In the introductory chapter of the QFT book by Mark Srednicki the author notes that
[p. 26] So now we have two different approaches to relativistic quantum theory [...] Which [one of those two] we use is a matter of convenience and taste. And quantum field theory, the formalism in which position and time are both labels on operators, is much more convenient and efficient for most problems.
The other approach is described as follows:
[p. 25] Let us discuss the second option [...] to promote time to an operator. [...] We can use the proper time $\tau$ of the particle [...] as the time parameter. The coordinate time $T$ [...] is then promoted to an operator. [...] Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so.
Does this "second option" have an own specific name (for distinction from quantum field theory, in Srednicki's sense)?