Quantum Time Travel Revisited: Noncommutative Möbius Transformations and Time Loops

We extend the theory of quantum time loops introduced by Greenberger and Svozil [1] from the scalar situation (where paths have just an associated complex amplitude) to the general situation where the time traveling system has multidimensional underlying Hilbert space. The main mathematical tool that emerges is the noncommutative Möbius Transformation and this affords a formalism similar to the modular structure well known to feedback control problems. The self-consistency issues that plague other approaches do not arise here, as we do not consider completely closed time loops. We argue that a sum-over-all-paths approach may be carried out in the scalar case but quickly becomes unwieldy in the general case. It is natural to replace the beam splitters of [1] with more general components having their own quantum structure, in which case the theory starts to resemble the quantum feedback network theory for open quantum optical models and indeed we exploit this to look at more realistic physical models of time loops. We analyze some Grandfather paradoxes in the new setting.