Green's Functions in Quantum Electrodynamics

Dyson has shown that the evaluation of the $S$ matrix for quantum electrodynamics can be reduced to the problem of evaluating certain quantities, $S^{\prime F}$, $D^{\prime F}$, and ${\Gamma }_{\mu }$. By making use of a formula relating the $T$ product of an operator with its corresponding $N$ product, integro-differential equations for $S^{\prime F}$ and $D^{\prime F}$ are obtained. These equations are identical in form with those given by Schwinger for his Green's functions, and hence it is concluded that the two formalisms are equivalent. In addition it is shown that all of the quantities introduced by Schwinger can be expressed in terms of a single quantity, $S_{\mathrm{vac}}$, the vacuum expectation value of the $S$ matrix. The renormalization problem is not discussed.