The intention of the present paper is to build the Dirac theory of the positron into the formalism of quantum electrodynamics. A requirement here is that the symmetry of nature in the positive and negative charges should from the very beginning be expressed in the basic equations of theory, and further that, except for the divergences which are caused by the well-known difficulties of quantum electrodynamics, no new infinities appear in the formalism, i.e. that the theory provides an approximation method for treating the group of problems which could also be treated according to the previous quantum electrodynamics. The latter postulate distinguishes the present attempt from the investigations of Fock, Oppenheimer and Furry, and Peierls, which it otherwise resembles; it is here rather closely connected with a paper by Dirac. Compared to Dirac's treatment, this paper emphasizes the significance of the conservation laws for the total system of radiation and matter, and the necessity of formulating the basic equations of the theory in a manner extending beyond the Hartree approximation.