Persistence of undecidability under quantization of billiards

Determine whether the algorithmic undecidability phenomena established for classical two-dimensional billiard dynamics—including Turing completeness and undecidable periodicity—persist after quantization; specifically, ascertain whether quantum billiards inherit analogous forms of undecidability.

Background

The paper proves that two-dimensional classical billiard systems are Turing complete by encoding Turing machine computations into billiard trajectories using Topological Kleene Field Theory. As a consequence, natural decision problems such as reachability and periodicity are algorithmically undecidable in these classical systems.

Given that billiard models also have quantum analogs (quantum billiards) where spectral and dynamical properties are influenced by classical trajectories, the authors raise the question of whether the classical algorithmic barriers—rooted in undecidability—carry over to the quantum regime under quantization.

Resolving this question would clarify whether limitations on predictability due to undecidability transcend the classical–quantum divide and appear in quantum billiard systems.