Operator Calculus in Quantized Field Theory

Abstract

Following the suggestion of Feynman, we analyze the operators corresponding to the el~tron. positron field by means of the operator calculus: Thus we can deduce operationally Feynman's theory from the second quantized theory. With the aid of the external spinors, which are considered as the analogues of the Schwinger's prescribed source, we obtain formally a closed form of the S·matrix. Using this result we can derive the dilfusion equation for the Dirac field very simply. The relations to the Schwinger's new theory of Green function are also discussed. Finally the formula for multiple boson or fermion pair.producrion are presented as illustrations for applications. § 1. Introduction and summary Recently Schwinger1) and many others have made attempts to obtain the closed forms of the S·matrix and the Green's functions of quantized fields, avoiding the use of the perturbation procedures. We have here discussed a closed form of the S·matrix formally, at least in the case of the external electromagnetic field, with the aid of the operator calculus. The calculus of quantized operator has been, originally and extensively, used by Feynman2), but we adopt here its developed scheme given by Fujiwara3). Using the operator calculus, in this paper two matters are intended; one is the proof of the equivallence of Feynman's theory and the usual quantized field theory of Tomonaga· Schwinger, and the other is the above mentioned closed form of the S·matrix. The first problem has already been discussed by many authors, Dyson proved this equivallence in the frame of perturbation expansion and Feynman's proof was attained by means of his operator calculus. But Feynman did not take up the quantized Dirac field, and used the results of his own positron theory, and carried -out the of the radiation field only. We now here disentangle the quantized Dirac field operators and provide the direct proof of the above equivallence operationally. Our aim is to deduce consistently as many results as possible from the definition of the transformation operator U and the commutation rela­ tions between the field variables. It means that without explicit applications of physical considerations except basic prescriptions set up above, we resort only to the means of the mathematical procedures of disentangling -avoiding the expansion with coupling constant-: for example, for one-body kernel of which we know only its definition: o (P(sif; (1) if; (2» )oE (1, 2), we seek the integral equation which it satisfies. The program was suggested by Feynman2), but contrally to his conjecture the analysis of this problem does not throw any new light on the problem of disentanglement of the form J~exp[i(rp+m) WJdW~ because, although in this form anticommuting quantity r