Nonlocal Quantum Electrodynamics Admits a Finitely Induced Gauge Field Action

The Letter reconsiders a result obtained by Chrétien and Peierls in 1954 within nonlocal QED in 4D [Proc. Roy. Soc. London A 223, 468]. Starting from secondly quantized fermions subject to a nonlocal action with the kernel [i 6∂x a (x)− m b (x)] and gauge covariantly coupled to an external U(1) gauge field they found that for a = b the induced gauge field action cannot be made finite irrespectively of the choice of the nonlocality a (= b). But, the general case a 6= b naturally to be studied admits a finitely induced gauge field action, as the present Letter demonstrates. ∗E-mail: kjsch @ qft.physik.uni-leipzig.d400.de Despite its inherent difficulties nonlocal quantum field theory has received continued although changing attention in past decades (for a review and references see [1]–[3]). Restricting ourselves to gauge theories let us mention that in recent years work in nonlocal theories has been performed within quantum field theory [4]–[8] as well as in the neighbouring field of the study of effective Lagrangians in nuclear theory [9]–[11]. The conceptual intentions in the few references selected are quite diverse, but they have in common to refer and to relate to earlier work of Chrétien and Peierls dealing with nonlocal QED in 4D [12]. This investigation explored a special Ansatz for nonlocal QED with the aim to remove the UV divergencies present in standard, local QED (for the Ansatz see Eq. (1) below, the special choice of Chrétien and Peierls was a = b, a(= b) arbitrary else). But, the attempt failed (see also [13]). In particular, this negative result is believed to have closed the door to a finitely induced gauge field action. However, because Chrétien and Peierls extended their consideration not to the most general setting possible (a 6= b), this belief is not correct in general. Contrary to this belief, in this Letter we will demonstrate that nonlocal QED in 4D admits a finitely induced gauge field action under appropriate choice of the nonlocal Ansatz applied. The purpose of the present Letter therefore is a purely technical one. Conceptual questions the calculation below grew from are dealt with in a separate comprehensive paper [14]. Consider within QED in 4D Minkowski space the following gauge covariant fermion action.