QED in the worldline representation

Abstract

Abstract. Simultaneously with inventing the modern relativistic formalism of quantum electrodynamics, Feynman presented also a first-quantized representation of QED in terms of worldline path integrals. Although this alternative formulation has been studied over the years by many authors, only during the last fifteen years it has acquired some popularity as a computational tool. I will shortly review here three very different techniques which have been developed during the last few years for the evaluation of worldline path integrals, namely (i) the "string-inspired formalism", based on the use of worldline Green functions, (ii) the numerical "worldline Monte Carlo formalism", and (iii) the semiclassical "worldline instanton" approach.Keywords: Quantum electrodynamics, perturbation theory, worldline, string inspired formalism PACS: 11.15. Bt,11.15.Kt,11.25.Db,12.20.Ds FEYNMAN'S WORLDLINE REPRESENTATION OF QEDIn 1950 Feynman presented, in an appendix to one of his groundbreaking papers on the modern, manifestly relativistic formalism of perturbative QED [1], also a first-quantized formulation of scalar QED, "for its own interest as an alternative to the formulation of second quantization". There he provides a simple rule for constructing the complete scalar QED S-matrix by representing virtual scalars and photons in terms of relativistic particle path integrals, and coupling them in all possible ways. Restricting ourselves, for simplicity, to the purely photonic part of the S-matrix (no external scalars), and moreover to the "quenched" contribution (only one virtual scalar), this "worldline representation" can be given most compactly in terms of the (quenched) effective action Γ[A]:Here T denotes the proper-time of the scalar particle in the loop, m its mass, and x(T )=x(0) Dx(τ) a path integral over all closed loops in spacetime with fixed periodicity in the proper-time. The worldline action S[x(τ)] has three parts,(see fig. 1). Of these, the kinetic term S 0 describes the free propagation of the scalar, S ext its interaction with the external field, and S int the corrections due to internal photon exchanges in the loop. The connection to a standard Feynman diagrammatic description is made simply by expanding out the two interaction exponentials.

Second order theory of $(j,0)\oplus (0,j)$ single high spins as Lorentz tensors

“…Second order fermion approaches have traditions in field theory [7], [8], and are of growing popularity in QED as well as in QCD [19], [20], [21].…”