Quantum theories of time travel#

The advent of CTCs and their associated issues in the semiclassical general theory of relativity prompted study into quantum theories of time travel. In the absence of a complete quantum theory of gravity, reconciling CTCs with standard quantum mechanics forms a compelling basis for research. Indeed, exploration into the interplay between quantum mechanics and CTCs, even in only a theoretical manner, may provide insight into a yet unknown full theory of quantum gravity. Thus far, research into this area has taken several main routes, all of which have produced incredibly fascinating results.

Perhaps the most immediately interesting research direction is the study of the various ways by which ordinary quantum mechanics can be extended to include antichronological time travel. Such theories, called prescriptions, are simply models of quantum state evolution on or near CTCs which employ self-consistency (often Novikov’s principle of self-consistency) in order to compute the evolved states and their histories. For so-called “paradoxical” interactions—those for which the interaction between the incoming chronology-respecting (CR) state and its chronology-violating (CV) counterpart on the CTC seemingly describes a temporal paradox—solutions for these states that satisfy the relevant prescription’s equations of motion for any given interaction are termed resolutions.

The two main such prescriptions of quantum time travel are the Deutsch model (giving Deutschian CTCs, or D-CTCs), in which self-consistency is applied to the density matrix itself, and postselected teleportation (giving P-CTCs), which is equivalent to a path-integral formulation. Note that alternative prescriptions of quantum time travel do exist, but none are as mature as either D-CTCs or P-CTCs, and so a select handful are discussed only briefly in Alternative formulations of quantum time travel. In any case, it is the exploration into these models and their consequences that is most pertinent to our concerns in this project, as this is perhaps the research direction in which the most conclusions can be reached in regards to the quantum information processing and computation abilities of CTCs.

Other research directions into the quantum mechanics of time travel take distinctly different approaches. These include various efforts to formulate a self-consistent version of quantum theory near or on CTCs [3], generalize relativistic quantum field theory to non-globally hyperbolic spacetimes (i.e., those containing CTCs) [], pose and solve the Cauchy problem for quantum fields on such spacetimes [], and experimentally simulate the various quantum models of closed timelike curves [4]. In particular, studies of interacting quantum systems near CTCs have concluded that, even on non-globally hyperbolic spacetimes, the interactions of quantum particles interpolate (depending on the interaction strength) between plane-wave scattering events (for weak interaction strengths) and hard-sphere collisions (for strong interaction strengths), exactly as one might expect for ordinary globally hyperbolic spacetimes. However, for large classes of fields and states, the corresponding quantum evolutions fail to be unitary [3], which is especially true in the case of strongly interacting fields. This jeopardizes the notion of obtaining a consistent probability interpretation of quantum mechanics—a serious consequence which arises specifically as ambiguities in the assignment of probabilities for events occurring before (or even spacelike-separated from) the region with CTCs.

On the other hand, recent studies of expressly unitary (albeit non-interacting) quantum mechanics on CTCs [] have yielded results which suggest that thermal fluctuations destroy causation and erase all memories of systems evolving along CTCs, even those of macroscopic observers. As a consequence, the resulting quantum theory is perfectly self-consistent, in the sense that all states undergo non-contradictory histories and all retro-causality paradoxes (i.e., time-travel paradoxes) are forbidden at the quantum level.

Less direct (yet still important) approaches of studying (antichronological) time travel and causality involve exploring various associated or complimentary aspects of these notions in both quantum and relativistic capacities. Such topics include:

Note however that these topics are mentioned purely for the sake of completeness. Since we are primarily interested in computing state evolutions using the aforementioned quantum prescriptions of time travel, all of the concerns in these topics are out of the scope of this project.

Contents