Quantum Time Travel Revisited: Noncommutative M\"{o}bius Transformations and Time Loops

TL;DR

This paper extends quantum time loop theory to multi-dimensional systems using noncommutative Möbius transformations, avoiding self-consistency issues and connecting to quantum feedback networks for more realistic models.

Contribution

It introduces a noncommutative Möbius transformation framework for quantum time loops with multi-dimensional Hilbert spaces, improving upon previous scalar models.

Findings

The formalism avoids self-consistency issues common in closed time loops.

It connects quantum time loops to quantum feedback networks.

Analyzes paradoxes within the new multi-dimensional framework.

Abstract

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 multi-dimensional underlying Hilbert space. The main mathematical tool which emerges is the noncommutative Mobius 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 in this approach 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 beamsplitters of [1] with more general components having their own quantum structure, in which case the theory starts to resemble the quantum feedback networks 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.