In an exactly solvable model — flat spacetime with a compact closed-timelike-curve (CTC) region reduced to two spatial points carrying one fermionic mode each — Politzer (Phys. Rev. D 49, 3981 (1994)) showed that the two available prescriptions for quantum dynamics with CTCs, the path-integral (sum-over-histories) formulation and Deutsch's density-matrix formulation, both exist but are inequivalent and each pathological in its own way: the former gives non-unitary, acausal evolution; the latter is causal but destroys coherence, and in both the handling of mixed states and information flow is ambiguous. The paper's own Conclusions pose the questions these idealizations leave open: in spacetimes with non-uniform metrics where particle production occurs, how do pure initial states such as the vacuum or ground state evolve under the various schemes; which conservation laws survive in the presence of CTCs; and is there a physically grounded measure of information or disorder for the resulting non-unitary evolutions, given that the von Neumann entropy fails once the density matrix no longer characterizes a statistical ensemble?

Background

General relativity admits solutions containing closed timelike curves (CTCs) — worldlines that return to their own past, so that a system can interact with its own earlier history. Morris, Thorne, and Yurtsever showed in 1988 that a traversable wormhole (a short throat connecting distant regions, held open by matter whose local energy density is negative for some observers and thus violates the weak energy condition) could, if its two mouths acquire a relative time shift, be converted into a 'time machine' whose interior supports CTCs. Whether such spacetimes can arise physically is contested: Hawking's chronology protection conjecture (1992) holds that quantum effects always intervene before CTCs form — the renormalized stress-energy tensor (the quantum expectation value of energy and momentum, including vacuum contributions) is expected to diverge at the 'chronology horizon' where closed causal curves first appear, a divergence made rigorous for linear fields by Kay, Radzikowski, and Wald (1997) — while Li and Gott (1998) exhibited a self-consistent vacuum for Misner space (the standard toy time-machine spacetime, obtained by identifying flat space under a boost) for which the renormalized stress-energy vanishes, keeping the question open.

A complementary line of inquiry, taken up by Politzer, asks: granting a spacetime with a compact CTC region, can it support any consistent quantum dynamics at all, and what is that dynamics like? Two inequivalent generalizations of quantum mechanics had been proposed. One is the path-integral (sum-over-histories) formulation: amplitudes are computed by a coherent, action-weighted sum over single-valued field histories on the spacetime, subject to the identification boundary conditions at the time machine. The other, due to Deutsch (1991), evolves density matrices — operators encoding both pure quantum states and statistical mixtures — with a consistency condition: the quantum state carried around a CTC must be a self-consistent fixed point of its own round trip. Deutsch's prescription is causal (physics before the CTC epoch — the time interval in which the closed timelike curves exist — is unaffected by the CTCs) but generically maps pure states to mixed states, destroying coherence.

Politzer's 1994 paper (with a companion 1992 analysis) solved both formulations exactly in a deliberately minimal model: flat spacetime in which a compact spatial region at time t=0t=0 is identified with the same region at t=Tt=T, further reduced to two spatial points — one outside the time machine, one inside — each carrying a single fermionic degree of freedom (a two-dimensional Hilbert space, spin up/down or occupied/unoccupied in second-quantized language), so the full system is a pair of coupled spins with an arbitrary four-by-four Hamiltonian. The results: under the path-integral prescription the evolution through the CTC epoch is non-unitary even in the exactly solved nonlinear (interacting) theory, and experiments completed before the CTC epoch become sensitive to CTCs in their future, violating the quantum superposition principle; under the density-matrix prescription pure states can emerge mixed. Strikingly, there are interactions that are non-unitary in one scheme but coherence-preserving in the other. Politzer further showed that the handling of mixed initial states is ambiguous in both formulations, exhibiting inequivalent alternatives for prescribing the information flow — ensembles that ordinary quantum mechanics would treat identically evolve differently.

This last phenomenon is sharpened by a simple observation. If XX is the (non-unitary) operator mapping states at t=0t=0 to states at t=Tt=T, then two statistical ensembles with the same density matrix at t=0t=0 can evolve to different states at t=Tt=T: the density matrix no longer characterizes the statistical state, and the traditional von Neumann entropy σ=−tr ρlog⁡ρ\sigma = -\mathrm{tr}\,\rho\log\rho — the standard measure of disorder or missing information for a quantum ensemble — 'cannot be an adequate measure of disorder or lack of information'. Politzer explicitly declined to propose a replacement, noting that any candidate must grapple with the fact that the non-unitary map itself transports any initial measure on state space, and that an 'arbitrarily technologically advanced civilization' exploiting the inverse map could in principle decode what the forward evolution apparently destroyed.

All of this was established in a model with three idealizations: a flat metric (so no particle production — the creation of quanta by a time-dependent or curved background, familiar from the Hawking and Unruh effects), a finite-dimensional Hilbert space, and the two-site spatial structure. In the Conclusions, Politzer states that 'There are certainly potentially interesting questions that one cannot address within models as simple as those analyzed here', and poses the questions that the present record takes as its problem.

Problem Statement

Granting a spacetime containing a compact region of closed timelike curves (a 'time machine'), the 1994 analysis of Politzer solved the flat, two-site model exactly in both the path-integral and Deutsch density-matrix prescriptions and showed that each exists but is pathological in its own way. In the Conclusions, Politzer states that 'There are certainly potentially interesting questions that one cannot address within models as simple as those analyzed here', and poses the questions this problem takes as its statement:

(1) Pure states with particle production. 'In a spacetime with a non-uniform metric there can be particle production. What now happens to pure initial states, e.g., the ground state, in the various schemes?' — determine how a pure initial state, in particular the vacuum or ground state, evolves through the CTC epoch under the path-integral prescription, under Deutsch's density-matrix prescription, and under the alternative information-flow prescriptions exhibited in the source, when the background metric is non-uniform and the dynamics produces particles.

(2) Conservation laws. 'What sort of conservation laws survive in the presence of CTC's?' — identify which conservation laws (for example energy or charge) hold under each of the dynamical schemes, and how the conserved quantities are accounted for when evolution is non-unitary or coherence is lost.

(3) Information for non-unitary dynamics. For the non-unitary evolutions these schemes produce, the density matrix at the initial time no longer characterizes the statistical state at the final time, and 'the traditional entropy σ≡−tr ρlog⁡ρ\sigma\equiv-\mathrm{tr}\,\rho\log\rho cannot be an adequate measure of disorder or lack of information'. Is there a satisfactory, physically grounded measure of information or disorder for such non-unitary dynamics, and what are its properties?

The common objective uniting the three components is a complete physical account of quantum evolution in CTC spacetimes beyond the exactly solvable flat model: component (1) supplies the state evolution, component (2) the conserved structure of that evolution, and component (3) the diagnostic in terms of which the non-unitary result is characterized. The source poses these questions generically, for spacetimes with non-uniform metrics in which particle production occurs and across 'the various schemes' — the path-integral prescription, Deutsch's density-matrix prescription, and the alternative information-flow prescriptions exhibited in the source. The same Conclusions raise two further questions — diagnostics for 'simple' versus 'complex' behavior in small Hamiltonian matrices, and the design of a time machine — which the source poses separately from the dynamical schemes; they are recorded as adjacent directions and are not parts of this problem.

References

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