### On the Relativistically Improved Integration in Perturbation Theory

Hiroomi Umezawa, Rokuo Kawabe
1951 Progress of theoretical physics
Previously,I)we proposed a new method to integrate in relativistically invariant manner the integral appearing in perturbation calculation. Hitherto, in various problems of the quantum field theory, the customary perturbation method has frequently given results evidently destroying the relativistic covariance; for example, in the problem of the self-energy of a moving electron. But, since the syste,m of quantum field theory is of a relativistically invariant structure, it must be concluded that
more » ... the cause of the failure of relativistic covariance lies in the course of the perturbation calculation. Our new calculation method (w-method) remedies this defect of perturbation calculation and gives results having the correct relativistic covariance. The advantage of the w-method are as follows; results having the correct relativistic covariance can be obtained; the calculation becomes much easier and the relation to the variouc; results obtained by means of usual perturbation calculation can be seen explicitly. However, we apply this method to the usual perturbation formalism which has not the covariant form. Of course, such patchy method will soon see its min as his own natural fate. It comes with the appearance of Feynman's theory!(2) But available parts of the past theory always still remain, and moreover, the perturbation calculation in the quantized field theory is a relativistic invariant method in spite of its non-relativistic form. In the calculation in Feynman theory, w-method can also be used and its meaning becomes clear. In the w-method, the domain 'of integral is determined in relativistically invariant manner. Considering the physical meaning of this method determining the domain, w-method is easily introduced into Feynman theory. § 2. 'w-method in Feynman theory. For simplicity, we consider the case when particles with jJ"., k". (mass ~, ..r;r; respectively) are created, annihilating a particle with energy momentum 4-vecter t". (m.ass Ie).