Transverse vertices and the gauge technique

This thesis is a self-contained treatise on the application of the Gauge Technique (GT) - a non-perturbative, self-consistent method for calculating gauge field theory Green functions - to Spinor and Scalar Electrodynamics. It includes a discussion of the elements that form the method, and a thorough review of the technique's development and uses. The simplest and most primitive version of the GT determines the \longitudinal\" components of the photon-amputated connected Green functions and efficiently reproduces the exact infra-red behaviour as well as the ultra-violet behaviour of the source 2-point functions; it is only gauge covariant in these asymptotic regimes. We present a new algorithm for extending the GT beyond the simplest version. The algorithm introduces transverse corrections into the GT in an intrinsically non-perturbative way and is a significant improvement on previously advocated methods. These transverse corrections (exact to order e4) are vital for restoring gauge covariance in subasymptotic momentum regimes and do not disturb the asymptotic results."