On the Heisenberg Treatment of the Field Variables

Abstract

The simultaneous integral equations of the vacuum expectation values in the Heisenberg representation are proposed, by t~ . . aid of which we reduced the equations to gain the expectation values. the S~matrix and the bound state solutions. The relation of these equations with the methods of Dyson, Bethe-Salpetel' and Tomonaga :s obvious. The pmctical calculations are carried out for the system containing one Fermion and the neutral s"lar mesons. The application to bther i;eneral systems may be readily possible. § 1. IntroductionRecently the progress in the formal treatment of the field theory seems to aim for the two directions. One of them is the covariant description of the field theory. It was advanced by the treatment in the interaction representation by Tomonaga,') Schwinger 2 ) and Dyson,'i) and has gained many successes for the problems of the relativistic phenomena and the relativistic subtraction. The description in the interaction representation is connected tightly with the perturbation method, because it gives the special roles to the free fields.Another direction of the formal progress is the attempt to overcome the perturbation method. It is well known that in the nucleon-1!" meson system as their mutual interaction is rather strong, the perturbation method is not suitable. As to this difficulty, the treatment based on the Fock's Schroedinger equations was proposed by Tomonaga 4 ) for the system of one ,l).udeon and by Tamm 5 ) and Dancoff6) for the system of two nucleons. Recently Levy has applied the .extended Tamm-Dancoff method to the deuteron problem and gained very reasonable results. Thus the treatment In the Schroedinger representation is found to be effective for overcoming the limit of the perturbation method. But· its non-covariant e~ression is incgnvenient for the relativistic invariant treatment. Especially for the high energy phenomena the covariant treatment will be unavoidable.

A Master Formula for Chiral Symmetry Breaking

“…The relations (14)(15) simply express the invariance of the S-matrix under gauge transformations connected with the identity, since (16)(17) are the generators of the local SU(2) × SU(2) symmetry:…”