Gauge Symmetry and its Implications for the Schwinger-Dyson Equations

Gauge theories have been a corner stone of the description of the world at the level of fundamental particles. The Lagrangian or the action describing the corresponding interactions is invariant under certain gauge transformations and contains the dependence on a covariant gauge parameter. This symmetry is reflected in terms of the Ward-Green-Takahashi (or the Slavnov-Taylor) identities (WGTI) which relate various Green functions among each other, and the Landau-Khalatnikov-Fradkin transformations (LKFT) which relate a Green function in a particular gauge to it in an arbitrary covariant one. As an outcome, all physical observables should be independent of the choice of gauge. The most systematic scheme to solve quantum field theories (QFT) is perturbation theory where the above-mentioned identities are satisfied at every order of approximation. This feature is exploited highly usefully as a verification of the results obtained after timeand effort-consuming exercises. As it stands, not all natural phenomena are realized in the perturbative regime of QFTs. Therefore, one is inevitably led to make efforts to solve these theories in a non perturbative fashion. A natural starting point for such studies in continuum are Schwinger-Dyson Equations (SDEs). One of the most undesirable features associated with their non-perturbative truncations is the loss of highly and rightly esteemed gauge invariance if special care is not taken. In this work, we present a summary of the modern approach to recover gauge invariance in a systematic manner invoking not only WGTI and LKFT but also requiring the truncation schemes to match onto the blindly trustable perturbative ones at every order of approximation. It is a highly difficult task but not impossible.