Renormalization and Quantum Correction of Soliton Solutions

Abstract

The renormalization procedure for soliton solutions in quantum systems in the framework of canonical formalism of quantum field theory is formulated. The steps of calculations are put in a systematic form and a detailed account of each step is clarified. It is shown that further renormalization is not needed in the presence of solitons; one can always obtain finite Yalues by the use of the renormalizecl quantities of the homogeneous case. § l. Introduction l\1athematical scientists have known for a long time that some non-linear clas­ sical field equations have a variety of solutions which behave as certain extended objects.]) It is clue to their particle-like behaviour that these solutions are called soliton solutions or solitary waves. In solid state physics the appearance of ex­ tended objects in various states is a common phenomenon. Due to their obvious quantum origin, these extended objects demand a quantum theory. The vortex-type solution in the Higgs mocleFl and the monopole solution in non-Abelian gauge the­ ory3l have attracted a great deal of attention among high energy particle physicists to extended objects created in field theoretical models. Though most of these analy­ ses were made at the classical level, they were aimed at a quantum theory for extended objects. There have been several attempts to study the quantum effects in soliton solutions. In Ref. 5), semiclassical quantization methods were developed using the \VKB approximation in the functional integral formalism. There the quantum. correction of the one-soliton energy was calculated in the (1 + 1) -dimensional J4/ model. The analysis began with the classical soliton solution and considered cor­ rections of order tz. A divergence appearing in the quantum correction of the energy is assumed to be cancelled by a mass counterterm. In Ref. 6) the method of collective coordinates was applied to th,e quantiza­ tion of the soliton sol uti on using the path integral formalism. The position of the soliton was treated as a collective coordinate and thus separated from the internal degrees of freedom. In Ref. 7) a soliton sector perturbation theory was developed. It was pointed out in Ref. 7a) that the ultraviolet divergence which arose in the one-loop computation of the soliton energy is removed, if one chooses the same