Abstract
In the first-quantised worldline approach to quantum field theory, a long-standing problem has been to extend this formalism to amplitudes involving open fermion lines while maintaining the efficiency of the well-tested closed-loop case. In the present series of papers, we develop a suitable formalism for the case of quantum electrodynamics in vacuum (part one and two) and in a constant external electromagnetic field (part three), based on second-order fermions and the symbol map. We derive this formalism from standard field theory, but also give an alternative derivation intrinsic to the worldline theory. In this first part, we use it to obtain a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N -photon kernel,” where off-shell this kernel appears also in “subleading” terms involving only N − 1 of the N photons. Although the parameter integrals generated by the master formula are equivalent to the usual Feynman diagrams, they are quite different since the use of the inverse symbol map avoids the appearance of long products of Dirac matrices. As a test we use the N = 2 case for a recalculation of the one-loop fermion self energy, in D dimensions and arbitrary covariant gauge, reproducing the known result. We find that significant simplification can be achieved in this calculation by choosing an unusual momentum-dependent gauge parameter.
Highlights
With the modern diagrammatic approach to perturbative QED, in the early fifties Feynman developed a representation of the QED S-matrix in terms of first-quantised relativistic particle path integrals [1, 2]
In the present series of papers, we develop a suitable formalism for the case of quantum electrodynamics in vacuum and in a constant external electromagnetic field, based on second-order fermions and the symbol map
The parameter integrals generated by the master formula are equivalent to the usual Feynman diagrams, they are quite different since the use of the inverse symbol map avoids the appearance of long products of Dirac matrices
Summary
With the modern diagrammatic approach to perturbative QED, in the early fifties Feynman developed a representation of the QED S-matrix in terms of first-quantised relativistic particle path integrals [1, 2]. The basic idea is quite simple, and was germinally presented already in [11]: by suitable series expansions, the path integrals are reduced to Gaussian ones, and evaluated by formal Gaussian integration as in a one-dimensional field theory, using appropriate “worldline Green’s functions.” In this formalism the calculation of the one-loop N -photon amplitude in scalar QED, starting from the path integral representation (1.1), proceeds as follows: after the above expansion of the interaction exponential, and truncation to N th order, the amplitude is represented as. Ημ is an external Grassmann Lorentz vector, and the “symbol map,” symb, converts products of η’s into fully antisymmetrised products of Dirac matrices; we will discuss the details in section 3 below Following this we perform the usual projection onto an N -photon background, and use the path integral representation (1.39) to derive master formulas for the N -photon kernel K both in configuration and in momentum space. The forthcoming second part of this series will be devoted to the use of the master formulas derived here for on-shell calculations such as cross-sections, the third part to the inclusion of a constant electromagnetic background field
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
