Quantum Theory of Path‐Dependent Field Operator Based on Characteristics of Displacement Operators

The path‐dependent operator formalism of quantum electrodynamics proposed by Mandelstam is reformulated through quantum field theory based on characteristics of displacement operators in Minkowski space. It is shown that total energy‐ and total angular‐momentum operators can generate inhomogeneous Lorentz transformations on any local operator including path‐independent bilinear forms constructed of path‐dependent electron operator Ψ( x , P ), but that generators for Ψ( x , P ) itself are only their Ψ( x , P )‐dependent parts. Such an unfamiliar feature is characteristic of the path‐dependent operator formalism. The present approach possesses unique merits in making the logic of the formalism transparent as described in the following: i) Quantum electrodynamics can be formulated but for the help of potential operator even as a tool for calculation up to a final step. ii) Some restriction, which can be used to discuss propriety of gauge conditions, can be figured out. iii) By introducing a path‐rearrangement operator, we can keep infinite variety of space‐like pathes with the same end point throughout our formulation as they stand. iv) Several points which must be modified in the presence of magnetic monopole are closed up.