Quantum Time Travel Revisited: Noncommutative Möbius Transformations and Time Loops

TL;DR

The paper extends quantum time-loop theory beyond scalars to general Hilbert spaces by introducing noncommutative Möbius transformations as the central tool, linking time-loop dynamics to quantum feedback networks in the SLH formalism. Time loops are modeled as open feedback systems with inputs and outputs, with well-posedness requiring the invertibility of $1 - S_{\text{back,back}} M$, yielding a unitary or contractive feedback transfer $S_{\text{fb}}$ through a fractional-linear map. It derives explicit transfer formulas, analyzes grandfather paradoxes via measurements and guardian-angel tuning, and develops both finite- and infinite-dimensional examples, including harmonic oscillators and continuous fields, showing the approach scales to realistic quantum systems. The framework enables controlled, monitorable past-future interactions by embedding loops in quantum networks, paving the way for practical modeling of time-loop phenomena in open quantum systems and continuous-signal regimes.

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.

Figures (12)

• Figure 1: A particle interacts with a future version of itself using two beamsplitters and time travel.
• Figure 2: The open loop unitary operator $S$.
• Figure 3: The time loop as a feedback system.
• Figure 4: Completely Closed Timelike Curves. (a) Deutsch closed timelike curve (CTC) with underlying Hilbert space $\mathfrak{h}_{\text{CTC}}$ It interacts with its external environment with Hilbert space $\mathfrak{h}_{\text{Ext.}}$ represented by several lines leading to a total unitary evolution $U$ on the joint Hilbert space. (b) The time loop is formed when a particle-antiparticle pair come into existence at the event labeled as a star. They then annihilate later. The antiparticle is interpreted as a copy of the original system traveling back in time.
• Figure 5: Path I comes about from two reflections at both $A$ and $B$. In path II it is all transmissions. Paths III has transmissions at $A$ and reflections at $B$.

Theorems & Definitions (4)

• Proposition 1
• Remark 2
• Remark 3
• Proposition 4