Operator and generalized gauge transformations in quantum electrodynamics

Abstract

Previous studies on the relations between Green’s functions are generalized. In the majority of the published literature dealing with this problem, gauge transformations are defined through the change of a generating functional, or equivalently, through certain specified changes of the Feynman rules. The connection between this definition, and the definition of defining a gauge transformation by the addition of the gradient of an operator-valued gauge function to the electromagnetic potential (operator gauge transformation), is studied in this paper. It is found that each operator gauge transformation from the radiation gauge may be defined through certain modifications of the Feynman rules in that gauge. However, usually it consists of more modifications than those assumed before. These modifications due to an operator gauge transformation are all of a special type. Modifications of other type may be included, which serves to define generalized gauge transformations. An example of a generalized gauge transformation is given in which the interacting electron propagator coincides with the free one outside any prespecified space-time domain. This example serves to illustrate the many other possibilities of gauge transformations that have not been considered before. A theorem giving a relation between the electron Green’s functions in different gauges is also proved.