Non-perturbative gauge transformations of arbitrary fermion correlation functions in quantum electrodynamics

Abstract

We study the transformation of the dressed electron propagator and the general N -point functions under a change in the covariant gauge of internal photon propagators. We re-establish the well known Landau-Khalatnikov-Fradkin transformation for the propagator and generalise it to arbitrary correlation functions in configuration space, finding that it coincides with the analogous result for scalar fields. We comment on the consequences for perturbative application in momentum-space.

The Generalised LKF Transformations for Arbitrary $N$-point Fermion Correlators

“…which is the direct generalisation of (2) to the spinor case reported in [62] (we emphasise that the exponential factor is identical to the scalar result). Moreover, we shall show that the exponential prefactor, once summed over k and l, is independent of the permutation, and thus factorises out of the sum over partial amplitudes, giving a simple multiplicative transformation for the N -point correlator itself, denoted S, S (x 1 , .…”

“…The main contribution of this paper is to provide additional details that prove the generalisation of the LKF transformation of the propagator to arbitrary correlators, expanding upon the results reported in [62]. To this end we generalise the propagator, that corresponds to the field theory correlator Ψ(x)Ψ(x ) , to the correlator of an arbitrary even number, N = 2n, of fields, Ψ(x 1 )…”

“…This corresponds to the transformation of the propagator caused by a change of gauge in the internal photon propagators that couple to the worldline trajectories. Aside from this there are two terms in (46) involving the source that provide the term denoted by ∆ ξ I M in [62]. It can be split up into the sum of…”

“…We are motivated by several main goals. Firstly this article expands upon the brief report of the main results given in [62] where we sought to compare the forms of the generalised transformation of the N -point functions between spinor and scalar QED; secondly the theoretical developments presented here for spinor QED are far from trivial and will serve as a stepping stone to the more complicated transformations in QCD or more general gauge theories (worldline techniques have been extended to the non-Abelian case in a series of recent articles [92][93][94][95][96][97]); finally a systematic study of the variation of N -point functions under a change of gauge is crucial for understanding how gauge invariant information can be extracted from calculations carried out in particular gauges -we have in mind, for instance, the truncation of Dyson-Schwinger equations to a particular order or numerical evaluation of such quantities on the lattice. Our use of the worldline formalism will be seen to simplify both the derivation of the LFK transformations and their implementation in perturbation theory.…”

“…which is the natural generalisation of (1), constructed now from functions ∆ ξ S (k,l) iπ to be defined below in (22). This same worldline formalism was recently applied to extend this work to the case of spinor QED [62]. In this companion paper we provide further calculational details on this worldline derivation of the LKF transformation of the fermion propagator and the N -point fermionic Green functions that enter the calculation of scattering amplitudes in perturbation theory and discuss various applications of these recent results.…”