Heisenberg Operators in Quantum Electrodynamics. I

Abstract

A new analytic continuation principle is described, by means of which the calculation of matrix elements of operators in any quantized field theory is greatly simplified. By a Heisenberg is meant an average over a finite space-time region of a field operator in the representation of the theory. The analytic continuation is made by varying the characteristic masses of the fields through real values. In this way a operator with the physically occurring masses is derived from an operator calculated with very large fictitious masses. In the region of large masses, where real creation of particles is impossible, the operator is identical with the $S$-matrix for a suitably chosen scattering problem. The calculation reduces to the calculation of an $S$-matrix, to which the techniques of Feynman are directly applicable.The $S$-matrix itself is a nonanalytic function of the masses in the region of values where thresholds for real processes occur. A special device is introduced in order to bypass the region of nonanalyticity. The operator is modified by the insertion of real exponential damping factors which have the effect of making all energy denominators in a perturbation expansion complex. The modified operator is an analytic function of the masses and of the damping coefficients, for all real positive values of the masses. The analytic continuation is made by varying the masses while the damping coefficients are non-zero, letting the damping coefficients tend to zero when the physical values of the masses have been reached.