Justify the use of nonphysical spacetime coordinates in quantum field theory

Determine why non-self-adjoint spacetime coordinates, which do not correspond to physical observables, must be introduced to construct quantum field theories when all physical operators can in principle be formulated without invoking them.

Background

The paper identifies a conceptual tension between quantum mechanics and conventional quantum field theory. Quantum mechanics requires every physical quantity to be represented by a self-adjoint operator, whereas the spacetime coordinates used as the background framework of quantum field theory do not correspond to self-adjoint operators. The unresolved issue is whether and why such nonphysical quantities are necessary for constructing a consistent theory, particularly when physical operators might be defined without them.