Quantum propagator for a nonrelativistic particle in the vicinity of a time machine

Dalia S. Goldwirth, Malcolm J. Perry, Tsvi Piran, and Kip S. Thorne
Phys. Rev. D 49, 3951 – Published 15 April 1994
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Abstract

We study the propagator of a nonrelativistic, noninteracting particle in any nonrelativistic ‘‘time-machine’’ spacetime of the following type: an external, chronal 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 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 the chronal 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 nonrelativistic time-machine spacetime and it is valid as long as the particle is not interacting with itself or any other quantum particle; it can, however, interact with an external field (gravitational or otherwise). This result is the nonrelativistic 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.

  • Received 12 August 1993

DOI:https://doi.org/10.1103/PhysRevD.49.3951

©1994 American Physical Society

Authors & Affiliations

Dalia S. Goldwirth

  • Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138

Malcolm J. Perry

  • DAMTP, University of Cambridge, Silver Street, Cambridge, CB3 9EW, England

Tsvi Piran

  • Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138
  • Racah Institute for Physics, The Hebrew University, Jerusalem 91907, Israel

Kip S. Thorne

  • Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125

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Issue

Vol. 49, Iss. 8 — 15 April 1994

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