integral which appears in each term of the perturbation series was successfully removed by the method of renormalization. · In this note we study the other important problem, that is, the summability of the power series, introducing from the beginning a so-called form factor in the interaction Hamiltonian and making each term of the series finite without using the renormalization technique. Some attempts have been made in this direction. For scalar boson :field if;(x) with self-interaction J.q}, it was shown 1 > that the Green function or the S-matrix can never be written in a power series of )., because the number of the Feynman diagrams in the n-th order is roughly proportional to n! and the lower bound of the absolute value of each matrix element corresponding to the Feynman diagram in that order is an with a being some positive number independent of n, and a fabulous cancellation of the contributions within the same order does not occur. For the boson field interacting with the fermion current, a considerable cancellation between the various terms seems to happen/> al