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arXiv:2411.08543 · Russian Journal of Mathematical Physics

Quantum Time Travel Revisited: Noncommutative Möbius Transformations and Time Loops

John Edward Gough

Published Jun 1, 2025• 0 citations

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 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. DOI 10.1134/S1061920824601691

DOI: 10.1134/s1061920824601691

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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. DOI 10.1134/S1061920824601691

It is claimed in the above paper that, if time travel were possible, quantum propagation would prevent classic time travel paradoxes by establishing consistent loops; an example circuit is used to demonstrate such a loop. It is argued here that established loops are not the framework in which the classic paradoxes arise; rather they arise via the establishment of a concrete initial history in which no disturbing time travel is allowed and then disturbing that history via the launch of the time traveller. It is shown that, operated in this two-pass fashion, if the first forward evolution of their circuit produces a definite triggering of a backwards time travelling state, the re-evolution thereby engendered may be organised so as to prohibit the triggering of this state, thereby creating a classic time travel paradox.

The mathematical theory of quantum feedback networks has recently been developed by Gough and James \cite{QFN1} for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, that their feedback reduction formula for the coefficients of the closed-loop quantum stochastic differential equation can be formulated in terms of Belavkin matrices. We show that the reduction formula leads to a non-commutative Mobius transformation based on Belavkin matrices, and establish a

\star

-unitary version of the Siegel identities.

In this paper, we analyze classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that it can be described by a closed-loop feedback problem. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be thought of as constraints in a non-commutative space, then the momentum must be dependent on the position for any generic wave function. This implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. For the sake of exposition, we give a short review of control theory, and some familiar examples at the classical and quantum levels are analyzed.

In this review we present the problem of time in quantum physics, including a short history of the problem and the known objections about considering time as a quantum observable. The need to deal with time as an observable is elaborated through some unresolved problems. The lack of a consistent theory of time is currently hindering the formulation of a full-fledged theory of quantum gravity. It is argued that the proposal set forth by several authors of considering an intrinsic measurement of quantum time, besides having the conventional external time, is compelling. Recently several suggestions have been put forward to revive the proposal of Page and Wootters (1983), elaborating and resolving some of the main ambiguities of the original proposal and opening new scope for understanding its content. The approach followed in these new contributions exposes the need to go beyond the limitations enforced by the conventional approach of quantum physics. The attitude of covariant loop quantum gravity, in which it is called to completely ignore time, is also discussed. This review could be a step forward in an endeavour to reform our outlook of the unification of the theory of relativity and quantum physics by furnishing the conceptual ground needed for this goal. Intentionally, some technical details are avoided since we aim to present the approaches to resolve the problem in a simple way with the clearest possible outlook. These can be looked up in the original references provided.