Goldwirth, Perry, Piran and Thorne (Phys. Rev. D 49, 3951, 1994) constructed the explicit path-integral propagator of a nonrelativistic, noninteracting particle in a two-temporal-wormhole time machine and proved that propagation between the chronal (causally well-behaved) regions is unitary, while propagation inside the dischronal region (the region containing the closed timelike curves) is not. Their paper leaves open four problems — two named extensions, a caution about their singularity-handling method, and a reported disagreement about the past chronal region — stated here as one problem: (i) extend the explicit propagator construction to relativistic free particles, handling light-cone constraints and particle-antiparticle zigzag processes; (ii) determine how self-interactions or interactions with other quantum particles modify the path integral and the local probability-current conservation law in the dischronal region, and identify the correct prescription for computing probabilities when unitarity fails; (iii) determine whether the singularity-excision-with-reflecting-boundary-conditions method underlying the Gauss-theorem unitarity proof is justified for more severe singularities; (iv) settle whether a Hamiltonian formulation can be recovered in the past chronal region or whether an alternative probability rule must be adopted there. The subsequent literature provides substantial partial and adjacent results: operator-versus-path-integral disagreements in discrete interacting models (Fewster-Higuchi-Wells 1996), a rigorous free-field construction on the Politzer identification spacetime (Tolksdorf-Verch 2017), mutually inconsistent quantum-circuit prescriptions for closed timelike curves (Deutsch's D-CTC density-matrix fixed-point condition, postselected CTCs, and transition-probability CTCs), and holographic analyses of interacting theories on closed-timelike-curve backgrounds — but no work that resolves any of the four questions for the spacetime path-integral setting. The problem is open.

Background

General relativity permits spacetimes containing closed timelike curves — future-directed worldlines that return to their own past. A 'time machine' in this literature is a spacetime whose region of such curves (the dischronal region) is preceded and followed by causally well-behaved chronal regions. A standard construction (Morris, Thorne and Yurtsever, 1988) turns a traversable spatial wormhole — a tunnel-like shortcut joining two distant regions of space — into a time machine by moving one wormhole mouth relative to the other, so that the mouths join events at different external times. Classically, the initial value problem in such spacetimes is ill-posed for interacting systems such as billiard balls that can collide with their older selves: self-consistent evolutions exist, but generally many of them, so classical predictability is lost. Quantum mechanically the difficulty is structural: a spacetime containing closed timelike curves is not globally hyperbolic — it admits no global foliation by Cauchy surfaces (spacelike surfaces that every inextendible causal curve crosses exactly once) — so there is no global 'state at time t' and the usual Hamiltonian (Schrödinger-picture) formulation fails. Feynman's path integral, the sum over all histories, is the only standard formulation that survives, and the propagator GjiG_{ji} — the amplitude to propagate from spacetime point ii to point jj — becomes the central object. It is called complete if amplitudes compose over an intermediate time slice, and unitary if the evolution it defines preserves total probability between the regions considered. Before 1994, unitarity of quantum propagation through time machines had been established only in special cases: Politzer (1992) proved it for a free nonrelativistic particle in flat spacetime with two finite regions at different times identified with each other, and Friedman, Papastamatiou and Simon (1992) proved it for a free relativistic scalar field in time machines that are initially and finally static, while showing in a companion paper that interacting fields on spacetimes with closed timelike curves can fail unitarity. Deutsch (1991) had proposed an alternative, density-matrix fixed-point ('D-CTC') formulation that abandons unitarity, and Boulware (1992) had studied quantum field theory in spaces with closed timelike curves. Goldwirth, Perry, Piran and Thorne (1994) then solved the general free nonrelativistic problem for a model time machine: a flat exterior spacetime in which a spatial region V−V_- at time t−t_- and a spatial region V+V_+ at time t+t_+ are connected by two temporal wormholes (wormholes whose mouths join events at different external times), one leading from the past side of V−V_- to the future side of V+V_+, the other from the past side of V+V_+ to the future side of V−V_-. They summed the path integral over all path classes distinguished by the number of wormhole traversals, resummed the resulting infinite series into a closed form built from the flat-space propagator KK and the two unitary wormhole transfer operators (via a geometric-series operator inverse), and proved that the propagator is complete and unitary in the initial and final chronal regions, that propagation from the initial to the final chronal region is unitary, and that neither property holds inside the dischronal region. They also gave a second, more general proof, valid for most nonrelativistic time-machine spacetimes, based on the local Schrödinger equation satisfied by the path-integral propagator, the associated local conservation law for the probability current, and Gauss's theorem; it assumes perfectly reflecting walls at the wormhole mouths and excludes self-interacting particles, since, as the source notes, self-interactions in the dischronal region would produce contributions to the action that invalidate the derivation of the local Schrödinger equation and presumably thereby invalidate the proof. Both proofs handle the 'mild' junction singularities where the wormhole mouths join the exterior by excising singular regions and imposing reflecting boundary conditions on the excision surface — the assertion that paths cannot begin or end on a singularity. The paper closes by recording a disagreement about the past chronal region: on the reading of Hartle and of Friedman, Papastamatiou and Simon, one cannot recover a Hamiltonian formulation there even though it is chronal, while the first three authors favor a picture in which any apparent loss of probability is attributed to particles becoming eternally trapped inside the time machine. In parallel, Hawking's chronology protection conjecture (1992) — the conjecture that the laws of physics prevent the formation of closed timelike curves — later supported by theorems of Kay, Radzikowski and Wald (1997) showing that the quantum theory of a free scalar field must break down at the base points of a compactly generated Cauchy horizon (the horizon that typically bounds the dischronal region), questions whether such time machines can form at all. The questions below are therefore best read as conditional on the given spacetime, which is how the source itself poses them.

