A gauge-invariant formulation in quantum electrodynamics, characterized by an arbitrary function ${\phi }_{\mu }\mathrm{}\left(x\right)\mathrm{}$, is reconsidered. Operators in a covariant case, however, are ill defined because of a ${\phi }_{\mu }\mathrm{}\left(x\right)\mathrm{}\sim {\partial }_{\mu }/\square$-type singularity in Minkowski space. We then build up a Euclidean path integral formula, starting with a noncovariant but well-defined canonical operator formalism. The final expression is covariant, free from the pathology, and shows that the model can be interpreted as the ${\phi }_{\mu }$-gauge fixing. Utilizing this formula we prove the gauge independence of the free energy as well as the $S$ matrix. We also clarify the reason why it is so simple and straightforward to perform gauge transformations in the path integral.

• Received 7 January 1997

DOI:https://doi.org/10.1103/PhysRevD.56.2281