0 QED in the Worldline Formalism

A survey is given of applications of the “string-inspired” worldline formalism to the computation of amplitudes and effective actions in QED. INTRODUCTION: QED IN FIRST QUANTIZATION The “worldline” or “string-inspired” formalism is an alternative to the usual second-quantized formalism in quantum field theory based on relativistic particle path integrals. Although it was invented by Feynman [1,2] simultaneously with modern relativistic second-quantized QED, until recently it was only occasionally used for actual computations [3]. During the last few years, however, certain computational advantages of the first-quantized approach were recognized which led to a sizeable number of nontrivial applications. Those recent developments were triggered by the work of Bern and Kosower, who derived new rules for the construction of one-loop QCD amplitudes from the infinite string tension limit of first-quantized string path integrals [4]. Strassler then showed [5] that the corresponding formulas for the QED case can also be derived from Feynman’s first quantized path integrals. For the case of scalar QED, the basic formula is given in the appendix A of [1]. It states that the amplitude for a charged scalar particle to move, under the influence of the external potential Aμ, from point xμ to x ′ μ in Minkowski space is given by ∫ ∞ 0 ds ∫ x(s)=x x(0)=x Dx(τ)exp ( − 2 ims ) exp [ − i 2 ∫ s 0 dτ( dxμ dτ ) 2 − i ∫ s 0 dτ dxμ dτ Aμ(x(τ)) − i 2 e ∫ s 0 dτ ∫ s 0 dτ ′ dxμ dτ dxν dτ ′ δ μν + (x(τ)− x(τ )) ] (1) That is, the amplitude can be constructed as a path integral over the set of all open trajectories running from x to x in the fixed proper time s. The action consists of 1) Talk given at QED 2000, 2nd workshop on “Frontier Tests of Quantum Electrodynamics and Physics of the Vacuum”, Trieste, Italy, Oct. 5-11, 2000. the familiar kinetic term, and two interaction terms. Of those the first represents the interaction with the external field, to all orders in the field, while the second one describes an arbitrary number of virtual photons emitted and re-absorbed along the trajectory of the particle. Here δ + denotes the photon propagator, e.g. in Feynman gauge, δ + (x, x ) = g μν (x−x′)2 . In second quantized field theory this amplitude would thus correspond to the infinite sequence of Feynman diagrams shown in fig. 1.