Problem Statement

Extend and secure the quantum-mechanical analysis of Goldwirth, Perry, Piran and Thorne (1994) for particle propagation through a wormhole time machine, beyond the free, nonrelativistic, mild-singularity case that they solved. The common objective is a complete and internally consistent quantum theory of particle propagation through a time-machine spacetime; four linked questions constitute it.

(1) Relativistic extension. Derive the explicit propagator for a relativistic free particle in the two-temporal-wormhole geometry (the flat exterior spacetime of the source's explicit construction, in which two temporal wormholes connect the spatial region V−V_- at time t−t_- to the spatial region V+V_+ at time t+t_+), accounting for the two features absent in the nonrelativistic case: propagation restricted to the future light cone, and particle-antiparticle creation and annihilation events that make trajectories zigzag in time. Establish whether completeness and the initial-chronal-region-to-final-chronal-region unitarity result survive when the flat-space propagator KK is replaced by the appropriate relativistic propagator, as the authors anticipated but did not carry out.

(2) Interacting systems and the probability prescription. Determine how self-interactions, or interactions with other quantum particles, in the dischronal region modify the path integral and the local probability-current conservation law on which the Gauss-theorem unitarity proof rests, and identify the correct quantum prescription for computing probabilities in the chronal regions when unitarity fails.

(3) Singularities and excision. Determine whether the procedure underlying the unitarity proof — excising singular regions and imposing reflecting boundary conditions on the excision surface — is physically and mathematically justified for more severe spacetime singularities than the mild junction singularities at the wormhole mouths, for which the authors state their method 'may fail'.

(4) Hamiltonian formulation in the past chronal region. Settle whether, and under what conditions, a Hamiltonian (Schrödinger-picture) formulation can be recovered in the past chronal region — as it can in the future chronal region for free particles — or whether an alternative probability rule must be adopted there, resolving the disagreement the source reports between the position of Hartle and of Friedman, Papastamatiou and Simon (no Hamiltonian recovery to the past) and the eternal-trapping picture favored by the source's first three authors, in which apparent probability loss is attributed to particles remaining forever inside the time machine.

The questions are not independent: (3) secures the foundations of the unitarity proof used in (1) and (2), and (4) is the interpretive core of (2) — whether the failure of probability conservation is physical or a limitation of the Hamiltonian bookkeeping. The source's authors call the second question 'vastly more complex' than the first.

The four questions are not alternatives but jointly posed parts of this one problem.

References

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