Formulating Schwinger-Dyson Equations for Qed Propagators in Minkowski Space

The Schwinger-Dyson equations (SDEs) are coupled integral equations for the Green\u27s functions of a quantum field theory (QFT). The SDE approach is the analytic nonperturbative method for solving strongly coupled QFTs. When applied to QCD, this approach, also based on first principles, is the analytic alternative to lattice QCD. However, the SDEs for the n-point Green\u27s functions involves (n+1)-point Green\u27s functions (sometimes (n+2)-point functions as well). Therefore any practical method for solving this infinitely coupled system of equations requires a truncation scheme. When considering strongly coupled QED as a modeling of QCD, naive truncation schemes violate various principles of the gauge theory. These principles include gauge invariance, gauge covariance, and multiplicative renormalizability. The combination of dimensional regularization with the spectral representation of propagators results in a tractable formulation of a truncation scheme for the SDEs of QED propagators, which has the potential to preserve the aforementioned principles and renders solutions obtainable in the Minkowski space. This truncation scheme is the main result of this dissertation

How gauge covariance of the fermion and boson propagators in QED constrain the effective fermion-boson vertex

We derive the gauge covariance requirement imposed on the QED fermion-photon three-point function within the framework of a spectral representation for fermion propagators. When satisfied, such requirement ensures solutions to the fermion propagator Schwinger-Dyson equation (SDE) in any covariant gauge with arbitrary numbers of spacetime dimensions to be consistent with the LandauKhalatnikov- Fradkin transformation (LKFT). The general result has been verified by the special cases of three and four dimensions. Additionally, we present the condition that ensures the vacuum polarization is independent of the gauge parameter. As an illustration, we show how the gauge technique dimensionally regularized in four dimensions does not satisfy the covariance requirement

Gauge covariance of the fermion Schwinger-Dyson equation in QED

Any practical application of the Schwinger-Dyson equations to the study of n-point Green\u27s functions in a strong coupling field theory requires truncations. In the case of QED, the gauge covariance, governed by the Landau-Khalatnikov-Fradkin transformations (LKFT), provides a unique constraint on such truncation. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the LKFT are linear operations on the spectral densities. We formally define these group operations and show with a couple of examples how in practice they provide a straightforward way to test the gauge covariance of any viable truncation of the Schwinger-Dyson equation for the fermion 2-point function. (C) 2017 The Author(s). Published by Elsevier B.V

Landau-Khalatnikov-Fradkin transformation for the fermion propagator in QED in arbitrary dimensions

We explore the dependence of fermion propagators on the covariant gauge fixing parameter in quantum electrodynamics (QED) with the number of spacetime dimensions kept explicit. Gauge covariance is controlled by the Landau-Khalatnikov-Fradkin transformation (LKFT). Utilizing its group nature, the LKFT for a fermion propagator in Minkowski space is solved exactly. The special scenario of 3D is used to test claims made for general cases. When renormalized correctly, a simplification of the LKFT in 4D has been achieved with the help of fractional calculus