Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f ( x ) and j μ ( x ) are explicitly expressed as local limits of sums of non‐local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term : A 3 ( x ) :.The corresponding field equations are used to derive renormalized Dyson‐Schwinger‐type integral equations for the renormalized proper part functions ∑, II μν , Λ μ , and X αβγδ (the four‐photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m ) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m , o ) = X (O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j μ ( x ) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.