The Quantum Propagator for a Nonrelativistic Particle in the Vicinity of a Time Machine

Abstract

We study the propagator of a non-relativistic, non-interacting particle in any non-relativistic ``time-machine'' spacetime of the type shown in Fig.~1: an external, flat spacetime in which two spatial regions, $V_{-}$ at time $t_{-}$ and $V_{+}$ at time $t_{+}$, are connected by two temporal wormholes, one leading from the past side of $V_{-}$ to t the future side of $V_{+}$ and the other from the past side of $V_{+}$ to the future side of $V_{-}$. We express the propagator explicitly in terms of those for ordinary, flat spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final ``chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any non-relativistic time-machine spacetime; it is the non-relativistic version of a theorem by Friedman, Papastamatiou and Simon, which says that for a free scalar field, quantum mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product.

References in corpus (2)

• Quantum Field Theory in Spaces with Closed Time-Like Curves
• Simple Quantum Systems in Spacetimes with Closed Timelike Curves

Cited by in corpus (12)

• Quantum Computational Complexity in the Presence of Closed Timelike Curves
• Energy-Momentum Restrictions on the Creation of Gott Time Machines
• Open timelike curves violate Heisenberg's uncertainty principle
• Unitarity of Quantum Theory and Closed Time-Like Curves
• Topological censorship and chronology protection
• Classical and Quantum Initial Value Problems for Models of Chronology Violation
• Propagators in the Correlated Worldline Theory of Quantum Gravity
• Quantum physics, fields and closed timelike curves: The D-CTC condition in quantum field theory
• Time travel without paradoxes: Ring resonator as a universal paradigm for looped quantum evolutions
• Causality Violation and Nonlinear Quantum Mechanics
• Gauge Invariant Propagators and States in Quantum Electrodynamics
• Quantum state tomography on closed timelike curves using weak measurements