>Personally, I think adding in the time component will be essential to completing this puzzle but all in all it makes for an avenue of investigation which is interesting.
Not a physicist, but I've often wondered if the basis of QFT got off on the wrong foot by making time a privileged coordinate instead of a quantum operator like it does for position.
While I sympathize with your unease about making time a privileged coordinate, even in conventional quantum mechanics an operator for time seems difficult. What would that operator measure? The time at which a given object "is"? The whole point of physics is to describe the dynamic nature of reality, parametrized by time.
Speaking of which, time(-of-arrival) measurements in quantum mechanics have recently attracted quite some interest: The classic Copenhagen formalism doesn't seem to give an answer here (or at least not a unique one – it depends on how you perform the calculation). Meanwhile, Bohmian mechanics does seem to make a precise prediction. It will be interesting to see what experiments will yield.
You just get into a twist of not being able to renormalize if you create a dependency chain like that. The same reason quantum gravity is such a problem - gravitons emit gravitons..
Not a physicist, but I've often wondered if the basis of QFT got off on the wrong foot by making time a privileged coordinate instead of a quantum operator like it does for position.