GAUGE INVARIANCE IN QUANTUM ELECTRODYNAMICS

A formulation of quantum electrodynamics based on finite local field equations is employed in order to prove and discuss the gauge invariance of the theory in a meaningful and rigorous way. The Dirac and Maxwell equations have the usual forms except that the current operators f(x) and jμ(x) are explicitly expressed as finite local limits of sums of nonlocal field products and suitable subtraction terms. The electric current, for example, involves the terms A(x) and : A3(x):. The field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions Σ, Πμν, Λμ, and Xαβγδ (the four-photon vertex function), etc. Application of the boundary conditions Σ(p = m) = Σ′(p = m) = Π(0) = Π′(0) = Π″(0) = Λ(p = m, 0) = X(0, 0, 0, 0) = 0 is shown to completely specify the current operators. It is shown that the theory is gauge invarian in the sense that the divergence conditions kμΠμν(k) = kαXαβγδ(k,...) = 0, etc. and the generalized Ward identities kμΛμ(p, k) = eΣ(p - k) - eΣ(p), etc. are all satisfied in each order or perturbation theory. This is shown to be equivalent to the invariance of the field equations under gauge transformations of the second kind. © 1969.