Prog. Theor. Phys. Vol. 12 No. 5 (1954) pp. 585-602

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On the Renormalization in Tamm-Dancoff Approximation for One-nucleon Problem, II

— Subtraction of Divergences in the Generalized Tamm-Dancoff Equations —

Kiyomi Itabashi

Physical Institute, Faculty of Science, Tohoku University

(Received July 17, 1954)

Abstract:

The subtraction of the divergences are carried out in the generalized Tamm-Dancoff equations derived in the previous paper (Part I). The essence of the method is the extension of Fubini's procedure to the arbitrarily higher order approximation. That is, first, we construct the formal solutions which satisfy the equations in question; they contain the infinities and accordingly are quite meaningless. Then, we separate the infinities (including the overlapping divergences) individually at each stage of the construction of these formal solutions. The forms of the nucleon propagation function or the vertex parts which have been made convergent by this method depend on the configurations to which they rafer. However, this seems to be inevitable for the present approximation method, i. e. the reduction of the infinite set of coupled integral equations to the finite one.

URL : http://ptp.ipap.jp/link?PTP/12/585/
DOI : 10.1143/PTP.12.585

References:

K. Itabashi, Prog. Theor. Phys. 12 (1954), 494 [PTP](in the following, referred to as Part 1).
S. Fubini, Nuovo Cim. 10 (1953), 851.
However, he has mentioned nothing as regards the method for the subtraction of the overlapping divergences.
This problem was solved by S. Chiba and H. Tanaka and D. Ito (references 3) and 4)).
S. Chiba, Soryushiron Kenkyu (Mimeographed Circular in Japanese) 6 (1954), 855; Prog. Theor. Phys. 11 (1954), 494[PTP].
Y. Tanaka and D. Ito, Soryushiron Kenkyu (Mimeographed Circular in Japanese) 6 (1954), 869.
D. Ito and H. Tanaka, Prog. Theor. Phys. 11 (1954), 501[PTP].
M. M. Lévy, Phys. Rev. 94 (1954), 460, [APS]
has also investigated the covariant treatment of pion-nucleon scattering and its renormalization, starting from the Salpeter-Bethe integral equation with the expansion in powers of g² or the “irreducible” kernels.
A. Salam, Phys. Rev. 82 (1951), 217[APS].
In this connection, see also T. Yoshimura, Prog. Theor. Phys. 11 (1954), 224[PTP].

Citing Article(s) :

Progress of Theoretical Physics Vol. 14 No. 2 (1955) pp. 151-165 :
On the Renormalization of Heisenberg Treatment

Ichie Watanabe